---
abstract: 'Transport, magnetic and thermal properties at high magnetic fields ($H$) and low temperatures ($T$) of the heavy fermion compound CeAuSb$_2$ are reported. At $H=0$ this layered system exhibits antiferromagnetic order below $T_N = 6$ K. Applying $B$ along the inter-plane direction, leads to a continuous suppression of $T_N$ and a quantum critical point at $H_c \simeq 5.4$ T. Although it exhibits Fermi liquid behavior within the Néel phase, in the paramagnetic state the fluctuations associated with $H_c$ give rise to unconventional behavior in the resistivity (sub-linear in $T$) and to a $T \ln {T}$ dependence in the magnetic contribution to the specific heat. For $H > H_c$ and low $T$ the electrical resistivity exhibits an unusual $T^3$-dependence.'
author:
- 'L. Balicas,$^1$ S. Nakatsuji,$^2$ H. Lee,$^3$ P. Schlottmann,$^1$, T. P. Murphy$^1$, and Z. Fisk$^3$'
title: 'Magnetic field-tuned quantum critical point in CeAuSb$_{2}$'
---

When the long-range order is suppressed to zero temperature by tuning an external variable, such as pressure, chemical composition or magnetic field $H$, the system is said to exhibit a quantum critical point (QCP)[@QCP; @stuart]. The magnetic field is an ideal control parameter, since it can be reversibly and continuously tuned towards the QCP. Two compounds with field-tuned QCP, YbRh$_2$Si$_2$ and Sr$_3$Ru$_2$O$_7$, reached prominence due to the non-Fermi liquid (NFL) behavior triggered by the quantum fluctuations associated with the QCP. In this letter we present a Ce-compound, CeAuSb$_2$, exhibiting a field-tuned QCP and unusual transport and thermodynamic properties. All three systems have a field-tuned QCP as a common thread, yet their behavior in high fields and low $T$ are considerably different.

At zero field YbRh$_2$Si$_2$ exhibits a second-order phase transition into an antiferromagnetic (AF) state at $T_N =70$ mK [@trovareli]. A magnetic field applied along the inter-plane direction drives $T_N$ to zero at a critical $H_c \simeq 0.66$ T, leading to NFL behavior, i.e., a logarithmic increase of $C_{e}(T)/T$ and a quasilinear $T$ dependence of the electrical resistivity $\rho$ below 10 K [@gegenwartprl]. Above $H_c$ Fermi liquid (FL) behavior is recovered ($\rho\propto AT^2$ and constant Sommerfeld coefficient $\gamma$), with $A(H)$ and $\gamma(H)^2$ displaying a $1/(H-H_c)$ divergence as $H \rightarrow H_c$ [@gegenwartnature]. A similar trend was recently found in YbAgGe [@budkoprb].

Field-tuned anomalous metallic behavior in the vicinity of metamagnetism (MM) was studied in detail in Sr$_3$Ru$_2$O$_7$ [@grigera]. At a MM transition the magnetization $M$ increases rapidly over a narrow range of fields. The transition is of first order, since there is no broken symmetry involved, and terminates in a critical “end" point $(H^{\star}, T^{\star})$ [@millisprl]. In the anisotropic MM transition of Sr$_3$Ru$_2$O$_7$, $T^{\star}$ is found to decrease continuously as $H$ is rotated towards the inter-plane c-axis [@perry], thus opening the possibility of a QCP in a first order transition [@grigera; @millisprl]. This scenario is supported by the $T$-linear dependence in $\rho$ [@grigera], a divergence of the coefficient $A$ of the resistivity [@grigera], the enhancement of the effective mass of the quasiparticles [@borzi], and the $\ln{T}$-dependence of the specific heat $\gamma$ [@Zhou]. Remarkably, at very low $T$ and very close to the critical field $H_c$, $\rho$ displayed a $T^3$-dependence [@grigera]. The QCP of a MM transition is also believed to cause the rich phase diagram of URu$_2$Si$_2$ at high fields [@marcelo].

Among Ce compounds, so far there is evidence for a field-tuned QCP only in CeCoIn$_5$ [@bianchiandpaglioni] and CeIrIn$_5$ [@capan]. In these systems the QCP is believed to give rise to a (possibly unconventional) superconducting (SC) phase, in addition to NFL behavior. In CeCoIn$_5$ the nature of the magnetic correlations at low $T$ is still unclear, in part due to the close proximity to SC, while in CeIrIn$_5$ the quantum critical behavior close to the metamagnetic (MM) transition is still under investigation.

In this letter we report on the anomalous properties of CeAuSb$_2$. CeAuSb$_2$ is a tetragonal metallic compound, which at $H=0$ orders AF [@onuki] with $T_N = 6.0$ K [@comment]. For $T < T_N$, $\rho(T)$ has the $A T^2$ dependence typical of a FL and the extrapolation of $C_e/T$ to $T=0$ yields a Sommerfeld coefficient of $\gamma \sim 0.1$ J/mol.K$^2$. Hence, CeAuSb$_2$ can be considered a system of relatively light heavy-fermions. Above $T_N$, on the other hand, $\rho(T)$ displays a $T^{\alpha}$ dependence with $\alpha \lesssim 1$ and, $C_e/T$ has a $-\ln T$ dependence, both characteristic of NFL behavior due to a nearby QCP. A magnetic field along the inter-plane direction leads to two subsequent metamagnetic transitions and the concomitant *continuous* suppression of $T_N$ to $T=0$ at $H_c = 5.3
\pm 0.2$ T. As the AF phase boundary is approached from the paramagnetic (PM) phase, $\gamma$ is enhanced and the $A$ coefficient of the resistivity diverges as $(H-H_c)^{-1}$. When $T$ is lowered for $H \sim H_c$, the $T$-dependence of $\rho$ is sub-linear and the one of $C_e/T$ is approximately $-\ln T$. These observations suggest the existence of a field-induced QCP at $H_c$. At higher fields an unconventional $T^3$-dependence emerges in $\rho$ and becomes more prominent as $H$ increases, suggesting that the FL state is *not* recovered for $H \gg H_c$.

Single crystals of CeAuSb$_2$ were grown by the self-flux method, as described in Ref. [@canfield], using high purity starting materials with excess Sb as a flux. Microprobe analysis confirms the stoichiometric composition of the samples as well as the absence of sub-phases. Electrical transport measurements were performed with the Lock-In technique in both, a Bitter and a superconducting magnet, coupled to cryogenic facilities . The magnetization as a function of $T$ and $H$ was obtained in a Quantum Design DC SQUID, as well as with a high field vibrating sample magnetometer. The heat capacity was measured in a Quantum Design PPMS system using the relaxation time technique. The heat capacity measurements were extended to higher temperatures in order to resolve the crystalline electric field (CEF) scheme. A Schottky-like peak centered around $T \simeq 50$ K was observed, suggesting an excitation energy of $\Delta = 110$ K between the ground and first excited doublets.

Figure 1a) displays the in-plane electrical resistivity $\rho$ as function of $T$ for $H=0$ T. The vertical arrow indicates the onset of AF order, below which a $T^2$-dependence of $\rho$ is obtained. Above $T_N$ a NFL-like $T^{\alpha}$-dependence with $\alpha \lesssim 1$ is observed. As $H$ increases the AF order is gradually suppressed and in the vicinity of the AF to PM phase boundary a linear dependence on $T$ emerges (see Fig. 1b)). At higher fields FL behavior, i.e., $\rho = \rho_{0} + AT^{2}$, is found over a limited range of $T$ as indicated in Fig. 1c) by the vertical arrows. The red lines are least square fits to the $T^2$-dependence. The slope, given by the $A$ coefficient, decreases as $H$ increases. At very low $T$ and very high $H$ $\rho$ displays an anomalous $T^3$-dependence (see Fig. 1d)), suggesting, on the one hand, an unusual scattering mechanism, and, on the other hand, that the $T^2$-dependence may just be a crossover regime between $T^{\alpha}$ with $\alpha < 1$ and the $T^3$ regions.

The $H$ dependence of $\rho$ sheds more light on the role of the magnetic fluctuations (see upper panel of Fig. 2). For $T > 6$ K, a crossover from positive to negative magnetoresistance is seen in $\rho(H)$ at $H \sim 5$ T. As $T$ is lowered below 6 K, two MM transitions emerge, with negative magnetoresistance developing at the AF to PM transition. The magnetic field suppresses the AF spin-correlations, giving rise to a gradual alignment of the spins and reducing this way the spin-flip scattering. The mechanism of the anomalous $T^3$-dependence in $\rho(H)$ is believed to be related to the alignment of the Ce-spins. As seen below, the saturation of the magnetization $m$ suggests that the system is nearly half-metallic for very large $H$.

The middle panel of Fig. 2 shows the phase diagram resulting from transport and magnetization measurements. The blue triangles indicate the boundary between the PM and the AF phases for $H \parallel$ c-axis, while the green ones define the phase boundary for $H$ in the a-b plane. Notice the remarkable anisotropy. Surprisingly no hysteresis was detected, neither in $T$ nor in $H$ scans, despite the abrupt changes in the slope of $\rho(H)$. This suggests that the transition between the PM and AF phase is a weak first order one or unexpectedly of second-order. The first MM-transition at $H_{\text{MM}} \simeq 2.8$ T, which could correspond to a spin-flop transition, is indicated with open circles.

The field dependence of the $A$ and $A_3$ coefficients (pre-factors of the $T^2$ and $T^3$ terms) of the resistivity is displayed in the lower panel of Fig. 2. Both coefficients diverge as $H \to H_c$ and the red line is a fit to $C/(H - H_c)$. The same dependence was reported for YbRh$_2$Si$_2$ [@gegenwartprl] and interpreted as a dramatic increase of the quasiparticles linewidth throughout the entire Fermi surface. The $A$-coefficient also increases at the first MM-transition due to an enhancement in the scattering of the quasiparticles.

The magnetic moment $m$ of a CeAuSb$_2$ single crystal divided by the field is shown in Fig. 3a) as a function of $T$. For small fields and higher $T$, $m/H$ follows a Curie-Weiss law, yielding an antiferromagnetic Weiss-temperature $\theta_C = 12.25$ K and an effective moment $p_{eff} \approx 2.26 \mu_B$, for the field applied along the c-axis. Notice that the AF transition is observed only for $H \parallel c$, which is the magnetic easy axis. At higher fields, $m/H$ saturates, suggesting the complete polarization of the Ce-spins. In Fig. 3b) both metamagnetic transitions are seen as steps in the magnetization $m$ as a function of $H$. Both traces contain field-up and down sweeps measured in a SQUID magnetometer. Again, as for the resistivity, no hysteresis is observed. The field of the first MM-transition is only weakly $T$-dependent. In contrast, as already seen in the resistivity data, the $T_N$ for the second MM-transition (AFM to PM) is markedly field-dependent. Fig. 3c) shows $m$ as a function of $H$ (up to 30 T) at $T = 1.5$ K. Above $H \sim 7$ T, $m$ saturates at $m \simeq 1.65 \mu_{B}$, a value that differs from the free ion moment of Ce due to crystalline electric field effects.

The upper panel of Fig. 4 shows $C(T)/T$ as a function of $T$ for CeAuSb$_2$ at $H=5$T, and for its isostructural non-magnetic analog LaAuSb$_2$ at $H=0$T. The large peak for CeAuSb$_2$ signals the AF transition. The subtraction of both curves yields the magnetic contribution to the heat capacity $C_{\text{e}}(T)/T$. For $3 < T < 20$ K, $C_{\text{e}}(T)/T$ displays a $-\ln(T)$ NFL-like dependence. The lower panel of Fig. 4 shows $C_{\text{e}}(T)/T$ as a function of $T$ for several values of $H$. In the PM phase all curves collapse into a single trace, suggesting that the origin of the $\ln(T)$-dependence is the magnetic pre-critical fluctuations to the AF order. Furthermore, the effective mass of the quasi-particles, as given by $\gamma_{\text{e}}$ ($C_{\text{e}}(T)/T$ for $T \rightarrow 0$) increases considerably as $H\rightarrow H_c$.

Fig. 5a) shows the $T^2$-dependence of $\rho$ down to $T
\simeq 25$ mK for $H = 5$ T $\lesssim H_c$. There is evidence for a pronounced reduction $\rho$ at the lowest $T$. In Fig. 5b) we display $C_{\text{e}}(T)/T$ at $T=1$ K as a function of $H$ by the black (open) markers. This shows that $\gamma$ dramatically increases at the critical field, but that it remains finite. In other words, the $-\ln(T)$-dependence does not continue to very low $T$. The blue (closed) represent the Kadowacki-Woods ratio, $A/\gamma^{2}$, as a function of $H$. The fact that this ratio is not constant indicates that there is more than one energy scale involved.

Fig. 5c) depicts a qualitative sketch of the phase diagram. It shows the dependence of the exponent $n
\simeq \partial \ln{(\rho(T) - \rho_{0})}/ \partial \ln{T}$ on $H$ and $T$. Here different values of $\rho_0$ were used in the PM and AF phases. The PM phase is indicated by the blue region which is influenced by the QCP leading to the anomalous NFL value $n \lesssim 1$. The FL state (in green) is recovered below the Néel temperature but is gradually suppressed as $H \rightarrow H_c$. As for YbRh$_2$Si$_2$ [@gegenwartnature; @gegenwartprl] a FL-like $n=2$ exponent is obtained above $H_c$ but only over a limited range of $T$. A spin-polarized metallic phase (in red) with an anomalous $T^3$-dependence is observed for $H >
H_c$ at low $T$. Besides Sr$_3$Ru$_2$O$_7$ [@grigera], such dependence has also been predicted for half-metallic ferromagnets in terms of a one magnon scattering process [@furukawa]. This could be a plausible explanation for our system, which is spin-polarized precisely in the region where $n=3$ is observed. Although CeAuSb$_2$ is not a FM in zero-field, it could have similar properties to a FM half-metal in a strong magnetic field.

In summary, electrical transport, magnetization and thermal properties indicate that CeAuSb$_2$ displays either a weak first-order or a second-order phase transition from a NFL-like PM metallic phase to a FL phase with long-range AF-order upon cooling at zero field. A magnetic field continuously reduces $T_N$, which vanishes for a field of $H_c = 5.3 \pm 0.2$ T along the inter-layer direction. The physical properties of the PM phase emerging at $H_c$ are anomalous, i.e., $\rho(T) \propto T^{1/2}$, and $C(T)/T \propto -\ln(T)$, and are a strong indication that $H_c$ corresponds to a field-tuned QCP. In addition, the $A$ and $A_3$ coefficients of the resistivity diverge as $H \rightarrow
H_c$. The consequences of the QCP in CeAuSb$_2$ are different and perhaps more dramatic than for other field-tuned QCP systems. This is the case because the FL phase is not recovered for $H$ and $T$ sufficiently far away from the QCP. The narrow region in Fig. 5 with $n=2$ should be considered a crossover region and not a FL phase. The field-tuned QCP systems represent a challenge from the theoretical perspective, since the different compounds have some common aspects, but do *not* seem to belong to the same universality class.

This work is sponsored by the National Nuclear Security Administration under the Stewardship Science Academic Alliances program through DOE grant DE-FG03-03NA00066. This work was performed at the NHMFL which is supported by NSF through NSF-DMR-0084173 and the State of Florida. LB acknowledges support from the NHMFL in-house research program and PS through grants (DOE) DE-FG02-98ER45707 and (NSF) DMR01-05431.

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