Thermodynamics of phase formation in the quantum critical metal Sr3Ru2O7

Abstract

The behavior of matter near zero temperature continuous phase transitions, or "quantum critical points" is a central topic of study in condensed matter physics. In fermionic systems, fundamental questions remain unanswered: the nature of the quantum critical regime is unclear because of the apparent breakdown of the concept of the quasiparticle, a cornerstone of existing theories of strongly interacting metals. Even less is known experimentally about the formation of ordered phases from such a quantum critical "soup." Here, we report a study of the specific heat across the phase diagram of the model system Sr(3)Ru(2)O(7), which features an anomalous phase whose transport properties are consistent with those of an electronic nematic. We show that this phase, which exists at low temperatures in a narrow range of magnetic fields, forms directly from a quantum critical state, and contains more entropy than mean-field calculations predict. Our results suggest that this extra entropy is due to remnant degrees of freedom from the highly entropic state above T(c). The associated quantum critical point, which is "concealed" by the nematic phase, separates two Fermi liquids, neither of which has an identifiable spontaneously broken symmetry, but which likely differ in the topology of their Fermi surfaces.

Fig. 1.

A schematic phase diagram for Sr₃Ru₂O₇ with magnetic field applied parallel to the crystallographic c axis, based on a combination of transport (7, 9, 10, 11, 13), thermal expansion (14), nuclear magnetic resonance (15) and quantum oscillations (16). Below a crossover temperature T^∗ sketched by the dotted white line, Fermi liquids are seen at both low and high magnetic fields (blue shading). T^∗, which is defined by thermodynamic measurements (, Fig. 3C below) is depressed towards T = 0 at a critical field H_c of approximately 7.9 T, accompanied by the appearance of non-Fermi liquid temperature dependence of transport and thermodynamic properties (red shading) and features in magnetization (7, 13). In zero applied field, T^∗ ∼ 10 K, and the material has a substantial specific heat coefficient of 110 mJ/molRuK², corresponding to large quasiparticle band renormalizations of a factor 10–30 compared with the predictions of LDA band calculations (17, 18). The solid white lines sketch the field dependence of the electronic specific heat at 250 mK. The specific heat coefficient rises sharply as H_c is approached from both the low and high-field sides (19). All of these observations are consistent with the existence of a QCP at 7.9 T, but in the highest purity samples, with mean-free paths of several thousand Å, the approach to the QCP is hidden by the formation of a new phase, which is associated with the onset of anisotropic transport (10, 11). The phase is indicated by the green shaded region, entered by first order phase transitions at low temperatures (solid white boundary) and a continuous transition at high temperatures (dashed white boundary). The temperature scale of this ordered phase has been multiplied by a factor of two for visual clarity. Approaching the quantum critical region at low temperatures from the low-field side, the effective mass determined from the specific heat has an apparent divergence, m^∗ ∼ [(H - H_c)/H_c]^-1 as a function of increasing H, which is then cut-off near where the nematic phase occurs. The formation of the nematic phase is then accompanied by a small jump in entropy, followed by a drop on exit at the high field side (19).

Fig. 3.

(A): Total specific heat divided by temperature for Sr₃Ru₂O₇ plotted against the square of temperature between 250 mK and 40 K in zero field (blue) and at 7.9 T (black). The red line is a fit to the field-independent data above 20 K. It is the sum of a noncritical Fermi liquid component and a phonon contribution (16). (B): The temperature dependence of the electronic specific heat (after subtraction of the phonon background) as the field is tuned through the quantum critical region. The hump seen in zero field sharpens and is depressed to progressively lower temperatures as the critical field is approached before reappearing on the high field side of the transition. The temperature at which the maximum occurs at each measured field is shown in box (C) (blue dots). At 7.9 T no maximum is seen and based on previous data from more disordered samples it is assumed to occur at T = 0 (red dot). In box (D) we show the evolution of the entropy at each of the fields for which data are displayed in (B).

Fig. 2.

Resistivity (blue) and electronic specific heat (black dots) data for Sr₃Ru₂O₇ on cooling at the critical field of 7.9 T. The resistivity was measured between 100 mK and 18 K in a continuous run using an adiabatic demagnetization refrigerator, and the results of both up and down sweeps are shown. The dotted gray line indicates the critical temperature of the nematic phase, and the red curve is a fit of the form C_el = T ln T to the data between 1.4 K and 18 K, extrapolated to 100 mK.

Fig. 4.

(A): Low temperature electronic specific heat divided by temperature for four fields above and below the nematic phase (red) and four within it (black). The transition into the phase is marked by the step-like feature centered on approximately 1.15 K. (B): The transition region on an expanded scale, showing the systematic trend for the transition to be depressed to lower temperatures as the field is increased, in agreement with the known phase diagram (10). (C): The entropy saving as a function of temperature at 7.9 T, expressed as a fraction of that at T_c (black) compared with the results of two example model calculations: a phase opening a full gap (red) and a gapless Pomeranchuk distortion (blue). Full details of the models and the comparison can be found in the SI Text: SI4.