Metric space formulation of quantum mechanical conservation laws

Sharp, Paul M.; D’Amico, Irene

doi:10.1103/physrevb.89.115137

“…which is the analogue of Eq. (14) for the Hamiltonian (A.1) and represents a well-defined L 1 norm when extended to the appropriate set [5]. From this, following the metric space approach to quantum mechanics [5], we derive the generalisation of D v1 to the external potential In a similar way the metric D v1 can be generalised to measure the distance between systems containing an arbitrary number of sets of different particles p a , p b .…”

Section: Discussionmentioning

confidence: 99%

“…In order to derive a metric for external potentials, we use the metric space approach to quantum mechanics [4][5][6], which allows us to derive metrics from conservation laws of the form…”

Section: Deriving Metrics For Potentialsmentioning

confidence: 99%

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

Sharp

¹

,

D’Amico

²

2016

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Understanding the behavior of quantum systems subject to magnetic fields is of fundamental importance and underpins quantum technologies. However, modeling these systems is a complex task, because of many-body interactions and because many-body approaches such as density functional theory get complicated by the presence of a vector potential into the system Hamiltonian. We use the metric space approach to quantum mechanics to study the effects of varying the magnetic vector potential on quantum systems. The application of this technique to model systems in the ground state provides insight into the fundamental mapping at the core of current density functional theory, which relates the many-body wavefunction, particle density and paramagnetic current density. We show that the role of the paramagnetic current density in this relationship becomes crucial when considering states with different magnetic quantum numbers, m. Additionally, varying the magnetic field uncovers a richer complexity for the "band structure" present in ground state metric spaces, as compared to previous studies varying scalar potentials. The robust nature of the metric space approach is strengthened by demonstrating the gauge invariance of the related metric for the paramagnetic current density. We go beyond ground state properties and apply this approach to excited states. The results suggest that, under specific conditions, a universal behavior may exist for the relationships between the physical quantities defining the system.

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“…In Refs. [5] and [8] it was demonstrated that the core theorems of DFT and CDFT, respectively, indeed represent mappings between metric spaces: This helps understanding the power of the metric space approach and why its use has already allowed for the discovery of additional properties of these core theorems. The results in Ref.…”

Section: Introductionmentioning

confidence: 97%

“…[9][10][11][12][13]). The metric space approach involves the derivation of "natural" metrics from conservation laws to assign a distance between two physically relevant functions [8]. In recent work these metrics were applied to the basic variables of both standard DFT [5][6][7]14] and CDFT [8].…”

Section: Introductionmentioning

confidence: 99%

Metric-space analysis of systems immersed in a magnetic field

Sharp

¹

,

D’Amico

²

2015

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Understanding the behavior of quantum systems subject to magnetic fields is of fundamental importance and underpins quantum technologies. However, modeling these systems is a complex task, because of many-body interactions and because many-body approaches such as density functional theory get complicated by the presence of a vector potential into the system Hamiltonian. We use the metric space approach to quantum mechanics to study the effects of varying the magnetic vector potential on quantum systems. The application of this technique to model systems in the ground state provides insight into the fundamental mapping at the core of current density functional theory, which relates the many-body wavefunction, particle density and paramagnetic current density. We show that the role of the paramagnetic current density in this relationship becomes crucial when considering states with different magnetic quantum numbers, m. Additionally, varying the magnetic field uncovers a richer complexity for the "band structure" present in ground state metric spaces, as compared to previous studies varying scalar potentials. The robust nature of the metric space approach is strengthened by demonstrating the gauge invariance of the related metric for the paramagnetic current density. We go beyond ground state properties and apply this approach to excited states. The results suggest that, under specific conditions, a universal behavior may exist for the relationships between the physical quantities defining the system.

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Characterizing Adiabaticity in Quantum Many‐Body Systems at Finite Temperature

Skelt

¹

,

D’Amico

²

2020

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The quantum adiabatic theorem is fundamental to time‐dependent quantum systems, but being able to characterize quantitatively an adiabatic evolution in many‐body systems can be a challenge. This work demonstrates that the use of appropriate state and particle‐density metrics is a viable method to quantitatively determine the degree of adiabaticity in the dynamic of a quantum many‐body system. The method applies also to systems at finite temperature, which is important for quantum technologies and quantum thermodynamics related protocols. The importance of accounting for memory effects is discussed via comparison to results obtained by extending the quantum adiabatic criterion to finite temperatures: it is shown that this may produce false readings being quasi‐Markovian by construction. As the proposed method makes it possible to characterize the degree of adiabatic evolution by tracking only the system local particle densities, it is potentially applicable to both theoretical calculations of very large many‐body systems and to experiments.

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Metric space formulation of quantum mechanical conservation laws

Cited by 12 publications

References 18 publications

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

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

Metric-space analysis of systems immersed in a magnetic field

Characterizing Adiabaticity in Quantum Many‐Body Systems at Finite Temperature

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