Strange metal state near a heavy-fermion quantum critical point

TL;DR

This paper proposes a theoretical mechanism explaining the strange metal state observed near a heavy-fermion quantum critical point, addressing non-Fermi liquid behaviors through a competition between RVB spin-liquid fluctuations and Kondo correlations.

Contribution

It introduces a field-theoretical renormalization group analysis of an effective model to explain the strange metal phenomena near quantum criticality in heavy-fermion systems.

Findings

Identifies the critical point via RG analysis.

Explains T-linear resistivity and specific heat anomalies.

Describes crossover behaviors in the strange metal region.

Abstract

Recent experiments on quantum criticality in the Ge-substituted heavy-electron material YbRh2Si2 under magnetic field have revealed a possible non-Fermi liquid (NFL) strange metal (SM) state over a finite range of fields at low temperatures, which still remains a puzzle. In the SM region, the zero-field antiferromagnetism is suppressed. Above a critical field, it gives way to a heavy Fermi liquid with Kondo correlation. The T (temperature)-linear resistivity and the T-logarithmic followed by a power-law singularity in the specific heat coefficient at low T, salient NFL behaviours in the SM region, are un-explained. We offer a mechanism to address these open issues theoretically based on the competition between a quasi-2d fluctuating short-ranged resonant- valence-bonds (RVB) spin-liquid and the Kondo correlation near criticality. Via a field-theoretical renormalization group analysis on an effective field theory beyond a large-N approach to an anti- ferromagnetic Kondo-Heisenberg model, we identify the critical point, and explain remarkably well both the crossovers and the SM behaviour.

Equations

$$S_{eff}$$

$$S_0$$

$$S_{\Phi }$$

$$S_J$$

$$S_4$$

$$\frac{d{j}_{\chi }}{dl}=-\left(\frac{d-z}{2}\right)j_{\chi }+\frac{1}{2}{j}_{\chi }^3+2j_{\Phi }^2{j}_{\chi }$$

$$\frac{d{j}_{\Phi }}{dl}=-\frac{d}{2}{j}_{\Phi }+4j_{\Phi }^3;\frac{d{u}_{\Phi }}{dl}=-(d-z)u_{\Phi }-3u_{\Phi }^2$$

$$\frac{d{u}_{\chi }}{dl}=-(d-z)u_{\chi }-4j_{\chi }^2{u}_{\chi }-3u_{\chi }^2$$

$$\frac{d}{dl}\left(\frac{1}{\Gamma }\right)=-z\left(\frac{1}{\Gamma }\right)+4j_{\Phi }^2;\frac{d{m}_{\chi }}{dl}=z{m}_{\chi }+j_{\chi }^2$$

$$\frac{d{m}_{\Phi }}{dl}=z{m}_{\Phi },$$

$$\displaystyle\gamma\left(\bar{T}\right)\equiv\frac{e^{-\eta l}}{T(l)A(l)}\,\frac{\partial\bar{E}_{G}(l)}{\partial T(l)}\Big{|}_{l=l_{0}}=\frac{\bar{T}^{\frac{\eta}{2}}}{4}\int^{\frac{\Lambda}{\bar{T}}}_{\frac{1}{\bar{T}}}\,dx\,\frac{x^{2+\eta/2}}{\sinh^{2}(x/2)}$$

Full text

Introduction. Magnetic field tuned quantum phase transitions (QPTs)SachdevQPT in heavy-fermion metals of both pure and Ge-substituted $\text{YbRh}_{2}\text{Si}_{2}$ (YRS) compounds Gegenwart2008 ; Custers2003 ; Grosche2001a ; Custers2010 ; Coleman2007 are of great interest both theoretically and experimentally. Near a quantum critical point (QCP), these systems show exotic non-Fermi liquid (NFL) electronic properties at finite temperatures, including a $T$-linear resistivity Custers2010 ; SiNature2008 ; Custers2003 and power law-in-$T$ at low $T$ followed by a $T$-logarithmic specific heat coefficient at higher $T$ Custers2003 , which still remain as outstanding open issues QMSi2001 ; Zhu2002 ; Si2003 ; Si2011 ; Senthil2003 ; Wolfle2011 ; Abrahams2012 ; Rosch1997 ; 2005Pepin .

For the QPT in pure YRS, the “Kondo breakdown” scenario QMSi2001 offers a general understanding: competition between the Kondo and antiferromagnetic RKKY couplings leads to a QCP, separating the antiferromagnetic metallic state from the paramagnetic Landau Fermi-liquid state (LFL) with enhanced Kondo correlation. There, a magnetic phase transition and the emergence (or the breakdown) of the Kondo effect occur simultaneously and in the ground state the system undergoes a jump from a small to a large Fermi surface PaschenNat2004 ; FriedemannPNAS . The understanding of the NFL properties near the QCP still remains an outstanding open issue though certain aspects have been addressed QMSi2001 ; Zhu2002 ; Si2003 ; Si2011 ; Senthil2003 ; Senthil2004 ; Wolfle2011 ; Abrahams2012 ; Rosch1997 ; 2005Pepin .

Recent experiments on Ge-substituted YRS, however, reveals intriguing distinct features. First, the magnetic phase transition at $g_{c}^{\prime}$ occurs at a lower field than the Kondo destruction at $g_{c}$, leading to a decoupling of the AFM and the LFL phases (see Fig. 1(a)) Custers2003 . Interestingly, similar NFL behaviour persists over a finite range in magnetic fields at the lowest temperatures, suggesting a possible exotic novel stable spin-disordered “strange-metal” (SM) ground state Custers2010 . Open questions to be addressed include: Dose this SM behaviours come from the SM ground state or from a single QCP? What is the role played by the magnetic field? What mechanism is behind the Kondo breakdown at the QCP?

A short-ranged resonanting-valence-bond (RVB) spin-liquid (SL) picture has recently been proposed to describe the metallic spin-liquid phase of heavy fermion metal with frustrated antiferromagnetic RKKY coupling Pixley2013 . A generic phase diagram was proposed in Ref. Pixley2013, in terms of magnetic frustration and Kondo correlation. There, a small Kondo coupling at low fields may co-exist with a small Fermi surface to form a (Kondo stablized) SL metal ColemanAndrei1989 . The disorder due to Ge substitution here makes this proposal more attractive to account for the SM state. It is of great interest to further explore the mechanism of this behaviour. In this work, we propose a mechanism for the Kondo breakdown and the quantum criticality in Ge-substituted YRS at $g_{c}$ via a fermionic large-N approach based on the symplectic group $Sp(N)$ symmetry on a quasi-$2d$ Kondo lattice. The competition between the Gaussian fluctuating RVB spin-liquid and the Kondo correlation (see Fig. 1) explains remarkably well both the transition and the NFL behaviours. The SM state indicated in experiments can be understood as quantum critical region near critical Kondo breakdown.

The Large-$N$ Mean-Field Hamiltonian. Our starting point is the fermionic $Sp(N)$ large-$N$ mean-field Hamiltonian of the Kondo lattice model Senthil2003 : $H_{Sp(N)}^{MF}=H_{0}+H_{\lambda}+H_{K}+H_{J}$, where $H_{0}=\sum_{\langle i,j\rangle;\sigma}\left[t_{ij}c^{\dagger}_{i\sigma}c_{j\sigma}+h.c.\right]-\sum_{i\sigma}\,\mu\,c_{i\sigma}^{\dagger}c_{i\sigma}$ , $H_{\lambda}=\sum_{i,\sigma}\lambda\left[f_{i\sigma}^{\dagger}f_{i\sigma}-2S\right]$, $H_{J}=\sum_{\langle i,j\rangle}J_{ij}{\bf S}_{i}^{imp}\cdot{\bf S}_{j}^{imp}=\sum_{\langle i,j\rangle;\alpha,\beta}\left[\Phi_{ij}\mathcal{J}^{\alpha\beta}f_{i\alpha}f_{j\beta}+h.c.\right]+\sum_{\langle i,j\rangle}N\frac{|\Phi_{ij}|^{2}}{J_{H}}$, $H_{K}=J_{K}\sum_{i}{\bf S}_{i}^{imp}\cdot{\bf s}^{c}=\sum_{i,~{}\sigma}\left[\left(c_{i\sigma}^{\dagger}f_{i\sigma}\right)\chi_{i}+h.c.\right]+\sum_{i}\,N\frac{|\chi_{i}|^{2}}{J_{K}}$, where the antiferromagnetic RKKY interaction $H_{J}$ is described by the fermionic $Sp(N)$ spin-singlet with $\left[Sp(1)\simeq SU(2)~{}\text{for}~{}N=2\right]$, $\mathcal{J}^{\alpha\beta}=\mathcal{J}_{\alpha\beta}=-\mathcal{J}^{\beta\alpha}$ being the generalization of the $SU(2)$ antisymmetric tensor $\epsilon_{\alpha\beta}=\epsilon^{\alpha\beta}=-\epsilon_{\beta\alpha}=i\sigma_{2}$ ReadSachdev1991 ; SachdevKagome ; Chung2001 ; Flint2008Nature . $H_{0}$ describes hopping of conduction $c$-electrons, while $H_{K}$ denotes Kondo interaction. We assume a uniform $J_{ij}=J_{H}$ RKKY coupling on a lattice with $i,~{}j$ being nearest-neighbour sites, $\sigma,\alpha,\beta\in\{-\frac{N}{2},\cdots,\frac{N}{2}\}$ with $N\to\infty$. $H_{\lambda}$ describes the local impurity $f_{i\sigma}$ electrons with $\lambda$ being the Lagrange multiplier to impose the local constraint $\langle\sum_{\sigma}f_{i\sigma}^{\dagger}f_{i\sigma}\rangle=N\kappa$ where a constant $\kappa\equiv 2S$ ensures the fully screened Kondo effect ColemanSchwingerboson ; foot-kappa . The mean-field Kondo hybridization and RVB spin-singlet are defined as $\chi_{i}\equiv\langle\frac{J_{K}}{N}\sum_{\sigma}f^{\dagger}_{i\sigma}c_{\sigma}\rangle$ and $\Phi_{ij}\equiv\langle\frac{J_{H}}{N}\sum_{\alpha,\beta}\,\mathcal{J}_{\alpha\beta}f^{\alpha\dagger}_{i}f^{\beta\dagger}_{j}\rangle$, respectively foot-SU2-gauge . Besides the RVB phase, this approach enables us to describe both the Kondo LFL and the superconducting phases scYRS2016 via Bose-condensing $\chi$-field and the fermionic $Sp(N)$ singlets, respectively SpNfootnote . Excluding superconductivity here (as it is likely suppressed by magnetic field), $H_{Sp(N)}^{MF}$ on a $2d$ lattice is shown to have a fractionalized Fermi-liquid (FL*∗*) phase Senthil2003 with $\chi_{i}=0$, $\Phi_{ij}\neq 0$ for $J_{K}\ll J_{H}$, and a Kondo-RVB spin-liquid co-existing heavy-Fermi-liquid (LFL) phase with $\chi_{i}\neq\Phi_{ij}\neq 0$ for $J_{K}\sim\mathcal{O}(J_{H})$ and a large Fermi surface due to Kondo hybridization Senthil2003 ; LFLfootnote . The mean-field result captures qualitatively the competition between RVB singlets and the Kondo correlations near criticality (see Fig. 1(b)). However, to account for the NFL and crossovers, we shall analyze the dynamics and fluctuations beyond mean-field level via a perturbative renormalization group (RG) approach.

The Effective Field Theory Beyond $Sp(N)$ Mean-Field. We consider here the Gaussian (amplitude) fluctuations around the mean-field order parameters $\chi_{i}$ and $\Phi_{ij}$ with dynamics in the FL*∗* phase close to the QCP foot-Higgs ; supp . The effective action $S_{eff}$ for a fixed $N=1$ reads:

[TABLE]

with $k=(\omega,{\bf k})$ and $dk=d^{d}{\bf k}d\omega$ and $\tau$ the imaginary time. The actions $S_{\chi}$ ($S_{K}$) and $S_{\Phi}$ ($S_{J}$) represents the Kondo hybridization and RKKY interaction at (beyond) the mean-field level, respectively, while $S_{G}$ ($S_{4}$) represents the action of the quadratic Gaussian (quartic) fluctuating fields, respectively. $\Phi_{\textbf{k}}$ ($\hat{\tilde{\Phi}}_{k}$) and $\chi_{\textbf{k}}$ ($\hat{\tilde{\chi}}_{k}$) are the Fourier-transformed mean-field variables (amplitude fluctuating fields above mean-field) $\Phi_{ij}$ ($\hat{\Phi}_{ij}\equiv\frac{1}{N}\Sigma_{\alpha\beta}f^{\alpha\dagger}_{i}f^{\beta\dagger}_{j}$) and $\chi_{i}$ ($\hat{\chi}_{i}\equiv\frac{1}{N}\Sigma_{\sigma}f_{i\sigma}^{\dagger}c_{i\sigma}$), respectively. $\epsilon_{c}({\bf k})$, $\epsilon_{\chi}({\bf k})$ and $\epsilon_{\Phi}({\bf k})$ are the quadratically dispersed kinetic energies of the itinerant electrons $c_{k}$, local singlet $\hat{\tilde{\Phi}}_{k}$ and the Kondo hybridization $\hat{\tilde{\chi}}_{k}$, respectively. The quadratic forms of $\epsilon_{\chi}({\bf k})$ and $\epsilon_{\Phi}({\bf k})$ are derived via integrating out $c$-electrons away from Fermi surface foot-U1 ; supp . We find $\hat{\tilde{\chi}}$-field is not Landau damped since the imaginary part of its self-energy vanishes: $\text{Im}\Sigma_{\chi}(\omega)\propto\int\,d\epsilon\delta(\omega+\epsilon-\bar{\lambda})=0$ as $\bar{\lambda}\gg\epsilon+\omega$ supp , leading to a jump in the Fermi volume at the critical point, consistent with experiments PaschenNat2004 ; FriedemannPNAS .

RG Analysis: Within our RG scheme $k$, $k_{F}$ and $\omega$ are rescaled as: $k^{\prime}=e^{l}k;~{}~{}k_{F}^{\prime}=e^{l}k_{F};~{}~{}\omega^{\prime}=e^{zl}\omega$ where the dynamical exponent $z$ is set to $2$. This scheme, distinct from the conventional one Yamamoto2010 ; ShankarRG , allows the Fermi momentum $k_{F}$ to flow the same way as the momentum variable $k$: $[k]=[k_{F}]=1$, effectively capturing the mixture of electron population in small and large Fermi surface (or the continuous evolution of the Hall coefficient) at finite temperatures ShankarRG ; YeCuO1991 ; PatchRGfootnote . At tree-level, $J_{\chi}$, $J_{\Phi}$, $u_{\chi}$ and $u_{\Phi}$ are irrelevant couplings for $d>z=2$, while $J_{\chi}$, $u_{\chi}$ and $u_{\Phi}$ become marginal for $d=z=2$ ($[J_{\chi}]=(z-d)/2,[u_{\chi}]=[u_{\Phi}]=z-d$), which allows a controlled perturbative RG analysis on the effective action on a quasi-$2d$ lattice: $d=2+\eta$ with $\eta=0^{+}$ foot-quasi2d .

The RG $\beta$-functions in the weak-coupling limit, $J_{\chi}=J_{\Phi}\to 0$, are readily obtained via diagrammatic perturbative approaches, which include coupling constants renormalization and the field (or the Green’s functions)-renormalization Zhu2002 ; Si2003 (see Ref. supp, ) :

[TABLE]

where $dl=-d\ln\Lambda$ with $\Lambda$ being the running energy cutoff within momentum-shell RG is used while the constant $\bar{\lambda}$ is the effective chemical potential of the local $f$-electron supp . Here, $j_{\chi,\Phi}\,(J_{\chi,\Phi})$ refers to the renormalized (bare) coupling supp . At two stable phases, the mass term $m_{\chi,\Phi}$ flows to a massive fixed point, $m_{\chi,\Phi}^{\ast}\to\infty$. Near the QCP, their bare values vanish linearly with distance to criticality: $m_{\chi,\Phi}\propto|g-g_{c}|$. Since the effective dimension $d+z>4$, greater than the upper critical dimension, the Gaussian fixed point $u^{\ast}=v^{\ast}=0$ is stable Ginzburg-footnote , and violation of hyperscaling is expected LJZhu . Two non-trivial intermediate critical fixed points are found at $(j^{2^{*}}_{\chi},\,j^{2^{*}}_{\Phi})=\left(\eta,0\right)$ ($P$) and $\left(0,\,d/8\right)$ ($Q$) (see Fig. 2). The fixed point at $Q$ controls the transition between the two FL∗ fixed points, while $P$ separates FL∗ phase at $j_{\chi}=0$ from the Kondo co-existing with spin-liquid (LFL) phase at $j_{\chi}\to\infty$.

To more precisely locate the QCP at finite values of $j_{\Phi},~{}j_{\chi}$, the RG equations for $j_{\chi},j_{\phi}$ are obtained near the fixed points $P$ (with $J_{\chi}$ being fixed at $J_{\chi}^{*}$) and $Q$ (with $J_{\Phi}$ being fixed at $j_{\Phi}^{*}$) (see Ref. supp, ): $\frac{d\tilde{j}_{\Phi}}{dl}=\left(-\frac{d}{2}+\frac{\eta}{2}\right)\,\tilde{j}_{\Phi}+4\,\tilde{j}_{\Phi}^{3},\,\frac{d\tilde{j}_{\chi}}{dl}=-(\frac{\eta}{2}+d/4-2(j_{\Phi}^{*})^{2})\,\tilde{j}_{\chi}+\frac{1}{2}\tilde{j}_{\chi}^{3}.$ The critical point $C_{r}$ in Fig. 2, which controls the FL*∗*-LFL QPT, is located at the intersect of the above two RG flows: $((j_{\chi}^{\ast})^{2},\,(j_{\Phi}^{\ast})^{2})=\left(\eta,d/8\right)$. Note that $C_{r}$ is an interacting QCP due to the presence of Kondo interaction. As a result, the $\omega/T$ scaling in dynamical observables is found there via the Kondo breakdown scenario FriedemannPNAS ; supp .

Critical Properties and Crossovers. The correlation length $\xi$ diverges near $C_{r}$: $\xi\sim|g-g_{c}|^{-\nu}$ with an exponent $\nu$. We find that $\nu$ is solely determined by the RG flow of $j_{\Phi}-j_{\Phi}^{\ast}$ via $\beta(\tilde{j}_{\Phi})$ supp . This yields $\nu=1/z$, leading to the linear SM-LFL (SM-FL∗) crossover scale $T_{LFL}$ ($T_{FL^{\ast}}$) below which $j_{\Phi}<j_{\chi}^{*}$ ($j_{\Phi}>j_{\chi}^{\ast}$): $T_{LFL},T_{FL^{\ast}}\propto|g-g_{c}|$, in perfect agreement with the experiment on Ge-substituted YRS Custers2003 ; Custers2010 . This suggests that the main effect of magnetic field $B$ in experiment (represented by the coupling $g=g_{0}-J_{\Phi}/J_{\chi}$) near the QCP is to suppress $J_{\Phi}$ while $J_{\chi}$ is near its critical value $J_{\chi}\sim J_{\chi}^{*}$ (see Fig. 2 and Fig. 1(a)). Careful analysis on the pre-factors gives supp $\frac{T_{FL^{\ast}}}{T_{LFL}}\sim(\frac{J_{\chi}}{|\chi|})^{4}\sim L^{-2\eta}\ll 1$ with $L$ being system size, giving rise to the difficulty to observe $T_{FL^{*}}$ in experiments (see Fig. 1(a)) foot-FL* . We further identify $T^{\ast}$ as the crossover scale for the onset of the condensate $\chi$ field below which $j_{\Phi}<\chi$. A sub-linear dependence in $g-g_{c}$ in found: $T^{*}\propto(g-g_{c})^{z/d}$ with $z/d\sim 1-\eta/z$, and $\frac{T^{\ast}}{T_{LFL}}\sim\frac{|\chi|}{J_{\chi}}>1$, consistent with the experiments Custers2003 ; Custers2010 . The experimentally observed inverse-in-field divergence in the $A$-coefficient of the $T^{2}$ term to the resistivity $\rho\sim AT^{2}$ close to the FL∗-LFL QPT is also reproduced: $A\propto\xi^{2}\sim|g-g_{c}|^{-2\nu}\sim|g-g_{c}|^{-1}$. The SM region in Ref. Custers2010, on Ge-substituted YRS can be interpreted here as the extended quantum critical regime down to $T_{FL^{*}}\to 0$.

NFL: Electrical Resistivity. The finite temperature electrical resistivity $\rho(T)$ near the QCP is obtained via the conductivity HewsonBook $\sigma(T)=\rho^{-1}(T)=-\frac{2e^{2}}{3}\int\frac{d{\bf k}}{(2\pi)^{3}}v_{k}^{2}\,\tau(k)\,\frac{\partial f}{\partial\epsilon_{k}}$ with $v_{k}$ being the electron group velocity, and $\tau^{-1}(\omega)$ the scattering rate of the electron, given by the imaginary part of the conduction electron $T-$matrix $\text{Im}\,T(\omega)$ : $\tau^{-1}(\omega)=-\frac{c_{imp}}{2}\sum_{{\bf k}}\text{Im}\,T_{{\bf k}{\bf k}}(\omega^{+})$. Remarkably, we find that the $T-$matrix contributed from the quasi-2d bosonic fluctuations in the Kondo hybridization $\hat{\tilde{\chi}}-$field leads to the observed $T$-linear resistivity at low temperatures: $\rho(T)=a^{-1}\,(1+c\,T/a)$, where $a$ and $c$ are constant pre-factors supp . We find the ratio in resistivity at low temperatures: $\Delta\rho/\rho(T=0)\equiv(\rho(T)-\rho(T=0))/\rho(T=0)$, in reasonable agreement with that measured in experiment Custers2010 .

NFL: Specific Heat Coefficient. We further compute the (normalized) scaling function of electronic specific heat coefficient in the SM region:

[TABLE]

(where $\bar{T}\equiv\frac{T}{T_{LFL}}$, $A(l_{0})\propto|g-g_{c}|^{-\zeta}$ with $\zeta\sim 3\eta/2+\eta^{2}$ near the QCP, and $W(J_{\Phi})$ is a non-universal constant), contributed dominantly from the kinetic energy $\bar{E}_{G}=\sum_{{\bf k}}\frac{\epsilon_{\Phi}({\bf k})}{e^{\beta\epsilon_{\Phi}({\bf k})}-1}$ of $\hat{\tilde{\Phi}}$ fields supp . Here, $l_{0}$ is a scale at which $m_{\Phi}(l_{0})\sim 1$, and $x=\epsilon_{\Phi}(l)/T(l)$ while $T(l)=Te^{zl}$ is the scale-dependent dimensionless temperature via the finite-temperature RG scheme MillisRG . As shown in Fig. 3, $\gamma(\bar{T})$ bears a striking similarity to that observed in Ref. Custers2003, : it exhibits a power-law scaling behaviour at low temperatures before it saturates at $T=0$, i.e. $\gamma(\bar{T})\propto\bar{T}^{-\alpha}$, followed by a logarithmic tail at higher temperatures $\gamma(\bar{T})\propto-\ln\bar{T}$ with exponents $\alpha\sim\zeta\sim 0.3(1)$ for an estimated $\eta\sim 0.18$, in excellent agreement with the experimental values $\alpha\sim\gamma\sim 1/3$ supp . The $T$-logarithmic behaviour in $\gamma(\bar{T})$ comes as a result of the Gaussian fixed point for $d=2+\eta$ foot-uphi .

NFL: Local Spin Susceptibility. Finally, the observed anomalous exponent in the divergent temperature dependence of the zero-field local spin susceptibility $\chi_{\text{loc}}\sim T^{-0.75}\sim\frac{1}{T^{1-\eta_{\chi}}}$ for Ge-substituted YRS Vojta2004 ; Fritz2004 ; Fritz2006 with $\eta_{\chi}\sim 0.25$ is reasonably accounted for within our approach supp : $\chi_{\text{loc}}(T)\propto\frac{J_{\chi}^{4}(T)}{T}\propto T^{2\eta-1}$, giving an estimated $\eta_{\chi}\sim 0.36$.

Conclusions. We have theoretically addressed the non-Fermi liquid and quantum critical properties of Ge-substituted YbRh2Si2 by a field-theoretical renormalization group analysis on an effective field theory based on the $Sp(N)$ approach to the Kondo-Heisenberg lattice model. The quantum phase transition and crossover scales are well captured in terms of a competition between a short-ranged fermionic resonant-valence-bond spin liquid and the Kondo effect near a quantum critical point. The agreement of our predicted critical properties with experiments is remarkable. The strange metal state can be interpreted as the extended quantum critical region to $T\to 0$ due to its proximity to critical Kondo breakdown. Our theory shed light on the open issues of the non-Fermi liquid behavior in field-tuned quantum critical heavy fermion.