## Abstract

Low-temperature phase transitions and the associated quantum critical points are a major field of research, but one in which experimental information about thermodynamics is sparse. Thermodynamic information is vital for the understanding of quantum many-body problems. We show that combining measurements of the magnetocaloric effect and specific heat allows a comprehensive study of the entropy of a system. We present a quantitative measurement of the entropic landscape of Sr_{3}Ru_{2}O_{7}, a quantum critical system in which magnetic field is used as a tuning parameter. This allows us to track the development of the entropy as the quantum critical point is approached and to study the thermodynamic consequences of the formation of a novel electronic liquid crystalline phase in its vicinity.

Quantum critical points (QCPs) are continuous phase transitions at zero temperature, where quantum mechanical zero point fluctuations play the role assumed by thermal fluctuations in the classical case (*1*). Their importance is not only academic because large portions of the low-temperature phase diagram can be affected by their presence, and the production of QCPs and the formation of new quantum phases in their vicinity has become a central theme of modern condensed matter physics. Prominent examples of the discoveries made by this route include various unconventional superconducting states (*2*–*4*) and the mysterious order seen in URu_{2}Si_{2} at high magnetic fields (*5*). The possibility that quantum critical physics is at the heart of the phase diagram of the high-temperature superconductors has also received considerable attention (*6*), as has the role of quantum criticality in producing non-Fermi liquid behavior in metals (*7*).

Associated with each discovery of novel quantum order is a phase diagram describing the relevant phases and their interrelationships. Ideally, such systems should be investigated by using thermodynamic probes, but, although substantial progress has been made (*7*–*9*), comprehensive, quantitative phase diagram mapping has remained an elusive goal. Of all the information that can be deduced from thermodynamic measurements, the entropy, *S*, is fundamental in terms of understanding phase diagrams. On the approach to a QCP, a nonlinear dependence of *S* is expected both for isothermal trajectories varying the tuning parameter (*p*) and if the temperature is lowered at fixed *p*. In particular, *S*(*p*) is expected to show a pronounced peak at low temperatures, centered on the position of the QCP (*10*). If phases form on the approach to the QCP, the naïve expectation is that this peak in *S* would be cut off because of the formation of a state or states that reduce the entropy associated with the bare QCP.

Demonstrating the entropic behavior outlined above should be the prerequisite to classifying phase formation in the vicinity of quantum criticality. Here, we present a comprehensive experimental study of Sr_{3}Ru_{2}O_{7}, a system in which magnetic field is used as the tuning parameter for the quantum criticality (*11*, *12*).

**Our model system.** Sr_{3}Ru_{2}O_{7} is an oxide metal with the layered perovskite structure (*13*). At ambient conditions, it is a strongly enhanced paramagnet close to ferromagnetism, and at low temperatures it undergoes a metamagnetic transition in an applied magnetic field (*14*). In samples whose mean free path (m.f.p.) is ~300 Å, the critical end-point of this transition lies below 0.1 K for magnetic fields applied parallel to the crystallographic *c* axis, and a number of signatures of quantum criticality are seen (*11*, *12*). If higher purity crystals with m.f.p. > 3000 Å are grown, the phase diagram changes, and a new phase with the transport anisotropy characteristic of a nematic electronic fluid (*15*–*19*) appears to intervene in the approach to the QCP (*20*–*22*).

Electronic liquid crystalline phases are rare and subtle, and the above observations have generated significant theoretical attention (*23*–*27*). Experimental evidence for their existence before the observations on Sr_{3}Ru_{2}O_{7} came in ultra–high purity two-dimensional electron gases at very high magnetic fields (*16*, *17*). However, their electron density is extremely low, so they offer little prospect of yielding the kind of entropy mapping that is, in principle, accessible in Sr_{3}Ru_{2}O_{7}.

**Thermodynamics.** Further motivation for a full thermodynamic study comes because the boundaries of the putative nematic fluid phase in Sr_{3}Ru_{2}O_{7} are unusual. A combination of magnetic susceptibility, resistivity, magnetostriction, and magnetization measurements indicate that, at low temperatures, the phase is bounded by two lines of first-order phase transitions with opposite and concave curvatures, linked by a line of anomalies suggestive of second-order phase transitions (*21*). The path in parameter space of lines of first-order phase transitions between equilibrium phases gives information about the entropy. The slope is given by the magnetic Clausius-Clapeyron relation_{0} is the permeability constant, *H*_{c} the critical field, *T*_{c} the critical temperature, *S* the entropy, and *M* the magnetization; Eq. 1 therefore predicts that in this case the entropy within the bounded region is higher than that outside. A key assumption of using the Clausius-Clapeyron equations is that the transitions identified by the susceptibility analysis are indeed delineating the boundaries between phases in thermodynamic equilibrium. This assumption is widely used when drawing phase diagrams for systems in the vicinity of QCPs, but it warrants detailed investigation because the slow dynamics expected in such situations make nonequilibrium effects potentially significant (*12*, *28*, *29*).

In setting out to measure the field and temperature dependence of the entropy in Sr_{3}Ru_{2}O_{7}, our goals were to determine the overall entropic background to the phase formation to see whether or not it is consistent with an underlying QCP and to check that the entropic behavior on entering the putative nematic phase is consistent with a Clausius-Clapeyron analysis of the previously determined first-order phase boundaries. Additionally, in order to achieve a complete characterization of the phase diagram, we need to ascertain the nature of the lines of singularities linking the first-order transition lines, to establish whether a thermodynamic phase boundary must always be crossed to enter the putative nematic phase for *H*//*c*. Achieving these goals would give unique insight into the properties of a phase of “electronic matter” that is of considerable topical interest (*23*–*27*) and demonstrate that it is possible to perform a comprehensive study of a fundamental thermodynamic potential, the entropy, in quantum critical systems. It would also advance our understanding of the key issue of whether genuine thermodynamic equilibrium is even attainable in the vicinity of a QCP.

**Measurement apparatus.** Instead of relying primarily on measurements of the specific heat, we constructed apparatus to study the magnetocaloric effect and the specific heat simultaneously, using the latter both in its own right and as a consistency check on the results obtained by the former. The magnetocaloric effect has a crucial advantage over the specific heat in that it gives instant qualitative information on the sign of the changes of the field derivative of the entropy (*30*). It is thus well suited for determining the sign of entropy jumps across first-order phase boundaries and will display a characteristic sign change in the vicinity of a QCP (*10*). The deduction of the entropy from the specific heat involves integration over a range of temperatures below the base temperature of measurement. In a Fermi liquid, this can be done without dangerous assumptions, but near a QCP, where the effective Fermi temperature is strongly depressed, assumptions are required that might lead to erroneous results. Using the magnetocaloric effect, the entropy can be calculated via an integration as a function of magnetic field, starting from a field a long way from a QCP where the system is a Fermi liquid whose entropy can be readily deduced by other means. Details of the apparatus along with a description of the data analysis procedures are given in (*31*).

**Measurements.** The magnetization of a high-quality single crystal of Sr_{3}Ru_{2}O_{7} is shown in Fig. 1A. As previously reported (*20*, *21*), three nonlinear changes of magnetic moment are seen, centered on 7.5, 7.8, and 8.1 T. Susceptibility and magnetization measurements give evidence for dissipation and hysteresis at very low temperatures for the two higher field features, leading to their identification as first-order phase transitions. The broader feature at 7.5 T shows no such dissipation and is thought to be a metamagnetic crossover.

Some sample traces of raw magnetocaloric effect data taken with the apparatus set to a highly nonadiabatic mode of operation (*31*) are shown in Fig. 1B. In this mode, the step changes in temperature that would occur at first-order phase transitions in adiabatic conditions are replaced by sudden rises or falls followed by decay back to ambient temperature in a time determined by the characteristics of the heat link to the bath. The crossover is again seen to be significantly broader than the sharp features associated with the putative phase transitions. Importantly, the signs of the temperature changes at 7.8 and 8.1 T immediately determine the signs of the entropy changes across those transitions and confirm that the entropy rises at 7.8 T before falling again at 8.1 T. This qualitative information can be converted to quantitative data with use of equation S1.1 in (*31*). Applying this procedure to field sweeps in the temperature range from 0.2 K to 1.5 K then allows the construction of Fig. 2, in which, in order to show the changes of entropy as a function of field, we plot the behavior of ∂(*S*/*T*)/∂*H* in the *H*-*T* plane, superposed with previous data (*21*) displaying the boundaries of the nematic “phase.”

Figure 2 contains several noteworthy features. Although ∂(*S*/*T*)/∂*H* goes through zero at low temperatures near 7.5 T, indicating a maximum in the entropy there, the low temperature features at 7.8 T and 8.1 T correspond to entropy jumps, consistent with the presence of first-order phase transitions. Each disperses with temperature in a manner consistent with that determined previously using other probes and depicted in Fig. 2 by the dashed white lines. The field derivative of the entropy also changes abruptly at higher temperatures (light blue to dark blue on the plot) at a locus of fields and temperatures identified in the previous work (*21*).

The sharp changes in entropy at low temperatures are qualitatively consistent with first-order phase transitions as previously postulated but do not constitute a proof. We have therefore performed a quantitative comparison of the entropy change at the ~7.8 T transition by using two methods: the direct magnetocaloric effect measurements and application of the Clausius-Clapeyron relationship (Eq. 1). We used the maximum in ∂(*S*/*T*)/∂*H* to give a precise measurement of {*H*_{c}, *T*_{c}} and magnetization data such as that in Fig. 1A to define Δ*M* at a series of temperatures (Fig. 3A). Not only are the two measurements of Δ*S* in close quantitative agreement, but the temperature dependence allows an estimate of the temperature of the critical point terminating the line of first-order transitions, where Δ*S* goes to zero, to be 0.8 ± 0.1 K, in excellent agreement with previous susceptibility studies (*21*). A qualitatively similar conclusion can be drawn for the transition at ~8.1 T, whose critical temperature is about 0.5 K (*21*). The detailed parameters are less important than the main conclusion, which is that these thermodynamic measurements provide very strong evidence that the two metamagnetic transitions are first-order transitions between equilibrium phases.

Crucial to understanding the boundaries of the phase is the nature of the feature at its “roof,” in particular whether it is entered by a phase transition on cooling. If it is a phase transition, previous experiments suggested it would be second order, so the specific heat is a better thermodynamic probe than the magnetocaloric effect to investigate it. In Fig. 3B we show the electronic part of *C*/*T* as a function of *T* in a fixed magnetic field of 7.9 T (black curve). A jump is seen, centered on 1.2 K, with a width of about 0.075 K. Measurements at nearby fields confirm that its temperature dependence is consistent with the line showing the roof in Fig. 2. The blue line shows an extrapolation of the logarithmic divergence of the high temperature *C*/*T* that was first reported in (*14*) and reconfirmed on the current higher quality crystals. Furthermore in red we show a linear fit to the data in the low-temperature phase. Intriguingly, the data do not become temperature-independent below *T*_{c} down to our lowest temperature of measurement but are consistent with the existence of a significant negative *T*^{2} correction to the specific heat. The inset in Fig. 3B shows the entropy difference between the low- and high-temperature phases based on the respective fits to the specific heat data. The fact that the entropy difference is zero at the transition temperature of 1.2 K within experimental resolution suggests that the unusual temperature dependence extends to below our lowest experimental temperature of 0.2 K. Outside the phase, the observed behavior is qualitatively different, with *C*/*T* showing the normal *T* independence below 1 K; the origin of the anomalous correction to Fermi liquid theory within the phase is not yet established.

**Discussion.** The data shown in Figs. 1 to 3 establish the low-temperature nematic region of the phase diagram of Sr_{3}Ru_{2}O_{7} as a distinct equilibrium phase bounded by first- and second-order phase transitions. Given that the phase was discovered by purifying a system in which there was good evidence for quantum criticality, we looked for signs of that criticality in a more “coarse-grained” picture of the phase diagram. We acquired magnetocaloric data over a much wider range of magnetic field and temperature, namely 4 T < μ_{o}*H* < 15 T and 0.15 K < *T* < 1.5 K. At 4 T the system is deep in the Fermi liquid state for all temperatures in this range, with a temperature-independent *C*/*T* that could be integrated to fix the entropy. For all other fields *S*(μ_{o}*H*) – *S*(4 T) was obtained by field integration of the magnetocaloric signal. The full resultant entropy landscape is shown in fig. S2 (*31*); in Fig. 4A we focus for sake of clarity on the critical region and show *S*/*T* for 5 T < μ_{o}*H* < 11 T. At low fields, the Fermi liquid prediction of temperature-independent *S*/*T* is fulfilled. As the field increases toward the critical region, *S*/*T* increases, and at 1.5 K is seen to go through a peak centered around the critical field, whose existence is a central expectation of the approach to quantum criticality (*10*). As the temperature is lowered, the peaking in *S*/*T* becomes progressively sharper, but at low temperatures the steep rise on the low field side is cut off.

The low-temperature behavior of *S*/*T* is shown in more detail by the blue line in Fig. 4B, in comparison with that of the change of the specific heat over temperature, Δ*C*/*T*, shown in black. Both quantities are seen to show similar behavior as a function of magnetic field on the approach to the region in which the phase formation takes place. On the low field side, the red line is a power-law fit to Δ*C*/*T* below 7.4 T at 0.25 K of the form *C*/*T* = *A*[(*H-H*_{c})/*H*_{c}]^{–α}*+B*, with *A*, *B*, *H*_{c}, and α as free parameters. This yields an exponent α = 1 ± 0.1, and it is evident from Fig. 4B that *S*/*T* also follows this functional form within experimental error. Detailed fitting to the high field data is complicated because of a further weak metamagnetic transition at ~12.5 T (*32*) and because the data contain long-period quantum oscillations in these very high purity samples. However, this same function, with a different value *B* but similar value of *A* still gives a good match to the data between 8.1 T and 10 T, as can be seen in the figure. As explained in more detail in (*31*), these observations are consistent with the prediction of a divergence of the magnetic Grüneisen parameter in the vicinity of a field tuned quantum critical point (*33*), which can be derived on quite general grounds (*10*, *34*).

A qualitative interpretation strongly suggested by the data in Fig. 4, A and B, is that the formation of the novel nematic phase in Sr_{3}Ru_{2}O_{7} takes place in a region of the phase diagram where *S*/*T* would diverge at the QCP if nothing else occurred. Detailed examination shows that the incipient divergence at low fields is first cut off when the system undergoes its metamagnetic crossover at ~7.5 T. The entropy then jumps up as the nematic phase is entered at 7.8 T, before jumping down again as it is exited at 8.1 T. However, the data suggest that all of this is taking place against the background of a large saving of the entropy that would have been developed had the steep rise continued. This qualitative behavior strongly supports the postulate that the phase formation is occurring on the approach to quantum criticality. In Fig. 4C we present a contour plot of isoentropes in the (*H*, *T*) plane that reinforces this picture. Near the critical region they are forced down in temperature, with this trend cut off at sufficiently low temperatures by the metamagnetic transitions.

To date, all theories of the nematic physics of Sr_{3}Ru_{2}O_{7} have incorporated mean-field approximations, and the assumption that the metamagnetism is driven by the proximity of the Fermi level to van Hove singularities (vHs) in bands based either on the quasi–two-dimensional (2D) Ru *d*_{xy} orbitals or the quasi-1D Ru *d*_{xz,yz} orbitals. Although that theoretical work has successfully captured some of the relevant physics, the measurements presented here highlight the need to go beyond mean field. First, a divergence of *C*/*T* as [(*H* – *H*_{c})/*H*_{c}]^{−1} on both low and high-field sides of the metamagnetic region cannot generically be accounted for by either form of vHs (*31*). Second, existing mean-field approaches give no simple way to understand the entropic behavior of the nematic phase (*26*). We have shown that the isothermal entropy jumps as a function of field within the phase at low *T* and that *C*/*T* contains an anomalous correction when tracked in the phase at constant field. Although it is possible that these observations could be associated with domain formation, the challenge would be to explain a term as large as 20 mJ/Ru mol K^{2} in *S*/*T* (Fig. 3A).

**Concluding remarks.** We have determined the complete entropic landscape for a system in which a novel quantum phase forms in the vicinity of quantum criticality: the putative electronic nematic phase. The simplest qualitative picture suggested by our observations is that the phase formation is driven by the need for the system to avoid an entropic singularity near a QCP. Specific heat data inside the new phase show anomalous corrections to Fermi-liquid behavior. As well as providing new insights into the specific physics of the electronic nematic state in Sr_{3}Ru_{2}O_{7}, we believe that the work opens the way to a new generation of experiments that will advance the overall understanding of quantum criticality, one of the most interesting issues in modern condensed matter physics. Our magnetocaloric technique can be applied in a straightforward manner to any field-tuned quantum critical system, such as URhGe (*4*), YbRh_{2}Si_{2} (*35*), or the high-field phase diagram of URu_{2}Si_{2} (*5*). We believe that it may also be much more widely applicable, because differential magneto-caloric measurement is, in principle, possible in any system for which there is a coupling between magnetic and charge or structural degrees of freedom.

## Supporting Online Material

www.sciencemag.org/cgi/content/full/1176627/DC1

Materials and Methods

SOM Text

Figs. S1 to S5

References

## References and Notes

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Zhu
*et al*. (*10*) stress the importance of a sign change of the magnetocaloric signal as a characteristic of the correspondence of proximity to a QCP. Our observation of this sign change is manifested by the correspondence in Fig. 4 of*S*/*T*deduced from the magnetocaloric signal and*C*/*T*from direct measurement. - ↵
- We thank R. A. Borzi, J. C. Davis, Z. Fisk, E. Fradkin, A. G. Green, C. A. Hooley, E.-U. Kim, S. A. Kivelson, and Q. Si for insightful discussions; A. S. Gibbs for providing Sr
_{2}RuO_{4}samples used for characterizing the experimental setup; and the SUPA Graduate School, the UK Engineering and Physical Sciences Research Council, and the Royal Society for financial support.