Critical behavior of the specific heat in Ti-Si amorphous alloys at the metal-insulator transition

In this paper, we report the measurements of specific heat of an amorphous Ti9.5Si90. 5 alloy located very close to the critical point of the metal-insulator transition. In the presence of a magnetic field, the specific heat is dominated by the Schottky anomaly caused by magnetic moments associated with the dangling bonds in the matrix of amorphous Si. Subtraction of this contribution exposes the behavior of the electronic specific heat coefficient γγ. The coefficient is temperature-independent above 2 K and is, in order of magnitude, close to the value expected in the absence of electron-electron interactions. In the temperature range 0.4-1.5 K, the coefficient γγ shows an anomalous downturn, which can be approximated by the dependence γ(TT) = γ0 ln(TT/TT0), with TT0 ≈ 0.2 K . In a companion paper, we found that the Hall coefficient in Ti-Si alloys is affected by the electron-electron interaction up to much higher temperature of 150 K and also varies critically across the metal-insulator transition. We compare our results with theoretical predictions for three models, which can potentially explain the anomalous behavior of the specific heat: generalized non-linear σσ model, Coulomb glass, and many-body localization. Introduction. Systems with strong disorder and interactions present one of the grand challenges in condensed matter physics [1]. The study of these systems has a long history marked by many still unresolved problems. In particular, unlike the case of several clean systems [2,3,4], the connection between the long-range behavior that determines a system response near a metal-insulator transition (MIT) and the corresponding microscopic physics has not been established. For example, it is not clear why the correlation length exponent, νν , and dynamical exponent, zz, are different in crystalline silicon doped with phosphorous (Si:P, νν = 1, zz = 3) [5] and with boron (Si:B, νν = 1.6 , zz = 2) [6]. Nor is it known why the experimental exponents for Si:B coincide with values predicted for a non-interacting disordered 3d system, in clear conflict with the tunneling experiments, which indicate the emergence of the Coulomb gap at the critical point of MIT [7]. For the insulating state, there is a long-standing question of what the nature of the current carriers is. Experimentally, the large body of the data on conductivity can be explained by Efros-Shklovskii variable-range hopping (VRH), based on the picture that current is carried by single-particle hops in the presence of the Coulomb gap. Mott and Pollak [8], however, argued that in the presence of interactions, a jump of an individual electron has to lead to the displacement of neighboring elections, so the true carriers are many-electron entities called dressed polarons. Extending this reasoning to thermodynamic quantities, one can see that the relaxation of the whole system requires a complicated re-arrangement of a gigantic number of elections, resulting in a dynamically frozen state known as a Coulomb glass. Contrary to these expectations, several measurements of specific heat in doped semiconductors and amorphous alloys revealed a simple smooth variation of the electron coefficient γ across MIT [9,10,11,12]. A modern approach to the problem of interacting particles in a strongly disordered potential takes a broader perspective by also considering systems in their excited states and completely isolated from their environment. There is strong evidence in these systems for the existence of a new, dynamical state of matter caused by many-body localization (MBL) [13,14], which does not thermalize and retains the memory of its initial state. It was first theoretically discovered in zero-dimensional systems [15] and one-dimensional systems with local interactions [16,17]. Recent theoretical works [18,19] and experiments on cold atoms