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Nature of an intermediate non-Fermi liquid state in Ge-substituted YbRh$_{2}$Si$_{2}$: Fermionized skyrmions, Lifshitz transition, Skyrmion liquid, and Gruneisen ratio

We propose a skyrmion liquid state for the non-Fermi liquid (NFL) phase in Ge-substituted YbRh$_{2}$Si$_{2}$, where skyrmions form their Fermi surface, argued to result from the strongly coupled nature between skyrmions and itinerant electrons. The fermionized skyrmion theory identifies the antiferromagnetic (AF) transition with the Lifshitz transition, where the quantum critical point (QCP) is characterized by the dynamical critical exponent $z = 2$. Nonlocal interactions between skyrmions allow a critical line above the AF QCP, which originates from the Kondo-coupling effect with itinerant electrons. This critical line is described by the skyrmion liquid state, which results in Landau damping for spin fluctuations, thus characterized by $z = 3$. As a result, the Gruneisen ratio is predicted to change from $\sim T^{-1}$ at the AF QCP to $\sim T^{-2/3}$ in the NFL phase.