Magnetic-field-tuned quantum criticality of the heavy-fermion system YbPtBi

In this paper, we present systematic measurements of the temperature and magnetic field dependencies of the thermodynamic and transport properties of the Yb-based heavy fermion YbPtBi for temperatures down to 0.02 K with magnetic fields up to 140 kOe to address the possible existence of a field-tuned quantum critical point. Measurements of magnetic-field- and temperature-dependent resistivity, specific heat, thermal expansion, Hall effect, and thermoelectric power indicate that the AFM order can be suppressed by an applied magnetic field of $H_c\sim 4$ kOe. In the $H-T$ phase diagram of YbPtBi, three regimes of its low-temperature states emerge: (I) AFM state, characterized by a spin density wave-like feature, which can be suppressed to $T=0$ by the relatively small magnetic field of $H_c\sim 4$ kOe; (II) field-induced anomalous state in which the electrical resistivity follows $\Delta \rho \left(T\right)\propto T^{1.5}$ between $H_c$ and $\sim$8 kOe; and (III) Fermi liquid (FL) state in which $\Delta \rho \left(T\right)\propto T^2$ for $H\ge 8$ kOe. Regions I and II are separated at $T=0$ by what appears to be a quantum critical point. Whereas region III appears to be a FL associated with the hybridized 4$f$ states of Yb, region II may be a manifestation of a spin liquid state.

Figure 1

(a) Inverse magnetic susceptibility, $H/M\left(T\right)$, of YbPtBi, where the magnetic field was applied along [100], [110], and [111] directions. The solid line represents the Curie-Weiss fit to the data for H $\parallel$ [100]. (Inset) $H/M\left(T\right)$ at low temperatures. (b) Magnetization isotherms of YbPtBi at $T$ $=$ 1.8 K for H $\parallel$ [100], [110], and [111] direction.

Figure 2

Temperature-dependent electrical resistivity, $\rho \left(T\right)$, of YbPtBi measured for different sample mount conditions for cooling. $\rho \left(T\right)$ curves are normalized to the sample #13 curve at $T=1$ K. Samples #13 and #14 were hanging in vacuum, thus cooled only through high-purity platinum voltage and current lead wires. Samples #3 and #10 were attached to the thermal bath by GE 7301 varnish. (Inset) $\rho \left(T\right)$ of sample #10, measured by the following temporal procedure. (i) Initially, the sample was mounted with Apiezon N grease in ${}^3$He cryostat and $\rho \left(T\right)$ was measured down to 0.34 K (circles) in order to see the onset of a sharp phase transition. After cleaning the N grease (ii) the sample was attached to the dilution refrigerator with GE varnish and $\rho \left(T\right)$ was measured (squares, inset and main figure). After cleaning the GE varnish (iii) the sample was mounted with N grease again in a dilution refrigerator and $\rho \left(T\right)$ was measured (triangles).

Figure 3

Temperature-dependent electrical resistivity, $\rho \left(T\right)$, of YbPtBi (for samples #3 and #13) at $H=0$ and 140 kOe. For $T>0.35$ K data, samples were mounted in PPMS ${}^3$He option with Apiezon N grease. Below 1 K, in a dilution refrigerator sample #3 was mounted to the heat sink with GE varnish and sample #13 was measured with the sample hanging in vacuum. The sample mounting configuration for sample #13 in a dilution refrigerator is illustrated in the upper left side. Sample #13 was held onto the heat sink with four Pt electrical contact wires and very thin dental floss (dashed line) to address the torque on sample; in order to cool down the sample through contact wires, Pt wires were glued to the heat sink using dilute Ge varnish (hatched area). (Inset) $\rho \left(T\right)$ of sample #13 for several selected magnetic fields.

Figure 4

Low-temperature electrical resistivity [$\rho \left(T\right)$, sample #13] of YbPtBi in various magnetic fields applied along the [100] direction (a) for $H\le 6$ kOe and (b) for $4\text{kOe}\le H\le 20\text{kOe}$. For comparison, $\rho \left(T\right)$ data at $H=4$ and 6 kOe are plotted in both figures. (a) Open and closed symbols correspond to the data taken with 3 and 30 $\mu$A excitation current, respectively.

Figure 5

(a) Resistivity for sample #13 at $H=0$ and 3 kOe, where, in order to subtract residual resistivity (${\rho }_0$), shown as solid circles on the y axis, 4-kOe curves (dashed lines) are shifted to match with curves at 0.6 K for $H=0$ and 3 kOe. ${\rho }_n$ and ${\rho }_g$, the resistivities associated with the normal and gapped states are inferred as shown. (b) The degree of Fermi surface gapping can be estimated from the relative change in conductivity, $({\sigma }_n-{\sigma }_g)/{\sigma }_n=\Delta \sigma /{\sigma }_n$, where ${\sigma }_n=1/{\rho }_n$, ${\sigma }_g=1/{\rho }_g$, and ${\sigma }_0=1/{\rho }_0$. Square and triangle symbols are based on $\Delta \sigma /({\sigma }_n-{\sigma }_0)$ (subtracting the residual resistivity) and $\Delta \sigma /{\sigma }_n$ (including the residual resistivity), respectively. Error bars represent uncertainty, primarily associated with determination of ${\rho }_0$ and magnetoresistive effects.

Figure 6

(a) Transverse magnetoresistivity of YbPtBi (sample #13) as plotted $\rho$ vs $H$ at various temperatures: H $\parallel$ [100] and I $\parallel$ [010] (H $\perp$ I). (Inset) An expanded plot in the low-field regime for $T=0.02$ K, 0.3 K, and 0.5 K. Open circles in both the main figure and the inset represent the residual resistivity taken from the power-law fit to $\rho \left(T\right)$ data (Fig. 4); $T^{1.5}$ fit for $H<8$ kOe and $T^2$ fit for $H\ge 8$ kOe. Vertical arrows in the inset indicate slope changes in the d$\rho \left(H\right)$/d$H$ curve. (b) Transverse magnetoresistance of YbPtBi (sample #13) as plotted $\left[\rho \right(H)-\rho (0\left)\right]/\rho \left(0\right)$ vs $H$ at various temperatures.

Figure 7

(a) d$\rho \left(T\right)$/d$T$ at various magnetic fields up to 4 kOe. Vertical arrows indicate the determined AFM phase transition temperature. (b) d$\rho \left(H\right)$/d$H$ at various temperatures. Up arrows indicate the AFM phase boundary and the down arrow corresponds to a local maximum. (Inset) d$\rho \left(H\right)$/d$H$ up to 0.5 K, where vertical arrows indicate the determined phase transition field.

Figure 8

(a) $\rho \left(T\right)$ vs $T^{1.5}$ at various magnetic fields, where $\rho \left(T\right)$ curves for $H=8$ and 10 kOe are shifted by $-$1 $\mu \Omega$ cm each for clarity. Down arrows indicate the temperature below which $\Delta \rho \left(T\right)=A{T}^{1.5}$, determined from a power-law fit [$\Delta \rho \left(T\right)=A{T}^n$] to the data. For $H=10$ kOe the line is the fit of the power law to the data and the up arrow indicates a deviation from $T^{1.5}$ dependence of $\Delta \rho \left(T\right)$. (b) $\rho \left(T\right)$ vs $T^2$ at various magnetic fields. The arrows mark the temperature where the fits [$\Delta \rho \left(T\right)=A{T}^2$] deviate from the data. These temperatures, $T_{\mathrm{FL}}$, are plotted in the $H-T$ phase diagram (see Fig. 12). For $H=20$ kOe the line is the fit of the power law to the data and the up arrow in the low-temperature side indicates a deviation from $T^2$ dependence of $\Delta \rho \left(T\right)$. (c) Double-logarithmic plots of $\Delta \rho \left(T\right)$ vs $T$ for $H$ $=$ 6, 8, 10, and 15 kOe. The solid lines represent the temperature dependence expected for the exponent $n=1.5$ and $n=2$.

Figure 9

Plots of the sum of the squares of the offsets (residual), ${\chi }^2$, from least-squares fitting of the power law, $\rho \left(T\right)={\rho }_0+A{T}^n$, to the data with fixed $n$ values for (a) $H=5$ kOe and (b) 10 kOe. The calculated resistivity with $n=1.5$ (solid line) and 2 (dashed line) for (c) $H=5$ kOe and (d) 10 kOe. The calculated resistivity curves are based on the fitting up to 0.2 K and 0.15 K for $H=5$ kOe and 10 kOe, respectively.

Figure 10

$\rho \left(T\right)$ of (a) sample #3 and (b) sample #14 at selected magnetic fields. $\rho \left(H\right)$ of (c) sample #3 and (d) sample #14 at selected temperatures. (b) Zero-field $\rho \left(T\right)$ of sample #14 was shifted by $+$3 $\mu \Omega$ cm for clarity. Open circles in (c) and (d) represent the residual resistivity obtained from the power-law fit to the $\rho \left(T\right)$ data. As discussed in text and the caption of Fig. 2, sample #14 was suspended in vacuum by its four Pt leads and sample #3 was mounted to the thermal bath with N grease.

Figure 11

Parameters obtained from power-law fits, $\rho \left(T\right)={\rho }_0+A{T}^n$, to the data for three different samples. Open and solid symbols correspond to fits with $n=1.5$ and $n=2$, respectively. For $n=1.5$, the obtained parameters are plotted only up to 10 kOe. (a) Temperatures of the fit range below which a $T^n$ dependence of $\rho \left(T\right)$ fit data well. For $8\text{kOe}\le H\le 10\text{kOe}$ the fit of $T^{1.5}$ dependence was performed above the temperature, satisfying $T^2$ dependence. The horizontal line for $H\ge 20$ kOe indicates the temperature below which $\rho \left(T\right)$ flattens. (b) Determined exponents $n$ from least-squares fits to the data. (c) Obtained ${\rho }_0$. Solid lines are guides for the eye. (d) Field dependencies of the coefficients, $A=[\rho \left(T\right)-{\rho }_0]/T^n$, with $n=1.5$ and 2, for three different samples.

Figure 12

$H-T$ phase diagram of YbPtBi constructed from the $\rho (T,H)$ results for three different samples; all circles, triangles, and squares correspond to the results of samples #13, #14, and #3, respectively. $T_N$ was derived from the sharp minimum in $\text{d}\rho \left(T\right)/\text{d}T$ (solid symbols) and $\text{d}\rho \left(H\right)/\text{d}H$ (open symbols). $H^{*}$ was derived from the broad local maximum in $\text{d}\rho \left(H\right)/\text{d}H$ (Fig. 7). $T_{\mathrm{FL}}$ represents the upper limit of the $T^2$ dependence of $\rho \left(T\right)$.

Figure 13

(a) Specific heat of YbPtBi as $C_p\left(T\right)$ vs $T$ for $H=0$ (circles) and 140 kOe (triangles), applied magnetic field along the [100] direction, and zero-field $C_p\left(T\right)$ of LuPtBi (squares). (Inset) $C_p/T$ vs $T^2$ for LuPtBi. The solid line is a fit of the equation $C_p\left(T\right)=\gamma T+\beta T^3$ to the data. (b) Zero-field specific heat for YbPtBi and LuPtBi below 10 K.

Figure 14

(a) Logarithmic temperature variation of the magnetic contribution $C_m\left(T\right)$ to the specific heat of YbPtBi at selected magnetic fields: $C_m\left(T\right)=C_p\left(T\right)\left(\text{YbPtBi}\right)-C_p\left(T\right)\left(\text{LuPtBi}\right)$. (Inset) Positions of maxima developed in $C_m\left(T\right)$. (b) Magnetic entropy, $S_m\left(T\right)$, for $H=0$ and 140 kOe, inferred by integrating $C_m/T$ starting from the lowest temperature measured. (Inset) The low-temperature $C_p\left(T\right)$ of YbPtBi (left axis, symbols) and the magnetic entropy ($S_m$) divided by $R$ln(2) (right axis, line). The dashed vertical line marks the peak position of the $\lambda$-shaped anomaly.

Figure 15

(a) Low-temperature specific heat as $C_p\left(T\right)/T$ (solid symbols) and $\Delta C\left(T\right)/T$ (solid lines) vs $\log \left(T\right)$ for YbPtBi at various magnetic fields, applied along the [100] direction; $\Delta C\left(T\right)=C_p\left(T\right)-C_N\left(T\right)$, where the nuclear Schottky contribution was subtracted by using $C_N\left(T\right)\propto 1/T^2$. (Inset) The electronic specific-heat coefficient, ${\gamma =\Delta C\left(T\right)/T|}_{T\to 0}$ (open squares), and $\Delta C\left(T\right)/T$ at $T=0.1$ K (solid circles) as a function of magnetic field. (b) $C_p/T$, $C_N/T$, and $\Delta C/T$ for $H=30$ kOe, plotted in a $\log \left(T\right)$ scale.

Figure 16

(a) The linear thermal expansion coefficient, ${\alpha }_{100}=\text{d}(\Delta L/L_{100})/\text{d}T$, of YbPtBi, where $L$ is the sample length along the [100] direction. The AFM ordering temperature is indicated by the arrow at 0.38 K. The arrow at 6 K represents the maximum observed in the $H=0$ specific-heat data. (Inset) ${\alpha }_{100}$ and $C_m$ at $H=0$. Dashed vertical lines indicate maximum positions observed in $C_m$. (b) ${\alpha }_{100}$ at selected magnetic fields up to 10 K for $H=0$, 2.5, and 5 kOe, measured with a longitudinal configuration $\Delta L/L\parallel \mathbf{H}\parallel \left[100\right]$. (Inset) ${\alpha }_{100}$ for $H=0$, 2.5, 5, and 10 kOe (bottom to top) below 1.5 K.

Figure 17

Magnetostriction and the coefficient of YbPtBi. The linear magnetostriction coefficient, ${\lambda }_{100}=\text{d}(\Delta L/L_{100})/\text{d}H$ vs $H$, at selected temperatures, where $L$ is the sample length along the [100] direction parallel to the magnetic field applied along the [100] (longitudinal configuration $\Delta L/L\parallel \mathbf{H}\parallel \left[100\right]$). The upper inset shows the magnetic field dependence of the magnetostriction $\Delta L/L_{100}$. The lower inset shows ${\lambda }_{100}$ at 0.02 K. The up- and down-arrow indicate the phase transition $T_N$ and the local minimum $H^{*}$, respectively.

Figure 18

(a) The linear magnetostriction coefficient, ${\lambda }_{100}=\text{d}(\Delta L/L_{100})/\text{d}H$ vs $H$, at selected temperatures. For clarity, the data sets are vertically shifted each by $5\times {10}^{-5}$/kOe. The up and down arrows indicate the phase transition $T_N$ and the local minimum $H^{*}$, respectively. (b) ${\lambda }_{100}$ up to 3 K. (Inset) ${\lambda }_{100}$ up to 10 K. Arrows indicate the minimum ($H^{*}$) and maximum in ${\lambda }_{100}$. (c) $H-T$ phase diagram of YbPtBi, constructed form ${\alpha }_{100}$ and ${\lambda }_{100}$. The dashed line is a guide for the eye.

Figure 19

Temperature dependence of the Hall coefficient, $R_H={\rho }_H/H$, of LuPtBi for $H=10$ kOe, applied along the [111] direction. (Inset) The zero-field resistivity.

Figure 20

Hall resistivity, ${\rho }_H$, of YbPtBi as a function of magnetic field, applied along the [100] direction, at various temperatures. The arrow pointing to the 0.06 K curve near 55 kOe indicates a deviation from linear field dependence of ${\rho }_H$. The dash-dotted line is a guide for the eye. (Inset) The low-temperature and low-field ${\rho }_H$ at selected temperatures.

Figure 21

Hall coefficient, $R_H={\rho }_H/H$, of YbPtBi as a function of magnetic field, applied along the [100] direction, at various temperatures. The arrow near 8 kOe indicates the position of local minimum shown in $R_H$ at $T=0.06$ K. (Inset) The high temperature $R_H$ for $T=3$, 5, 10, 25, 50, 100, and 300 K (top to bottom).

Figure 22

(a) ${\rho }_H/H$ at selected temperatures. For clarity, the data sets have been shifted by different amounts vertically. Arrows indicate the position of local minimum. The positions of $H^{*}$ are plotted in the $H-T$ plane in the inset. For comparison, $\text{d}{\rho }_H/\text{d}H$ at selected temperatures are plotted in (b) and (c). Vertical arrows in (b) indicate the position $H^{*}$ used in (a) as the local minimum.

Figure 23

(a) Temperature dependence of the Hall coefficient ($R_H={\rho }_H/H$) of YbPtBi at various magnetic fields, applied along the [100] direction. Closed and open symbols are taken from temperature and field sweeps of ${\rho }_H$, respectively. The open-diamond symbols ($\diamond$) of $R_H(T\to 0)$ are obtained from the initial slope of ${\rho }_H$ vs $H$. The dashed line is a guide for the eye. (b) $R_H$ for $H=10$, 40, 90, and 140 kOe in linear scale of $T$.

Figure 24

Temperature-dependent TEP, $S\left(T\right)$, of YbPtBi at selected magnetic fields, applied along the [100] direction. (Inset) The zero-field $S\left(T\right)$ of LuPtBi.

Figure 25

(a) Low-temperature $S\left(T\right)$ of YbPtBi at selected magnetic fields for $H\le 15$ kOe. Vertical arrows indicate a local minimum $T_0$. (Inset) The zero-field $S\left(T\right)$ below 1 K. The vertical arrow represents the AFM ordering temperature at which the slope, $\text{d}S\left(T\right)/\text{d}T$, is changed. (b) Low-temperature $S\left(T\right)$ for $15\le H\le 90$ kOe. Vertical arrows indicate the characteristic features corresponding to a local minimum temperature $T_0$, a slope change at $T_{\mathrm{FL}}$, and a local maximum $T_{\max }$ for $H\ge 70$ kOe.

Figure 26

Temperature-dependent TEP divided by temperature, $S\left(T\right)/T$, in a logarithmic scale; $S\left(T\right)/T$ vs $\log$($T$). (a) The dashed line on the curve for $H=2.5$ kOe is a guide for the eye, representing a logarithmic increase of $S\left(T\right)/T$ below 4 K. (b) Dashed lines on the curves for $H=30$, 40, and 50 kOe indicate a saturation of $S\left(T\right)/T$, which corresponds to the linear temperature dependence of $S\left(T\right)$ below $T_{\mathrm{FL}}$, shown in Fig. 25b.

Figure 27

Magnetic field dependence of TEP, $S\left(H\right)$, of YbPtBi at selected temperatures.

Figure 28

$S\left(H\right)$ of YbPtBi below 1.5 K. For clarity, the data sets have been shifted by every $-$3 $\mu$V/K vertically. Solid circles $H_{\mathrm{SR}}$ indicate a sign change of TEP from positive to negative. Down arrows $H^{*}$ represent the determined position of the local minimum. Up arrows indicate a slope change, $\text{d}S\left(H\right)/\text{d}H$, above which $S\left(H\right)$ follows a linear field dependence.

Figure 29

Features from $S(T,H)$ measurements plotted in a $H-T$ diagram: $T_{\mathrm{SR}}$ and $H_{\mathrm{SR}}$ represent the sign reversal extracted from the position of $S(T,H)=0$; $H^{*}$ marks the position of the local minimum in $S\left(H\right)$; $T_0$ indicates the position of the local minimum in $S\left(T\right)$; $T_{\max }$ represents the position of the local maximum developed at low temperatures for $H\ge 70$ kOe; and $T_{\mathrm{FL}}$ and $H_{\mathrm{FL}}$ represent the slope change in $S\left(T\right)$ (Fig. 25) and $S\left(H\right)$ (Fig. 28), respectively. The bottom panels show the horizontal and vertical cut through the $H-T$ plane. (Left) Below 30 kOe $S\left(H\right)$ at $T=1$ K hits all three characteristic lines of $H_{\mathrm{SR}}$, $H_0$, and $H^{*}$. (Right) Below 2.5 K $S\left(T\right)$ at $H=15$ kOe indicates both $T^{*}$ and $T_0$ line. See details in the text.

Figure 30

$H-T$ phase diagram of YbPtBi in configuration $\mathbf{H}\parallel \left[100\right]$. The $T_N$ was derived from $\text{d}\rho \left(T\right)/\text{d}T$, $\text{d}\rho \left(H\right)/\text{d}H$, ${\alpha }_{100}$, and ${\lambda }_{100}$. For the phase boundary the $\rho (T,H)$ results for sample #13 are only included. The solid line on the AFM phase boundary represents the fit of equation $T_N={[(H-H_c)/H_c]}^{0.33}$ to the data. The dashed line represents the fit of equation $T_N={[(H-H_c)/H_c]}^{2/3}$ to the data. The $T_{\mathrm{FL}}$ represents the upper limit of the $T^2$ dependence of $\rho \left(T\right)$, where the results of samples #13, #14, and #3 are plotted. The solid line is a guide for the eye. The local maximum of $\text{d}\rho \left(H\right)/\text{d}H$ and the local minimum of ${\lambda }_{100}$ are assigned to $H^{*}\left(T\right)$. It should be noted that the width of each of these $H^{*}$ features decreases with decreasing temperature, as shown by horizontal bars. For each data set the width monotonically decreases with decreasing temperature.

Figure 31

Criteria for determining $T_N$. (a) Zero-field specific heat $C_p$ and the coefficient of linear thermal expansion ${\alpha }_{100}$. (b) Zero-field electrical resistivity $\rho \left(T\right)$ and the derivative $\text{d}\rho \left(T\right)/\text{d}T$. (c) Linear magnetostriction $\Delta L/L$ and the coefficient ${\lambda }_{100}=\text{d}(\Delta L/L)/\text{d}H$ at $T=0.02$ K. (d) Magnetoresistivity $\rho \left(H\right)$ and the derivative $\text{d}\rho \left(H\right)/\text{d}H$ at $T=0.02$ K. Solid lines are guides for the eye.

Figure 32

High temperature $H-T$ phase diagram for YbPtBi. The $S(T,H)=0$ and the slope change from ${\rho }_H/H$ are assigned to $T_{\mathrm{SR}}\left(H\right)$. The local maximum in $\text{d}\rho \left(H\right)/\text{d}H$, the local minimum of ${\lambda }_{100}$, the local minimum of ${\rho }_H/H$, and the local minimum in $S\left(H\right)$ are assigned to $H^{*}\left(T\right)$. The $T_{\mathrm{FL}}$ was derived from the upper limit of the $T^2$ dependence of $\rho \left(T\right)$ and the upper limit of the $T$ dependence of $S\left(T\right)$. The slope change in $S\left(H\right)$ is also assigned to $T_{\mathrm{FL}}$. All lines and shaded area are guides for the eye.

Figure 33

(a) Fermi liquid coefficient $A=\Delta \rho \left(T\right)/T^2$ and ${\gamma =C\left(T\right)/T|}_{T\to 0}$. The solid line through the higher field $A$ values represents a fit of equation $A-A_0\propto 1/(H-H_c)$, performed up to 50 kOe with the constant offset $A_0\simeq 0.03$$\mu \Omega$ cm/K${}^2$ and $H_c=4.2$ kOe. The vertical line represents the critical field ($H_c$). (b) $A^{-1}$ (left axis) vs $H$ for three different samples (samples #3, #13, and #14 in Fig. 11) and ${\gamma }^{-0.5}$ (right axis) vs $H$. Solid lines represent the linear fit to the data. See text for details.

Figure 34

The temperature dependence of the resistivity, $\Delta \rho \left(T\right)=\rho \left(T\right)-{\rho }_0$, of YbPtBi for (a) sample #13 and (b) sample #3. (a) The solid and dash-dotted lines represent the calculated $\Delta \rho \left(T\right)$ with $A\simeq 76.7$ $\mu \Omega$ cm/K${}^2$ for $H=6$ kOe and with $A\simeq 32.4$ $\mu \Omega$ cm/K${}^2$ for $H=8$ kOe, respectively. (b) The solid line and dashed line represent the calculated $\Delta \rho \left(T\right)$ with $A\simeq 76.7$ $\mu \Omega$ cm/K${}^2$ for $H=6$ kOe and with $A\simeq 46.7$ $\mu \Omega$ cm/K${}^2$ for $H=7$ kOe, respectively. The $A$ values used to generate $\Delta \rho \left(T\right)$ were obtained from the power-law fit [$A\propto 1/(H-H_c)$] to the $A$ values shown in Fig. 33. See text for details.

Figure 35

(a) $\log -\log$ plot of $A$ vs $\gamma$. Solid lines represent the Kadowaki-Woods (K-W) ratio for different ground-state degeneracy (Ref. 71) for $N=2–8$. (b) $A/{\gamma }^2$ vs $H$, where the horizontal arrow indicates the K-W ratio for $N$ $=$ 2. The solid line is a guide for the eye.

Figure 36

Global phase diagram for HF materials, adopted from Refs. 33 and 34, displaying the combined effects of Kondo coupling ($J_K$) and magnetic frustration ($G$). The large circle is the hypothetical location of YbPtBi. In order to destroy the AFM order, a small applied magnetic field ($\sim$4 kOe) is required. Beyond the AFM QCP, the $T^{1.5}$ dependence of resistivity is observed passing through a finite magnetic-field range ($4\text{kOe}<H<8\text{kOe}$). For $H\ge 8$ kOe, passing through the $f$-electron localized-to-delocalized line, the $T^2$ dependence of resistivity is clearly observed.