## Abstract

Fermi liquid theory, the standard theory of metals, has been challenged by a number of observations of anomalous metallic behavior found in the vicinity of a quantum phase transition. The breakdown of the Fermi liquid is accomplished by fine-tuning the material to a quantum critical point by using a control parameter such as the magnetic field, pressure, or chemical composition. Our high-precision magnetization measurements of the ultrapure *f*-electron–based superconductor β-YbAlB_{4} demonstrate a scaling of its free energy that is indicative of zero-field quantum criticality without tuning in a metal. The breakdown of Fermi liquid behavior takes place in a mixed-valence state, which is in sharp contrast with other known examples of quantum critical *f*-electron systems that are magnetic Kondo lattice systems with integral valence.

Quantum phase transitions occur at zero temperature as a consequence of quantum rather than thermal correlations. Generally, a quantum critical point (QCP) can be reached by driving a finite-temperature critical point (*1*, *2*) or a first-order critical end-point (*3*) to absolute zero. The breakdown of Fermi liquid (FL) behavior in metals observed near a magnetic QCP challenges our current understanding of strongly correlated electrons. Although the mechanism of unconventional quantum criticality is actively debated, there is a growing consensus that the underlying physics involves a jump in the Fermi surface volume associated with a partial electron localization (*4*–*8*). To date, the FL breakdown has only been observed by fine-tuning a material to a QCP through use of a control parameter such as magnetic field, pressure, or chemical composition.

Recent work reporting the discovery of superconductivity in an ytterbium (Yb)–based heavy fermion material β-YbAlB_{4} has raised the interesting possibility that this system may exhibit quantum criticality without tuning (*9*). In this compound, signatures of quantum criticality were observed to develop above a tiny superconducting (SC) dome, with a SC transition temperature of *T*_{c} ~ 80 mK and an upper critical field μ_{0}*H*_{c2} ≈ 30 mT (*9*, *10*). Although this observation motivated the possibility of a zero-field QCP, it did not rule out a QCP located near the upper critical field *H*_{c2}, as observed in the heavy fermion superconductor CeCoIn_{5} (*11*).

In this report, we present clear and quantitative evidence that quantum criticality develops at zero field without tuning in β-YbAlB_{4}, buried deep inside the SC dome. Moreover, we report a simple *T*/*B*–scaling form of the free energy spanning almost four decades in magnetic field, revealing that the signatures of the putative quantum critical point extend up to temperatures *T* and fields *B* more than 100 times larger than *T*_{c} and μ_{0}*H*_{c2}, respectively.

To quantify the free energy *F*(*T*,*B*), we used high-precision measurements of the magnetization *M* = –∂*F*/∂*B*. Measurements were made on ultrahigh-purity single crystals with a mean free path exceeding 1000 Å and residual resistivity of less than 0.6 microhm·cm, which were carefully etched to fully remove surface impurities (*12*). Our measurements revealed a simple *T*/*B* scaling over a wide range of temperature and field, governed by a single quantum-critical (QC) scaling exponent previously masked (*9*) by a limited experimental resolution and the impurity effects caused by surface and bulk impurities (*12*). The *T*/*B* scaling leads to the following meaningful consequences. First, the QC physics is self-similar over four decades of *T*/*B*, with no intrinsic energy scale. Second, the field-induced FL is characterized by a Fermi temperature that grows linearly with the field, determined by the Zeeman energy of the underlying critical modes. Lastly, the scaling allowed us to determine an upper bound on the magnitude of the critical field |*B*_{c}| < 0.2 mT, which is well inside the SC dome and comparable with Earth’s magnetic field; this indicates that β-YbAlB_{4} is intrinsically quantum critical, without tuning the magnetic field, pressure, or composition.

These results are surprising given the fluctuating valence nature of this material, with valence Yb^{+2.75} significantly far from integral, revealed by recent experiments (*13*). All QC heavy-fermion intermetallics known to date have an almost integral valence that stabilizes the local moments (*1*, *2*). Such so-called Kondo lattice systems are characterized by a small characteristic scale *T*_{0}, below which the moments are screened to form a paramagnetic heavy FL. Various types of order, such as superconductivity and antiferromagnetism (AFM), compete with the heavy FL, leading to quantum criticality, as seen in, for example, CeCu_{5.9}Au_{0.1} (*T*_{0} = 6.2 K) (*14*) and YbRh_{2}Si_{2} (24 K) (*2*). In contrast, mixed-valence compounds display a much larger *T*_{0}, below which they typically behave as stable FLs with moderate quasiparticle effective masses and no competing order. For example, YbAl_{3}, with nonintegral valence Yb^{+2.71}, is characterized by *T*_{0} ~ 300 K (*15*).

A remarkable feature of β-YbAlB_{4} (Fig. 1A) is that it is quantum critical (*9*), yet the scale *T*_{0} ~ 250 K, obtained from the resistivity coherence peak, is one or two orders of magnitude larger than in other known QC materials. This is confirmed by the scaling behavior of the magnetic specific heat: *S*_{0} is a constant (Fig. 1B, inset). The −ln*T* dependence of *C*_{M}/*T* in the three QC materials CeCu_{5.9}Au_{0.1} (*14*), YbRh_{2}Si_{2} (*2*, *16*), and β-YbAlB_{4} collapse onto one curve after setting *T*_{0} for β-YbAlB_{4} ~ 200 K. The recent observation of intermediate valence (Yb^{+2.75}) in β-YbAlB_{4} at 20 K by use of hard x-ray photoemission spectroscopy (*13*) is consistent with this large *T*_{0}.

The quantum criticality in this valence-fluctuating state is accompanied by several properties reminiscent of an integral-valence Kondo lattice. To understand their origin, it is useful to compare them with those of α-YbAlB_{4} (Fig. 1A) (*17*), which is a locally isostructural polymorph of β-YbAlB_{4} and a FL with a similarly intermediate valence (Yb^{+2.73}) (*13*). Instead of Pauli paramagnetism normally seen in a valence-fluctuating material, the magnetic susceptibilities of both materials display Curie-Weiss behavior with Weiss temperature Θ_{W}, χ = *C*/(*T* + Θ_{W}), indicating the existence of local moments (Fig. 2A). In addition, both materials have a maximum in –*dM*/*dT* at *T** ~ 8 K, signaling a crossover from local moment behavior (fig. S1). Below *T**, *C*_{M}/*T* of the α phase levels off to a constant characteristic of heavy FL behavior, whereas that of the β phase continues to diverge (Fig. 1B). Thus, the fate of local moments found above *T** is different in these locally isostructural systems: Yb spins are fully screened in the α phase but may well survive down to lower temperatures in the β phase and produce the quantum criticality. In both phases, strong correlation effects are manifest—for example, in the strongly enhanced ^{2}, which is two orders of magnitude larger than the band calculation estimate (~ 6 mJ/mol K^{2}) (*18*).

These signatures indicate that both phases are governed by two distinct energy scales: a high-energy valence fluctuation scale *T*_{0} ~ 200 K and a low-energy scale *T** ~ 8 K, characterizing the emergence of Kondo lattice physics. A possible origin of this behavior is the presence of ferromagnetic (FM) interactions between Yb moments, manifested by the large Wilson ratio *R*_{W} between χ and *C*_{M}/*T*, observed in both α- and β-phases (minimum estimate *R _{W}* ≵7) (

*9*) and further corroborated by the observation of an electron spin resonance signal (

*19*), which is generally only seen in the presence of FM correlations (

*20*). Such FM interactions are known to give rise to Kondo resonance narrowing (

*21*) in

*d*-electron systems in which Hund’s coupling causes a marked reduction in the Kondo temperature (

*21*,

*22*).

In the present case, the role of Hund’s coupling is played by FM intersite RKKY interactions, probably along the short Yb-Yb bonds that form chains along the *c* axis. The moments of a few *n* neighboring Yb ions may thus become aligned, forming a fluctuating “block” spin *S* = *n J*. The observed valence Yb^{+2.75} could then be understood in terms of Yb^{3+} >⇌ Yb^{2+} fluctuations because at any one time, approximately 1/4 of the Yb atoms along the chains are in a singlet Yb^{2+} configuration, forming the ferromagnetic blocks of approximately *n* ~ 3 spins. The effect of these block spins is to exponentially suppress the characteristic spin fluctuation scale (*21*), resulting in localized-moment behavior. The absence of long-range magnetic order in the α or β phase points to the presence of competing magnetic interactions. Indeed, the Weiss temperature Θ_{W} ~ −110 K (Fig. 2A), which is characteristic of an AFM, indicates the importance of magnetic frustration. The competing interplay of FM interactions and valence fluctuations thus leads to Kondo lattice–like behavior in a mixed-valent material, setting the stage for quantum criticality to emerge at lower temperatures in β-YbAlB_{4}.

A prominent feature of the quantum criticality in β-YbAlB_{4} is the divergence of the magnetic susceptibility χ as *T* → 0. By examining the field evolution of magnetization *M* = −∂*F*/∂*B* as a function of both *T* and *B* (*12*), we can accurately probe the free-energy *F* near quantum criticality. Figure 2A shows the *T* dependence of χ(*B*) = *M*/*B* for different values of *B* ‖ *c*. Spanning four orders of magnitude in *T* and *B*, the data show a systematic evolution from a non-FL (NFL) metal with divergent susceptibility at zero field (χ ~ *T*^{−1/2}) to a FL with finite χ in a field *g*μ_{B}*B* ‰ *k*_{B}*T*, where *k*_{B} is the Boltzmann constant.

Intriguingly, the evolution of *M*/*B* found in the region *T* ≲ 3 K and *B* ≲ 2 T (Fig. 2A, inset) can be collapsed onto a single scaling function of the ratio *T*/*B*:*k*_{B}*T* /*g*μ_{B}*B* ~ 1, marking a cross-over between the FL and NFL regions, showing that *k*_{B}*T*_{F} ~ *g*μ_{B}*B* plays the role of a field-induced Fermi energy, as shown in the Fig. 2B inset. Integrating both parts of Eq. 1, one obtains the following scaling law for the free energy (*12*):*f* is a scaling function of the ratio *T*/*B* with the limiting behavior *f*(*x*) ∝ *x*^{3/2} in the NFL regime (*x* >> 1) and *f*(*x*) ∝ const + *x*^{2} in the FL phase (*x* << 1). Indeed, the observed scaling of *dM*/*dT* in Eq. 1 is best fitted with *12*), resulting in a particularly simple form of the free energy:*g*μ_{B} = 1.94 μ_{B} and the energy scale *12*). This means that the free energy depends only on the distance from the origin in the (*T*,*B*) phase diagram, similarly to the *T*/*B* scaling established in the Tomonaga-Luttinger liquids in one-dimensional metals (*23*, *24*). Equation 3 implies that the effective mass of the quasi-particles diverges as *m** ~ *B*^{−1/2} at the QCP (*12*). This divergence in a three-dimensional material, together with the *T*/*B* scaling, cannot be accounted for by the standard theory based on spin-density-wave fluctuations (*25*, *26*). Instead, it indicates a breakdown of the FL driven by unconventional quantum criticality.

The *T*/*B* scaling suggests that the critical field *B*_{c} of the quantum phase transition is actually zero. A finite *B*_{c} would require that the argument of the scaling functions *f*(*x*) and ϕ(*x*) is the ratio *x* = *T*/|*B*−*B*_{c}|, as seen for instance in YbRh_{2}Si_{2} (*16*). To place a bound on *B*_{c}, we substituted this form for *x* into Eq. 1, seeking the value of *B*_{c} that would best fit the experimental data. The Pearson’s correlation coefficient *R* obtained for this fit (Fig. 2B, inset), indicates that *B*_{c} is optimal at −0.1 ± 0.1 mT. The uncertainty is only a few times larger than Earth’s magnetic field (~0.05 mT). More importantly, it is two orders of magnitude smaller than μ_{0}*H*_{c2} = 30 mT and six orders of magnitude smaller than valence fluctuation scale *T*_{0} ~ 200 K. Thus, β-YbAlB_{4} provides an example of essentially zero-field quantum criticality.

Further evidence for zero-field quantum criticality is obtained from an analysis of the magnetocaloric ratio, *C* is the total specific heat (*12*). Our results show a clear divergence of Γ_{H}/*B* as *T* → 0 in the NFL regime, which is a strong indicator of quantum criticality (*27*). From the NFL regime, we can extract the critical field |*B*_{c}| < 0.2 mT, which is consistent with the estimate of *B*_{c} obtained from the scaling behavior of *M* in Eq. 1 (fig. S2).

The simple *T*/*B* scaling in the thermodynamics enables us to characterize the QC excitations of β-YbAlB_{4}. In particular, the collapse of all magnetization data in terms of the dimensionless ratio *r* = *k*_{B}*T*/(*g*μ_{B}*B*) between the Boltzmann energy *k*_{B}*T* and the Zeeman energy *g*μ_{B}*B* indicates an absence of scale in the zero-field normal state. Furthermore, the appearance of a field-induced Fermi energy—linear over more than three decades in *B*—shows that the underlying critical modes are magnetic in character.

Using the Heisenberg energy-time uncertainty principle (Δ*t*Δ*E* ≳ *ћ*, where *ћ* is Planck’s constant *h* divided by 2π), we can reinterpret the *T*/*B* scaling in the time domain, visualizing the field-induced FL as a kind of “quantum soda” of bubbles of quantum critical matter of finite duration τ_{Q} = *ћ*/*g*μ_{B}*B*, immersed in a FL. At finite temperatures, thermodynamics averages the physics over a thermal time scale τ* _{T}* =

*ћ*/

*k*

_{B}

*T*; thus, the quantity

*r*= τ

_{Q}/ τ

*~*

_{T}*T*/

*B*in the scaling is the ratio of the correlation time τ

_{Q}to the thermal time-scale τ

*. At low temperatures*

_{T}*r*<< 1 (τ

*>> τ*

_{T}*), thermodynamics probes the FL exterior of the bubbles, but when*

_{Q}*r*>> 1 and τ

*<< τ*

_{T}_{Q}, it reflects the QC interior of the bubbles. This accounts for the cross-over between FL and QC behaviors at

*r*~ 1. Moreover,

*T*/

*B*scaling over a wide range

*r*~ 10

^{−1}to ~10

^{3}indicates that the quantum fluctuations in the ground state are self-similar down to 1/1000th of the correlation time τ

_{Q}.

The observation of zero-field quantum criticality in valence-fluctuating β-YbAlB_{4} cannot naturally be interpreted as a conventional QCP, which would require a fortuitous combination of structure and chemistry to fine-tune the critical field *B*_{c} to within 0.2 mT of zero. A more natural interpretation of the results is that β-YbAlB_{4} forms a quantum critical phase that is driven into a FL state by an infinitesimal magnetic field. The *T*/*B* scaling requires that the critical modes are Zeeman-split by a field, and as such, various scenarios—such as critical Fermi surfaces (*28*) or local quantum criticality with *E*/*T* scaling (*6*)—may be possible contenders for the explanation, provided they can be stabilized as a phase. Established theoretical examples of a critical phase with *T*/*B* scaling include the Tomonaga-Luttinger liquid in half-integer spin chains and the one-dimensional Heisenberg ferromagnet (*23*, *24*). Experimentally, the *d*-electron metal MnSi is a candidate for a quantum critical phase, with anomalous transport exponents observed over a range of applied pressure (*29*). Although present work provides a strong indication for existence of such a phase in β-YbAlB_{4}, future studies—in particular, under pressure—are necessary in order to establish it definitively.

## Supporting Online Material

www.sciencemag.org/cgi/content/full/331/6015/316/DC1

Materials and Methods

SOM Text

Figs. S1 to S4

References

## References and Notes

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Materials and methods, as well as the details of the scaling analysis and the difference from the previously reported χ(
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- We thank H. Ishimoto, D. E. MacLaughlin, K. Miyake, T. Senthil, Q. Si, T. Tomita, K. Ueda, and S. Watanabe for useful discussions. This work is partially supported by grants-in-aid (21684019) from the Japan Society for the Promotion of Science; by grants-in-aid for Scientific Research on Priority Areas (19052003) and on Innovative Areas (20102007 and 21102507) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan; by Global Centers of Excellence Program “the Physical Sciences Frontier,” MEXT, Japan; by Toray Science and Technology Grant; and by a grant from NSF DMR-NSF-0907179 (P.C. and A.H.N.). P.C. and A.H.N. acknowledge the hospitality of the Aspen Physics Center.