Evidence for charge delocalization crossover in the quantum critical superconductor CeRhIn5

The nature of charge degrees-of-freedom distinguishes scenarios for interpreting the character of a second order magnetic transition at zero temperature, that is, a magnetic quantum critical point (QCP). Heavy-fermion systems are prototypes of this paradigm, and in those, the relevant question is where, relative to a magnetic QCP, does the Kondo effect delocalize their f-electron degrees-of-freedom. Herein, we use pressure-dependent Hall measurements to identify a finite-temperature scale Eloc that signals a crossover from f-localized to f-delocalized character. As a function of pressure, Eloc(P) extrapolates smoothly to zero temperature at the antiferromagnetic QCP of CeRhIn5 where its Fermi surface reconstructs, hallmarks of Kondo-breakdown criticality that generates critical magnetic and charge fluctuations. In 4.4% Sn-doped CeRhIn5, however, Eloc(P) extrapolates into its magnetically ordered phase and is decoupled from the pressure-induced magnetic QCP, which implies a spin-density-wave (SDW) type of criticality that produces only critical fluctuations of the SDW order parameter. Our results demonstrate the importance of experimentally determining Eloc to characterize quantum criticality and the associated consequences for understanding the pairing mechanism of superconductivity that reaches a maximum Tc in both materials at their respective magnetic QCP.

The Kondo singlet is a quantum state in which spins of surrounding conduction electrons collectively screen a local moment through their antiferromagnetic (AFM) exchange.
Theoretically expected and experimentally confirmed 1,2 , the cloud of screening conduction electrons around a Kondo impurity extends radially to a distance ξ = ℏvF/kBTK, where vF is the conduction electron Fermi velocity and kBTK is the energy scale of singlet formation.ξ can be up to micrometers and certainly greater than interatomic spacing.In the many-body process of Kondo-singlet formation, the spin of the local moment becomes part of the conduction electron Fermi volume.For a periodic lattice of Kondo impurities, typified by f-electron heavy-fermion metals, quantum coherence among Kondo singlets (qualitatively, a Bloch state of Kondoscreening clouds) results in the formation of highly entangled composite heavy quasiparticles as T  0 K and an increase in the Fermi volume that counts both the local moments and conduction electrons.Interactions within the narrow (of order meV) quasiparticle bands can lead to an instability, often a spin-density-wave (SDW), of the large Fermi volume.Tuning the SDW transition to T = 0 K by a non-thermal control parameter, such as pressure, magnetic field, or chemical substitution, allows access to a quantum critical point (QCP) in which quantum fluctuations of the SDW order parameter control physical properties to temperatures well above T = 0 K [3][4][5] .The transition from a SDW order to a paramagnetic state at T = 0 K has no effect on the Fermi volume.This picture of a SDW QCP ignores the role of a long-range Rudermann-Kittel-Kasuya-Yosida (RKKY) interaction I among local moments that is mediated by the same conduction electrons that produce a Kondo singlet.The RKKY interaction induces dynamical correlations among the local moments, inhibiting Kondo singlet formation and thus preventing the emergence of a large Fermi volume.The competition between Kondo and RKKY interactions can be characterized by a non-thermal tuning parameter  = kBTK/I 6,7 .Relative to the Kondo scale, I, though always finite, is relatively insensitive to pressure, field, etc. so that the T- phase diagram illustrated in Fig. 1 for heavy-fermion systems is controlled primarily by changes in TK.
The nature of quantum critical fluctuations at c where the magnetic boundary TN() reaches T = 0 K changes qualitatively depending on the location of a crossover scale Eloc() that separates an electronic state in which static Kondo entanglement breaks down because of RKKY interactions and the Fermi surface is small (FSS) and a state with fully intact composite quasiparticles that produce a large Fermi surface (FSL).Depending on the position of Eloc(), the nature of the magnetic QCP at c is either of the SDW or Kondo-breakdown type.In the SDW scenario (Fig. 1a), Eloc remains finite at the QCP and terminates inside the ordered phase; only magnetic degrees-of-freedom are quantum critical at c 3 .In a Kondo-breakdown scenario (Fig. 1b), however, Eloc reaches zero temperature at the magnetic QCP and the concomitant reconstruction of the FS from small-to-large coincides with the onset of magnetic order, thus incorporating both charge and magnetic quantum fluctuations at the Kondo-breakdown QCP 8- 11 .The distinct difference between SDW and Kondo-breakdown criticality has fundamental consequences for interpreting the origin of new phases of matter that frequently emerge around a QCP as the system relieves the buildup of entropy.
Evidence for a Kondo-breakdown type of quantum criticality has been found in a few heavy-fermion materials, notably CeCu6-xAux 12 , YbRh2Si2 13,14 , and CeRhIn5 15,16 ; whereas, a phase diagram like that in Fig. 1a appears in CeIn3 17 , Co-doped YbRh2Si2 18 and Ir-doped CeRhIn5 19,20 .Among examples representative of Fig. 1b physics, a FS change at their QCP has been inferred from Hall measurements.In CeRhIn5, on the other hand, evidence for an abrupt change in FS as T  0 K has come from a pressure-dependent quantum oscillation study that directly probes the FS of CeRhIn5 in which an abrupt reconstruction of the FS coincides with an AFM QCP 15,21,22 .Evidence for the crossover scale Eloc(), characteristic of Kondo breakdown, has yet to be reported in CeRhIn5, opening the possibility that FS reconstruction could be a consequence of a change in FS topology resulting from the loss of magnetic order 23 .
Here, we probe the change in Hall effect via systematic control of external pressure to identify Eloc() in CeRhIn5, thereby demonstrating clearly that its criticality is of the Kondo-breakdown type.Replacing 4.4% of the In atoms by Sn shifts Eloc() such that it intersects the TN(P) boundary at finite temperature (Fig. 1a) and the magnetic criticality changes to the SDW type.
Not only imposing a far more unambiguous interpretation of criticality in CeRhIn5, these discoveries point to a change in the nature of fluctuations leading to Cooper pairing in pure and Sn-doped CeRhIn5 and to the importance of Eloc for confirming a theoretically proposed beyond-Landau framework to understand quantum criticality 4,6 .

Results
Observation of Eloc in CeRhIn5.Previous Hall measurements of CeRhIn5 were limited to 2.6 GPa close to the critical pressure, Pc = 2.3 GPa, at which the AFM transition TN extrapolates to T = 0 K and the superconducting transition temperature Tc reaches a maximum 21,22,24 .Our measurements of the Hall coefficient RH at P < Pc, Fig. 2a, are consistent with the primary features reported earlier 24 .A local minimum in RH at T * (as indicated by purple arrows) signals the onset of short-range AFM spin correlations above TN 25,26 and is suppressed together with TN under pressure.Figure 2b shows RH in the high-pressure regime (P > Pc) where a previously unidentified local minimum in RH appears at TL (as marked by the orange arrows) and moves to higher temperature as the system is tuned away from magnetic order with increasing pressure.
(TL appears well above the temperature below which the resistivity assumes a Fermi-liquid T 2 dependence 16 .)The Hall coefficient in a multiband system with magnetic ions, particularly a compensated heavy-fermion metal like CeRhIn5, is not straightforward to interpret 15,24,27 .Prior experiments rule out any significant asymmetric (skew) scattering in CeRhIn5 at low temperatures and pressures below Pc 24,27 , leaving temperature-and pressure-dependent changes in the ordinary Hall contribution as the most likely origin of TL.In a system like CeRhIn5, the magnitude of the ordinary term can be influenced not only by changes in carrier density but also by scattering rates on different parts of the Fermi surface, details of the surface topology, and the presence of spin fluctuations.Likely, all contribute to some extent to the unusual temperature-and pressure-dependence of an ordinary Hall contribution at P > Pc and to a change in the character of charge carriers at TL.
A color-contour map of the absolute value of RH (|RH|) for CeRhIn5 is displayed in the temperature-pressure (T-P) plane in Fig. 2c.In the low-pressure AFM regime, T * and TN smoothly extrapolate to T = 0 K at Pc, at which Tc and |RH| are a maximum.The existence of an AFM QCP at Pc, hidden by the pressure-induced superconducting dome, has been revealed explicitly by applying a magnetic field sufficient to suppress superconductivity 21,22 .As shown in the figure, TL(P) extrapolates smoothly on the paramagnetic side of the diagram to T = 0 K at the magnetic QCP where, in the limit T  0 K, the FS reconstructs, de Haas-van Alphen (dHvA) frequencies increase, and the quasiparticle effective mass diverges 15 .These are all essential characteristics of a Kondo-breakdown QCP.The phase diagram in Fig. 2c coincides with the theoretically predicted phase diagram (Fig. 1b) if we identify TL(P) as Eloc(P), i.e., a change in the nature of charge carriers at TL is accompanied by a crossover from small-to-large Fermi surface with increasing pressure at finite temperature.Theoretically, Eloc appears at some temperature below the onset of heavy quasiparticle formation 6 , typically taken experimentally to be signaled by the temperature Tmax where the magnetic resistivity reaches a maximum.In CeRhIn5, Tmax(P) is roughly 6.5 times TL(P) at P > Pc 28 .Multiple experiments, including dHvA measurements 15 , suggest that CeRhIn5 at P = Pc is equivalent to the isostructural heavy-fermion superconductor CeCoIn5 at P = 0 GPa 29 .Recent Hall measurements argue that CeCoIn5 at P = 0 GPa is very close to a QCP characterized by a localization/delocalization of 4f-electrons at a transition connecting two Fermi surfaces of different volumes 30 .A minimum in its Hall coefficient evolves with pressure increasing from P = 0 GPa in very much the same way as TL(P) shown in Fig. 2b for CeRhIn5 at P > Pc 24 .Further, the pressure-dependent temperature of a Hall minimum tracks the delocalization crossover temperature determined by nuclear quadrupole resonance measurements in CeIn3 17,31 , providing additional support for associating the Hall minimum in CeRhIn5 and CeCoIn5 with a delocalization crossover at finite temperature.Finally, similar to T *27 , the influence of magnetic field on TL (Supplementary Fig. 1) is negligible even though |RH| is suppressed, an observation arguing against a substantial contribution of magnetic fluctuations to determining TL.Each of these further compels the identification of TL with Eloc.
Quantum criticality in Sn-doped CeRhIn5.We turn to the case of CeRhIn5 with a Sn concentration of 0.044, CeRh(In0.956Sn0.044)5,labelled as Sn-doped CeRhIn5 in the following.
Figures 3a,b show the temperature dependence of the Hall coefficient RH for Sn-doped CeRhIn5 at representative pressures up to 2.25 GPa.RH was obtained by applying a magnetic field of 5 T that completely suppresses superconductivity over the whole pressure range studied.Similar to pure CeRhIn5, two characteristic temperatures T * and TL are revealed by a local minimum in RH.In the low-pressure regime, T * decreases monotonically with increasing pressure and becomes unresolvable at 1.0 GPa (Fig. 3a).At pressures higher than 1.0 GPa, another local minimum in RH appears at TL and increases with increasing pressure (Fig. 3b).We note that the field effects on T * and TL in Sn-doped CeRhIn5 are negligible (Supplementary Fig. 3).A color-contour map of the amplitude of RH (|RH|) for Sn-doped CeRhIn5 at 5 T is shown in the T-P plane in Fig. 3c, which is overlaid with the phase boundaries.The slight Sn doping leads to a decrease of TN from 3.8 K for pure CeRhIn5 to 2.1 K in the Sn-doped case at ambient pressure 32 .With applying pressure, TN decreases gradually and extrapolates to a terminal critical pressure Pc2 (~1.3 GPa) where pressure-induced superconductivity reaches a maximum Tc.Accompanying the suppression of TN, T * (P) decreases with pressure and also extrapolates to T = 0 K at Pc2, a response qualitatively similar to pure CeRhIn5 in which T * (P) is associated with the development of short-range AFM correlations.With TL(P) << Tmax(P) (Supplementary Fig. 5 and TL/P > 0 at P  1.2 GPa (Fig. 3b), as in CeRhIn5 at pressures above Pc, we associate TL(P) with Eloc(P) in Sn-doped CeRhIn5.In contrast to CeRhIn5, Eloc(P) extrapolates to T = 0 K at a distinctly lower pressure Pc1 (~1.0 GPa) than the critical pressure Pc2 where AFM transition is suppressed to T = 0 K.The termination of Eloc at Pc1 indicates that a local-toitinerant transformation of 4f degrees-of-freedom and concomitant reconstruction of FS take place within the AFM state of the Sn-doped material.
The existence of two critical pressures in Sn-doped CeRhIn5 is supported by the lowtemperature resistivity ab measured parallel to the Ce-In plane under a magnetic field of 4.9 T (Supplementary Fig. 6).The color contour of isothermal resistivity in the T-P plane, illustrated in Fig. 4a, shows a funnel of enhanced scattering centered at Pc1.The local temperature exponent n derived from n = (ln)/(lnT), in contrast, reveals a funnel of non-Fermi-liquid behavior centered near Pc2 where the resistivity exhibits a linear-in-T dependence, illustrated in Fig. 4b.Figures 4d,g show ab as a function of temperature at representative pressures of 0.20, 1.38, 1.45, and 2.30 GPa, respectively.A Landau-Fermi-liquid T 2 dependence is observed at low-and high-pressure regimes in Figs.4d,g, but the linear-T dependence is prominent near Pc2 in Figs.4e,f. Figure 4c summarizes the dependence on pressure of the residual resistivity 0 on the left ordinate and the temperature coefficient A on the right ordinate estimated from a fit to ab = 0 + AT n .With increasing pressure, 0 increases by over a factor of two, reaches a maximum at 1.0 GPa (=Pc1), and decreases with increasing pressure, reflecting enhanced scattering of critical charge fluctuations around Pc1 8-11 .The coefficient A, which is related to the effective mass of quasiparticles 19,33 , however, gradually increases at low pressures, goes through a local minimum at Pc1, and peaks sharply at Pc2, which is consistent with the divergence of effective mass predicted at an SDW QCP.

Discussion
Despite several quantum-critical heavy-fermion candidates, the delocalization energy scale Eloc has been identified in a limited number of compounds, including YbRh2Si2 13,14,18 , Ce3Pd20Si6 34 , and CeIn3 17,31 .In the case of YbRh2Si2, Eloc manifests as anomalies in isothermal measurements, such as a step-like crossover in the field-dependent Hall coefficient and magnetoresistivity that sharpens with decreasing temperature or a smeared kink in the field-dependent Hall resistivity, magnetostriction, and magnetization 13,14,18 .The locus of Eloc points extrapolates to a field-tuned zero-temperature boundary of magnetic order, as in CeRhIn5 under applied pressure.Similar analysis of Hall data for Ce3Pd20Si6 finds, however, that Eloc extrapolates inside the ordered part of its field-dependent phase diagram 34 , analogous to Sn-doped CeRhIn5 (Fig. 3c).This also is the case with pressure-tuned CeIn3, discussed earlier, where a localization/delocalization scale intersects its antiferromagnetic phase at finite temperature 17,31 .
In these other examples, the signature for Eloc in various physical quantities has been used to infer a Fermi-surface change, but dHvA measurements unambiguously establish a jump in dHvA frequencies at the critical pressure where TL extrapolates to T = 0 K in CeRhIn5.A simple interpretation of RH would anticipate an associated step-like jump in carrier density at Pc, but as shown in Fig. 5a, instead of having a sharp jump, the pressure-dependent isothermal Hall coefficient peaks strongly at Pc. Perhaps measurements have not been made at sufficiently low temperatures to reveal a jump.More likely it is obscured by effects of critical charge and spin fluctuations on the Hall resistivity and/or only a small net change in the sum of hole and electron contributions in this nearly compensated metal.Though present experiments cannot make a definitive distinction, they do reflect the approach to FS reconstruction detected in dHvA that is coincident with a magnetic QCP.In contrast to the sharp peak in |RH(P)| in CeRhIn5, there is a broad maximum |RH(P)| that peaks at Pc1 but encompasses both Pc1 and Pc2 in CeRh(In0.956Sn0.044)5(Fig. 5b).Such a broad maximum relative to that in CeRhIn5 is not surprising because of the close proximity of two critical pressures, each with their own spectrum of critical spin/charge fluctuations, and smearing these spectra by disorder inherent to the Sn substitution.
The distinctly different critical behaviors of pure and Sn-doped CeRhIn5 are captured in a generalized quantum-critical phase diagram that includes both Kondo-breakdown and SDW criticality 8,35 .In this theory, the nature of quantum criticality is determined by two quantities,  and G (Supplementary Fig. 9), where, as before,  = kBTK/I, and G is a parameter that reflects magnetic frustration or effective spatial dimensionality.Our results are consistent with Kondo breakdown coinciding in pure CeRhIn5 under pressure with a T = 0 K transition from an AFM state with a small FS (AFMS) to a paramagnetic state with a large FS (PL) at Pc, i.e., Kondo breakdown and the AFM QCP coincide.Similar to Sn-doped CeCoIn5 36 , Sn substitution for In in CeRhIn5 enhances hybridization between 4f-and conduction electron wave functions, i.e., increases TK and thus .Stronger hybridization also reduces frustration among magnetic exchange pathways 8,37 .Analysis of magnetic neutron-diffraction experiments finds a clear change in magnetic structure and decrease of the ordered moment for a Sn concentration x  0.052, comparable to CeRh(In0.956Sn0.044)5under a modest pressure, that is attributed to an abrupt modification of the Fermi surface 38 .With these changes induced by Sn doping, the system follows a different trajectory that goes through an intermediate magnetically ordered state with a large FS (AFML), i.e., Kondo breakdown occurs inside the AFM region, and thus the corresponding AFM QCP is of the SDW type as illustrated in Fig. 1a and Supplementary Fig. 9 and found in the prototypical Kondo-breakdown system YbRh2Si2 when Rh is replaced by a small amount of Co 18 .
Before identifying the crossover scale Eloc in pure and Sn-doped CeRhIn5, their criticality was ambiguous, possibly either Kondo-breakdown or SDW 32,39 .That ambiguity now is removed, with consequences for an interpretation of the origin of their pressure-induced superconductivity.At the SDW QCP in Sn-doped CeRhIn5, which is decoupled from the Kondo breakdown and where Tc reaches its maximum, quantum fluctuations of the magnetic order parameter are the prime candidate for mediating Cooper pairing 40 .Fluctuations around a Kondo-breakdown QCP, however, are far more complex and involve not only quantum-critical fluctuations of a magnetic order parameter but also of the Fermi surface, i.e., charge degreesof-freedom [8][9][10][11] .Initial model calculations show that critical Kondo-breakdown fluctuations can produce a superconducting instability 41,42 , but much remains to make these calculations directly testable by experiment.Interestingly, Tc  2.3 K of CeRhIn5 at Pc and of CeCoIn5 at P = 0 GPa is among the highest of any rare-earth-based heavy-fermion superconductor.
The theory of Kondo-breakdown criticality in heavy-fermion materials has two essential signatures -an abrupt change from small-to-large Fermi surface coincident with magnetic criticality and the charge delocalization crossover scale Eloc that extrapolates from the paramagnetic state to the QCP 6 .Without both, Kondo-breakdown criticality cannot be established with certainty.Our Hall measurements provide compelling evidence for Eloc in pure and Sn-doped CeRhIn5 and, in light of dHvA results 15 , for Kondo-breakdown criticality in CeRhIn5.Finally, we speculate that critical charge fluctuations at a Kondo-breakdown QCP, evidenced by /T scaling of the frequency ()-dependent optical conductivity 10 , should play a non-trivial role in these signatures for Eloc.Making this connection experimentally and theoretically would mark a significant advance.

Methods
Single crystals of pure and Sn-doped CeRhIn5 were synthesized using the standard self-flux technique 32 .The high-pressure resistivity and Hall measurements on CeRhIn5 were carried out using the Van der Pauw method 43 in a diamond-anvil cell made of Be-Cu alloy.NaCl powder was applied as the pressure medium to obtain a quasihydrostatic pressure environment.The pressure in the diamond-anvil cell was determined by the ruby fluorescence method 44 .The high-pressure resistivity and Hall measurements on CeRh(In0.956Sn0.044)5were measured using the standard six-probe method in a Be-Cu/NiCrAl hybrid clamp-type cell.Daphne oil was employed as the pressure medium to obtain a hydrostatic pressure environment.The pressure dependence of the superconducting transition temperature of Pb was used to determine the pressure inside the clamp-type cell 45 .All measurements were performed with a low-frequency resistance bridge from Lake Shore Cryotronics.A 4 He cryostat without magnetic field and a Physical Property Measurement System with a maximum magnetic field of 9 T were used in the temperature range of 1.8 to 300 K.A HelioxVL system with a maximum magnetic field of 12 T was used to control temperature down to 0.3 K.

Fig. 1
Fig. 1 Schematic phase diagrams of two classes of QCPs in AFM heavy-fermion metals.Under the variation of the non-thermal parameter , an AFM transition is continuously suppressed to zero temperature at a critical value c.Consequently, an AFM QCP appears at c and the ground state changes from an AFM to a paramagnetic Fermi-liquid (FL) state.TN and TFL represent the AFM transition temperature and the onset temperature of the FL regime, respectively.Eloc denotes the Kondo destruction energy scale that is associated with delocalization of the local moment.In a sufficiently low-temperature regime, Eloc() separates the Kondo-screened regime with a large FS (FSL, 4f-electrons delocalized, right side of the Eloc() line) and the Kondo-destruction regime with a small FS (FSS, 4f-electrons localized, left side of the Eloc() line).The ground state is divided into three phases: an AFM phase with a small FS (AFMS), an AFM phase with a large FS (AFML), and a paramagnetic phase with a large FS (PL).a, For the conventional SDW type QCP, Eloc() terminates inside the AFM regime.b, For a Kondo-breakdown type QCP, Eloc() terminates at the AFM QCP.Top and bottom insets show cartoon pictures of the FS and the spin fluctuations in the different phases.Olive, orange, and red arrows indicate the local moments, the itinerant conduction electrons, and the magnetic moments screened by the conduction electrons, respectively.

Fig. 2
Fig. 2 Evolution of the Hall coefficient with pressure and the T-P phase diagram for CeRhIn5.a,b, Temperature dependence of Hall coefficient RH for CeRhIn5 at representative pressures measured at temperatures above the superconducting transition temperature Tc and under a magnetic field of 1 T applied along the c-axis.The field is much lower than those required to observe quantum oscillations 15 and to suppress superconductivity 21 .The purple and orange arrows represent the onset of short-range AFM spin correlations at T * and the 4felectron delocalization crossover temperature TL, respectively (see text for details).c, T-P phase diagram of CeRhIn5 at 0 T overlaid with a contour plot of the amplitude of the Hall coefficient |RH| at 1 T. T * , TN, TL, Tc, and TFL are denoted by purple circles, navy circles, orange triangles, violet circles, and blue circles, respectively.Kondo breakdown and the AFM QCP coincide at the critical pressure Pc where Tc reaches a maximum, the Fermi surface reconstructs from small-to-large, and temperature TL of the local extremum in |RH| extrapolates to zero temperature.The purple squares and olive diamonds are data adopted from Hall 24 and nuclear quadrupole resonance 26 measurements, respectively.The dashed and solid lines are guides to the eyes.AFM, SC, and FL stand for antiferromagnetic, superconducting, and Fermi liquid regions, respectively.Error bars on the T * and TL represent the uncertainties in determining the minimum in the Hall coefficient.

Fig. 3
Fig. 3 Evolution of the Hall coefficient with pressure and the T-P phase diagram for Sndoped CeRhIn5.a,b, Hall coefficient RH of Sn-doped CeRhIn5 measured at 5 T is plotted as a function of temperature at representative pressures.The purple and orange arrows represent the onset of short-range magnetic correlations at T * and the 4f-electron delocalization crossover temperature TL, respectively (see text for details).c, T-P phase diagram of Sn-doped CeRhIn5 at 0 T overlaid with a contour plot of the amplitude of the Hall coefficient |RH| at 5 T. T * , TN, TL, Tc, and TFL are denoted by purple circles, navy circles, orange triangles, violet circles, and gray circles, respectively.Pc2 (~1.3 GPa) is the critical pressure where TN extrapolates to zero temperature, corresponding to an SDW QCP, and Tc reaches a maximum.Pc1 (~1.0 GPa) is another critical pressure where TL(P) extrapolates to zero temperature, indicating that destruction of the Kondo effect occurs within the AFM phase.The dashed and solid lines are guides to the eyes.AFM, SC, and FL stand for antiferromagnetic, superconducting, and Fermi liquid regions, respectively.Error bars on the T * and TL represent the uncertainties in determining the minimum in the Hall coefficient.

Fig. 4
Fig. 4 Quantum criticality of Sn-doped CeRhIn5 under pressure.a, Contour plot of resistivity ab for Sn-doped CeRhIn5 measured parallel to the Ce-In plane under a magnetic field of 4.9 T. The significant enhancement of ab is centered around Pc1. b, Colours represent the local temperature exponent n derived from n = (ln)/(lnT), where  = ab -0 = AT n and 0 is the residual resistivity.A funnel regime with linear-T dependence of ab is observed around Pc2, a characteristic of the non-Fermi liquid behavior near the AFM QCP, as shown in (e) and (f) at representative pressures of 1.38 and 1.45 GPa, respectively.c, Pressure dependence of the residual resistivity 0 (left-axis, blue circles) and the coefficient A (rightaxis, red circles) determined by fitting the low-temperature resistivity to ab = 0 + AT n .d-g show fits to representative data from which (c) is constructed.The dashed red lines in (a-c) are guides to the eyes.The red lines in (d-g) are least-squares fits to the low-temperature data.

Fig. 5
Fig. 5 Isothermal pressure dependence of the Hall coefficient.a, Pressure dependence of Hall coefficient |RH(P)| for CeRhIn5 at 2.4 K. Pc indicates the magnetic QCP.b, Pressure dependence of Hall coefficient |RH(P)| for CeRh(In0.956Sn0.044)5 at 0.3 K. Pc2 denotes the magnetic QCP.Pc1 indicates the critical pressure where TL extrapolates to zero temperature and is lower than the magnetic QCP Pc2.