Title Renormalization Group Analysis of Nonequilibrium PhaseTransitions in Driven Disordered Systems( Digest_要約 )
Author(s) Haga, Taiki
Issue Date 2018-03-26
Right 学位規則第9条第2項により要約公開; 許諾条件により本文は2020-08-01に公開; 許諾条件により要旨は2018-04-01に 公開
Type Thesis or Dissertation
Summary of thesis: Renormalization Group Analysis of
Nonequilibrium Phase Transitions in Driven Disordered Systems
In the thesis, I report a renormalization group analysis of phase transitions in disordered systems driven by an external force. The effect of quenched disorder on the large-scale structure of interacting systems remains an unsettled issue. How the spatially inhomogeneous disorder affects the properties of phase transitions and critical phenomena has been extensively discussed in statistical mechanics. A more challenging problem concerning the effects of disorder is to understand the critical behavior of disordered systems driven out of equilibrium in the presence of an external force. In this thesis, I developed renormalization group formalism for driven disordered systems and the critical behavior of a simple model was investigated.
The peculiarity of disordered systems comes from the fact that the competition between disorder and interaction leads to multiple meta-stable states, which are local minima of the mesoscopic free energy. One of the most remarkable phenomena associated with meta-stable states is the breakdown of the so-called “dimensional reduction property” in random field spin models. Standard perturbation theory predicts that the critical exponents of the 𝐷-dimensional random field spin models are the same as those of the (𝐷 − 2)-dimensional pure spin models. However, it is known that this dimensional reduction breaks down in low enough dimensions due to a non-perturbative effect associated with multiple meta-stable states. A promising approach to overcome the dimensional reduction is the functional renormalization group. In this approach, one considers the evolution of the whole functional form of the cumulant for the disorder. For a particular range of parameters, the renormalized cumulant corresponding to a fixed point exhibits a cusp as a function of the field. Such a non-analytic behavior is a consequence of multiple meta-stable states and leads to the breakdown of the dimensional reduction.
In this thesis, I consider phase transitions and critical phenomena of disordered systems driven out of equilibrium by an external force. To obtain a clear insight into the underlying physics of such driven disordered systems, it is a natural starting point to introduce a nonequilibrium counterpart of the dimensional reduction property. We first introduce a simple model of driven
disordered systems, the driven random field O(N) model, which is the random field O(N) model driven at a uniform and steady velocity. From an intuitive argument, we derive a novel type of dimensional reduction property. It predicts that the critical exponents of the 𝐷-dimensional driven random field O(N) model are the same as those of the (𝐷 − 1)-dimensional pure O(N) model. Unfortunately, as in equilibrium cases, this dimensional reduction can break down in low enough dimensions due to a non-perturbative effect.
To elucidate the condition that the dimensional reduction holds or breaks down, I performed a functional renormalization group analysis for the driven random field O(N) model. For driven systems, perturbative approaches are not useful to derive a functional renormalization group equation. Therefore, I employed the non-perturbative renormalization group formalism, which is based on an exact flow equation of the effective action. The flow equation for the renormalized disorder correlator of the driven random field O(N) model was derived and I calculated the critical exponents corresponding to its fixed point. By studying the non-analytic behavior of the renormalized disorder correlator, I determined the region in the parameter space wherein the dimensional reduction breaks down.
Remarkably, the dimensional reduction predicts that the driven random field XY model exhibits the Kosterlitz-Thouless (KT) transition in three dimensions. I found that this expectation is certainly correct. First, we introduce the spin-wave model that describes the large-scale behavior of the model in the weak disorder regime. By using the functional renormalization group approach, it was shown to exhibit a quasi-long-range order, wherein the correlation function shows power-law decay with an exponent that depends on the disorder strength and driving velocity. However, the spin-wave model is not reliable in the strong disorder regime because it cannot describe vortices (topological defects). Therefore, we next attempted to establish a phenomenological theory of the KT transition by taking into account the effect of the vortices. The main idea of this theory is to introduce an effective elastic constant, which is assumed to flow according to a renormalization group equation similar to that of the two-dimensional pure XY model. From this theory, the exponent of the correlation function was calculated as a function of the disorder strength and the driving velocity. We also discussed the structural change in the vortices at the three-dimensional KT transition with the aid of the dimensional reduction.