Fig. 1: Key thermodynamic response of the trimer lattice Ba_₄Nb_1-xRu_3+xO_₁₂.

a Schematic phase diagram as a function of Nb content x, illustrating the heavy spinon Fermi surface underpinning both the HFSM and QSL phases. The extraordinarily large Sommerfeld coefficient γ (blue) and exchange energy θ_CW (red, right scale) are shown¹. Inset: The crystal structure of the trimer lattice. Note that the heavy-fermion strange metal (HFSM, 1-x = 0.81 or Nb_0.81) and the insulating quantum spin liquid (QSL, 1-x = 1.16 or Nb_1.16) are adjacent to each other. b, c Heat capacity C(T) for the HFSM and QSL, respectively, at 0 T and 14 T for both field orientations and for 50 mK ≤ T ≤ 1 K. The black arrows mark the onset of the upturn in C, T_s. d Temperature-dependent entropy difference, ΔS(T), between 14 T and 0 T for the QSL, calculated as ΔS(T) = \({\int }_{T}^{0.46K}\left[C\left(14T\right)-C\left(0\right)\right]/{T}^{{\prime} }d{T}^{{{\prime} }}\), where T is the argument of ΔS, i.e., the lower bound of the integration that varies, and \({T}^{{\prime} }\) is the integration variable. The upper bound of the integration is set at 0.46 K. e Comparison with 9R-BaRuO₃ and Nb₂O₅ shows no upturn in C, confirming the intrinsic nature of the effect in Ba₄Nb_1-xRu_3+xO₁₂. Note that the anomaly in BaRuO₃ at 0.3 K may signal an emergent state.