Quantum critical scaling of the geometric tensors

Berry phases and the quantum-information theoretic notion of fidelity have been recently used to analyze quantum phase transitions from a geometrical perspective. In this paper we unify these two approaches showing that the underlying mechanism is the critical singular behavior of a complex tensor over the Hamiltonian parameter space. This is achieved by performing a scaling analysis of this quantum geometric tensor in the vicinity of the critical points. In this way most of the previous results are understood on general grounds and new ones are found. We show that criticality is not a sufficient condition to ensure superextensive divergence of the geometric tensor, and state the conditions under which this is possible. The validity of this analysis is further checked by exact diagonalization of the spin-1/2 XXZ Heisenberg chain.Comment: Typos correcte

Statistical Mechanics and Information-Theoretic Perspectives on Complexity in the Earth System

Peer reviewedPublisher PD

Field-theoretic methods

Many complex systems are characterized by intriguing spatio-temporal structures. Their mathematical description relies on the analysis of appropriate correlation functions. Functional integral techniques provide a unifying formalism that facilitates the computation of such correlation functions and moments, and furthermore allows a systematic development of perturbation expansions and other useful approximative schemes. It is explained how nonlinear stochastic processes may be mapped onto exponential probability distributions, whose weights are determined by continuum field theory actions. Such mappings are madeexplicit for (1) stochastic interacting particle systems whose kinetics is defined through a microscopic master equation; and (2) nonlinear Langevin stochastic differential equations which provide a mesoscopic description wherein a separation of time scales between the relevant degrees of freedom and background statistical noise is assumed. Several well-studied examples are introduced to illustrate the general methodology.Comment: Article for the Encyclopedia of Complexity and System Science, B. Meyers (Ed.), Springer-Verlag Berlin, 200