Metric Space Approach Research Articles

Metric-Space Approach for Distinguishing Quantum Phase Transitions in Spin-Imbalanced Systems

Metric spaces are characterized by distances between pairs of elements. Systems that are physically similar are expected to present smaller distances (between their densities, wave functions and potentials) than systems that present different physical behaviors. For this reason metric spaces are good candidates for probing quantum phase transitions, since they could identify regimes of distinct phases. Here we apply metric space analysis to explore the transitions between the several phases in spin imbalanced systems. In particular we investigate the so-called FFLO (Fulde-Ferrel-Larkin-Ovchinnikov) phase, which is an intriguing phenomenon in which superconductivity and magnetism coexist in the same material. This is expected to appear for example in attractive fermionic systems with spin-imbalanced populations, due to the internal polarization produced by the imbalance. The transition between FFLO phase (superconducting phase) and the normal phase (non-superconducting) and their boundaries have been subject of discussion in recent years. We consider the Hubbard model in the attractive regime for which Density Matrix Renormalization Group calculations allow us to obtain the exact density function of the system. We then analyze the exact density distances as a function of the polarization. We find that our distances display signatures of the distinct quantum phases in spin-imbalanced fermionic systems: with respect to a central reference polarization, systems without FFLO present a very symmetric behavior, while systems with phase transitions are asymmetric.

Measuring adiabaticity in nonequilibrium quantum systems

Understanding out-of-equilibrium quantum dynamics is a critical outstanding problem, with key questions regarding characterizing adiabaticity for applications in quantum technologies. We show how the metric-space approach to quantum mechanics naturally characterizes regimes of quantum dynamics, and provides an appealingly visual tool for assessing their degree of adiabaticity. Further, the dynamic trajectories of quantum systems in metric space suggest a lack of "ergodicity", thus providing a better understanding of the fundamental one-to-one mapping between densities and wavefunctions.

Testing density-functional approximations on a lattice and the applicability of the related Hohenberg-Kohn-like theorem

We present a metric-space approach to quantify the performance of approximations in lattice density-functional theory for interacting many-body systems and to explore the regimes where the Hohenberg-Kohn-type theorem on fermionic lattices is applicable. This theorem demonstrates the existence of one-to-one mappings between particle densities, wave functions and external potentials. We then focus on these quantities, and quantify how far apart in metric space the approximated and exact ones are. We apply our method to the one-dimensional Hubbard model for different types of external potentials, and assess the regimes where it is applicable to one of the most used approximations in density-functional theory, the local density approximation (LDA). We find that the potential distance may have a very different behaviour from the density and wave function distances, in some cases even providing the wrong assessments of the LDA performance trends. We attribute this to the systems reaching behaviours which are borderline for the applicability of the one-to-one correspondence between density and external potential. On the contrary the wave function and density distances behave similarly and are always sensitive to system variations. Our metric-based method correctly predicts the regimes where the LDA performs fairly well and the regimes where it fails. This suggests that our method could be a practical tool for testing the efficiency of density-functional approximations.

Metric-space approach to potentials and its relevance to density-functional theory

External potentials play a crucial role in modelling quantum systems, since, for a given inter- particle interaction, they define the system Hamiltonian. We use the metric space approach to quantum mechanics to derive, from the energy conservation law, two natural metrics for potentials. We show that these metrics are well defined for physical potentials, regardless of whether the system is in an eigenstate or if the potential is bounded. In addition, we discuss the gauge freedom of potentials and how to ensure that the metrics preserve physical relevance. Our metrics for potentials, together with the metrics for wavefunctions and densities from [I. D'Amico, J. P. Coe, V. V. Franca, and K. Capelle, Phys. Rev. Lett. 106, 050401 (2011)] paves the way for a comprehensive study of the two fundamental theorems of Density Functional Theory. We explore these by analysing two many- body systems for which the related exact Kohn-Sham systems can be derived. First we consider the information provided by each of the metrics, and we find that the density metric performs best in distinguishing two many-body systems. Next we study for the systems at hand the one-to-one relationships among potentials, ground state wavefunctions, and ground state densities defined by the Hohenberg-Kohn theorem as relationships in metric spaces. We find that, in metric space, these relationships are monotonic and incorporate regions of linearity, at least for the systems considered. Finally, we use the metrics for wavefunctions and potentials in order to assess quantitatively how close the many-body and Kohn-Sham systems are: We show that, at least for the systems analysed, both metrics provide a consistent picture, and for large regions of the parameter space the error in approximating the many-body wavefunction with the Kohn-Sham wavefunction lies under a threshold of 10%.

Neural coding of spatial phase in V1 of the macaque monkey.

We examine the responses of single neurons and pairs of neurons, simultaneously recorded with a single tetrode in the primary visual cortex of the anesthetized macaque monkey, to transient presentations of stationary gratings of varying spatial phase. Such simultaneously recorded neurons tended to have similar tuning to the phase of the grating. To determine the response features that reliably discriminate these stimuli, we use the metric-space approach extended to pairs of neurons. We find that paying attention to the times of individual spikes, at a resolution of approximately 30 ms, and keeping track of which neuron fires which spike rather than just the summed local activity contribute substantially to phase coding. The contribution is both quantitative (increasing the fidelity of phase coding) and qualitative (enabling a 2-dimensional "response space" that corresponds to the spatial phase cycle). We use a novel approach, the extraction of "temporal profiles" from the metric space analysis, to interpret and compare temporal coding across neurons. Temporal profiles were remarkably consistent across a large subset of neurons. This consistency indicates that simple mechanisms (e.g., comparing the size of the transient and sustained components of the response) allow the temporal contribution to phase coding to be decoded.

A metric for traces

We have derived an ultrametric suitable for the domain of traces, which constitutes a natural generalization of the standard metric used in the complete metric space of strings. The results presented here have applications in the denotational semantics for concurrency. As opposed to the domain-theoretic approach, the metric-space approach guarantees unique solutions to reflexive domain equations