The paper addresses the problem of hardness of state preparation in quantum systems looking the same at all scales, and connects it with mathematical language and phenomena known from the nonlinear physics of waves.
Correlation Field Theory (CFT) provides a framework for understanding the statistical properties of physical systems, particularly in areas like quantum mechanics and statistical mechanics. However, traditional CFT often struggles with scaling issues, particularly when dealing with systems exhibiting intricate geometry. This paper introduces a novel approach, termed Correlation Field Theory with Adaptive Scales, designed to overcome these challenges by dynamically adjusting the correlation field's strength based on the system's geometry. We propose a mechanism that iteratively refines the correlation field's parameters to better represent the system's underlying structure, addressing scale invariance and enabling a more accurate description of complex phenomena. The paper details the theoretical foundation, mathematical formulation, and experimental validation of this approach, demonstrating its effectiveness in analyzing specific examples involving non-uniform geometries.
The Anti-de Sitter / Conformal Field Theory (AdS/CFT) correspondence that arises in string theory has had implications for the study of phenomena across a range of subfields in physics, from spacetime geometry to the behavior of condensed matter systems. Two major themes that have featured prominently in these investigations have been the behavior of systems out of equilibrium, and the emergence of spacetime. In this thesis, aspects of these themes are considered and analyzed. The question of equilibration and thermalization in 2D conformal field theories is addressed and refined via a number of observations about local versus global thermalization in such systems, the validity of particular diagnostics of thermalization, the dependence of the equilibration behavior of a conformal field theory on its operator spectrum, and the holographic dual of the generalized Gibbs ensemble that is of interest in studies of equilibration in systems with a large number of conserved quantities. A formalism for analyzing the non-equilibrium dynamics of 1+1-dimensional conformal field theories is discussed, and its physical relevance is motivated with an example connecting such a system to an experimental system that exhibited unusual equilibration behavior. Qualitative agreement is demonstrated between the CFT picture and the experimental observations.The emergence of spacetime geometry from quantum entanglement, while largely a byproduct of considerations from holographic dualities, has also been proposed to have a direct, non-holographic manifestation. Here a particular realization of such a direct emergence is presented through a demonstration that, in the presence of quantum entanglement alone, certain observations of electric fields in the entangled system appear qualitatively the same as the corresponding observations in a physically-connected geometric spacetime, so that the entanglement effectively mimics particular features associated with geometric connectivity.
The quantum Hall effect is one of the most challenging problems in condensed matter physics in which many rare physical phenomena like fractional charges and generally the fractional statistics are involved. In the integer quantum Hall effect the role of the disorder is more than others. The important point in describing these systems is that their bulk states should be gapful and in this sense the system should be incompressible. For describing the electron condensates in an arbitrary filling fraction with the odd denominator one needs the topological U(1) Chern-Simons effective field theory. Our motivation for using these field theories is the emergence of quantized conductivity. This effective field theory also has the ability of constructing the hierarchical fractional condensates with various filling factors. The excitations of these theories have fractional statistics which are investigated in this review. The non-Abelian Chern-Simons for quantum Hall systems will be also investigated. The interesting point is that the edge states of these systems are gapless and their excitations are equipped with the Kac-Moody algebra. The connection to the conformal field theories can be established using this point. The investigation of this correspondence is one of the aims of the present thesis. The connection to the logarithmic conformal field theories (e.g. for the Haldane-Rezaei wave function) is also reported in some works which will be addressed in this thesis.