We elucidate the geometry of quantum adiabatic evolution. By minimizing the deviation from adiabaticity, we find a Riemannian metric tensor underlying adiabatic evolution. Equipped with this tensor, we identify a unified geometric description of quantum adiabatic evolution and quantum phase transitions that generalizes previous treatments to allow for degeneracy. The same structure is relevant for applications in quantum information processing, including adiabatic and holonomic quantum computing, where geodesics over the manifold of control parameters correspond to paths which minimize errors. We illustrate this geometric structure with examples, for which we explicitly find adiabatic geodesics. By solving the geodesic equations in the vicinity of a quantum critical point, we identify universal characteristics of optimal adiabatic passage through a quantum phase transition. In particular, we show that in the vicinity of a critical point describing a second-order quantum phase transition, the geodesic exhibits power-law scaling with an exponent given by twice the inverse of the product of the spatial and scaling dimensions.

We apply the fidelity metric approach to analyze two recently introduced models that exhibit a quantum phase transition to a topologically ordered phase. These quantum models have a known connection to classical statistical mechanical models; we exploit this mapping to obtain the scaling of the fidelity metric tensor near criticality. The topological phase transitions manifest themselves in divergences of the fidelity metric across the phase boundaries. These results provide evidence that the fidelity approach is a valuable tool to investigate novel phases lacking a clear characterization in terms of local order parameters.PACS numbers: 03.65. Vf, 64.70.Tg, 24.10.Cn Introduction.-This is an exciting period for condensed matter physics, when novel phases of matter that defy traditional understanding are being observed and predicted. Examples include topological phases [1] which cannot be described by Landau-Ginzburg-Wilson paradigm [2]. Absence of local order parameters and symmetry breaking mechanisms are among the most remarkable features of these systems. These novel phases arise, for example, in collective phenomena exhibited in strongly correlated systems of two dimensional electrons at very low temperature, like in the fractional quantum Hall effect [3,4]. In such systems, the motion of electrons is highly constrained, and the fluctuations are entirely quantum in nature. In this situation Landau's theory, which is essentially a theory of classical order, can fail.It is compelling to find new ways to analyze such phases. Using tools from quantum information, it has been possible to characterize topological order using the concept of topological entropy [5]- [7]. Here, we call for a new informationtheoretic tool for studying quantum phase transitions (QPTs) [8] to topological phases. The new notion is the fidelity of ground states, whose role in the study of QPTs has been developed in [9]-[33]. The basic idea is that near a quantum critical point there is a drastic enhancement in the degree of distinguishability between two ground states, corresponding to slightly different values of the parameter space that defines the hamiltonian. This distinguishability can be quantified by the fidelity, which for pure states reduces to the amplitude of inner product or overlap. This approach is suitable for detecting QPTs and analyzing topological phases, since the method does not rely on constructing an order parameter, nor on the symmetries of the system. The overlap of two nearby ground states is a global quantity of the system that does not depend on local features like the existence of a local order parameter. Therefore, it should contain all the information that describes topological order. The capability of fidelity to spot a topological QPT has been shown in [22] by numerical analysis. Since topological order is a property of the ground state wavefunction alone, knowledge of the ground state of the system is sufficient in order to carry out this analysis.

Symmetric couplings among aggregates of n chromophores increase the transfer rate of excitons by a factor n 2 , a quantum-mechanical phenomenon called 'supertransfer'. In this work, we demonstrate how supertransfer effects induced by geometrical symmetries can enhance the exciton diffusion length by a factor n along cylindrically symmetric structures, consisting of arrays of rings of chromophores, and along spiral arrays. We analyse both closed-system dynamics and open quantum dynamics, modelled by combining a random bosonic bath with static disorder. In the closed-system case, we use the symmetries of the system within a short-time approximation to obtain a closed analytical expression for the diffusion length that explicitly reveals the supertransfer contribution. When subject to disorder, we show that supertransfer can enhance excitonic diffusion lengths for small disorders and characterize the crossover from coherent to incoherent motion. Owing to the quasi-one-dimensional nature of the model, disorder ultimately localizes the excitons, diminishing but not destroying the effects of supertransfer. When dephasing effects are included, we study the scaling of diffusion with both time and number of chromophores and observe that the transition from a coherent, ballistic regime to an incoherent, random-walk regime occurs at the same point as the change from supertransfer to classical scaling.