Multiple Metamagnetic Quantum Criticality in ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$

Bilayer strontium ruthenate ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$ displays pronounced non-Fermi liquid behavior at magnetic fields around 8 T, applied perpendicular to the ruthenate planes, which previously has been associated with an itinerant metamagnetic quantum critical end point (QCEP). We focus on the magnetic Grüneisen parameter ${\mathrm{\Gamma }}_H$, which is the most direct probe to characterize field-induced quantum criticality. We confirm quantum critical scaling due to a putative two-dimensional QCEP near 7.845(5) T, which is masked by two ordered phases $A$ and $B$, identified previously by neutron scattering. In addition, we find evidence for a QCEP at 7.53(2) T and determine the quantum critical regimes of both instabilities and the effect of their superposition.

Figure 1

Phase diagram of ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$ for $H\parallel c$ with color coding of the magnetic Grüneisen parameter ${\mathrm{\Gamma }}_H$. Solid green symbols mark positions of sharp anomalies in ${\mathrm{\Gamma }}_H(T,H)$, related to metamagnetism [10] and the spin-density-wave phases $A$ and $B$ [12]. Open green symbols indicate positions of maxima in the field dependence of specific heat. The dotted black line marks ${\mathrm{\Gamma }}_H=0$, corresponding to a local entropy maximum. The stars on the $x$ axis show the positions of the two metamagnetic quantum critical end points QCEP1 and QCEP2. Gray, white, and black solid lines bound different regimes. Labels “QC1” and “QC2” denote regions where quantum critical scaling with respect to QCEP1 and QCEP2 is observed. Within the $\mathrm{QC}1+\mathrm{QC}2$ regime, scaling fails due to the superposition of criticality from both instabilities, see Supplemental Material [14]. Anomalies in isothermal ${\mathrm{\Gamma }}_H(H)$ scans are indicated as the yet unidentified regime “$C$”.

Figure 2

Specific heat divided by temperature of ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$ as a function of magnetic field applied parallel to $c$ axis at different constant temperatures. The solid blue line indicates $C/T=\alpha /[{\mu }_0(H_c-H){]}^{1/3}+\gamma$ with $\alpha =(0.073\pm 0.002)\mathrm{J}/\mathrm{Ru}\text{-\hspace{0.17em}}\mathrm{mol}{\mathrm{K}}^2{\mathrm{T}}^{1/3}$, ${\mu }_0{H}_c=7.57(4)\mathrm{T}$ and $\gamma =0.058(1)\mathrm{J}/\mathrm{Ru}\text{−}\mathrm{mol}{\mathrm{K}}^2$, in accordance with a two-dimensional metamagnetic QCEP [3, 24].

Figure 3

Magnetic field dependence of the magnetic Grüneisen parameter ${\mathrm{\Gamma }}_H$ at 0.2 K of ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$. The field is applied parallel to the $c$ axis. Arrows at ${\mu }_0{H}_{M1}=7.5\mathrm{T}$, ${\mu }_0{H}_{M2}=7.85\mathrm{T}$, and ${\mu }_0{H}_{M3}=8.0\mathrm{T}$ indicate metamagnetic anomalies. The black arrow at 8.3 T marks the anomaly related to the SDW-$B$ phase [12, 15]. Green arrows indicate anomalies which correspond to open circles in Fig. 1, enclosing an anomalous regime “$C$”. The inset shows a plot of $1/{\mathrm{\Gamma }}_H$ vs ${\mu }_{0}H$. The solid red line represents a linear fit, $1/{\mathrm{\Gamma }}_H=-{\mu }_0(H-H_c)/G_r$ with $G_r=-0.17(1)$ and ${\mu }_0{H}_c=7.51(2)\mathrm{T}$.

Figure 4

Magnetic Grüneisen parameter ${\mathrm{\Gamma }}_H$ of ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$ as a function of temperature at different magnetic fields, applied parallel to the $c$ axis. The red and green arrows indicate the transitions to the spin-density-wave phases $A$ and $B$ [12].

Figure 5

Metamagnetic quantum critical scaling of the magnetic Grüneisen parameter in ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$. The $y$ axis displays ${\mathrm{\Gamma }}_{H}h$, while the $x$ axis shows $T^{8/3}/h^2$, with $h={\mu }_0(H-H_{c2})$ and the critical field ${\mu }_0{H}_{c2}=7.845\mathrm{T}$ [11]. Panels (a) and (b) display regimes below and above the critical field. Red and blue dotted lines indicate predicted asymptotic quantum critical and FL behavior for a $d=2$ metamagnetic QCEP [24]. In panel (a), data on the left of the black arrow have been excluded (for failed scaling, see Supplemental Material [14]).