Quantum information metric and Berry curvature from a Lagrangian approach

Alvarez‐Jimenez, Javier; Dector, Aldo; Vergara, J. David

doi:10.1007/jhep03(2017)044

Cited by 25 publications

(53 citation statements)

References 39 publications

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“…Furthermore, in the quantum case the components of the Berry curvature for the ground state coming from Equation are (see ref. []):

F_{12}^{(0)} false(x false) = - \frac{Z}{8 ω^{3}}, F_{13}^{(0)} false(x false) = \frac{Y}{8 ω^{3}}, F_{23}^{(0)} false(x false) = - \frac{X}{8 ω^{3}}

By comparing and , and taking into account the Bohr–Sommerfeld quantization rule for action variable

I = ℏ / 2

, it is direct to check that these curvatures satisfy the relation .…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…We can also compare Equation (30) with the quantum metric tensor for the ground state obtained by using Equation (4) (we refer the reader to ref. [18] where the computations are done):…”

Section: Generalized Harmonic Oscillatormentioning

confidence: 99%

“…Furthermore, in the quantum case the components of the Berry curvature for the ground state coming from Equation (5) are (see ref. [18]):…”

Section: Generalized Harmonic Oscillatormentioning

confidence: 99%

“…The component g 22 follows by substituting Equation (86c) into Equation (18). After some algebra, we get , 0 ≤ z ≤ 1.…”

Section: Singular Euclidean Oscillatormentioning

confidence: 99%

“…where t 12 = t 1 − t 2 , and we have used ⟨e i( a0 − b0 ) ⟩ = ab and ⟨e i( a0 − b0 + c0 − d0 ) ⟩ = ab cd + a1 b2 c2 d1 + a2 b1 c1 d2 . Then, plugging Equation (121) into Equation (18) and integrating, we arrive at the classical metric…”

Section: Spin-half In a Magnetic Fieldmentioning

confidence: 99%

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Geometry of the Parameter Space of a Quantum System: Classical Point of View

Alvarez‐Jimenez¹,

González²,

Gutiérrez‐Ruiz³

et al. 2019

Annalen der Physik

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The local geometry of the parameter space of a quantum system is described by the quantum metric tensor and the Berry curvature, which are two fundamental objects that play a crucial role in understanding geometrical aspects of condensed matter physics. Classical integrable systems are considered and a new approach is reported to obtain the classical analogs of the quantum metric tensor and the Berry curvature. An advantage of this approach is that it can be applied to a wide variety of classical systems corresponding to quantum systems with bosonic and fermionic degrees of freedom. The approach used arises from the semiclassical approximation of the Berry curvature and the quantum metric tensor in the Lagrangian formalism. This semiclassical approximation is exploited to establish, for the first time, the relation between the quantum metric tensor and its classical counterpart. The approach described is illustrated and validated by applying it to five systems: the generalized harmonic oscillator, the symmetric and linearly coupled harmonic oscillators, the singular Euclidean oscillator, and a spin‐half particle in a magnetic field. Finally, some potential applications of this approach and possible generalizations that can be of interest in the field of condensed matter physics are mentioned.

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“…Furthermore, in the quantum case the components of the Berry curvature for the ground state coming from Equation are (see ref. []):

F_{12}^{(0)} false(x false) = - \frac{Z}{8 ω^{3}}, F_{13}^{(0)} false(x false) = \frac{Y}{8 ω^{3}}, F_{23}^{(0)} false(x false) = - \frac{X}{8 ω^{3}}

By comparing and , and taking into account the Bohr–Sommerfeld quantization rule for action variable

I = ℏ / 2

, it is direct to check that these curvatures satisfy the relation .…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…We can also compare Equation (30) with the quantum metric tensor for the ground state obtained by using Equation (4) (we refer the reader to ref. [18] where the computations are done):…”

Section: Generalized Harmonic Oscillatormentioning

confidence: 99%

“…Furthermore, in the quantum case the components of the Berry curvature for the ground state coming from Equation (5) are (see ref. [18]):…”

Section: Generalized Harmonic Oscillatormentioning

confidence: 99%

“…The component g 22 follows by substituting Equation (86c) into Equation (18). After some algebra, we get , 0 ≤ z ≤ 1.…”

Section: Singular Euclidean Oscillatormentioning

confidence: 99%

Section: Spin-half In a Magnetic Fieldmentioning

confidence: 99%

See 3 more Smart Citations

Geometry of the Parameter Space of a Quantum System: Classical Point of View

Alvarez‐Jimenez¹,

González²,

Gutiérrez‐Ruiz³

et al. 2019

Annalen der Physik

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Berry phases on Virasoro orbits

Oblak

2017

J. High Energ. Phys.

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We point out that unitary representations of the Virasoro algebra contain Berry phases obtained by acting on a primary state with conformal transformations that trace a closed path on a Virasoro coadjoint orbit. These phases can be computed exactly thanks to the Maurer-Cartan form on the Virasoro group, and they persist after combining left-and right-moving sectors. Thinking of Virasoro representations as particles in AdS 3 dressed with boundary gravitons, the Berry phases associated with Brown-Henneaux diffeomorphisms provide a gravitational extension of Thomas precession.

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Geometric phases characterise operator algebras and missing information

Banerjee,

Dorband,

Erdmenger

et al. 2023

J. High Energ. Phys.

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We show how geometric phases may be used to fully describe quantum systems, with or without gravity, by providing knowledge about the geometry and topology of its Hilbert space. We find a direct relation between geometric phases and von Neumann algebras. In particular, we show that a vanishing geometric phase implies the existence of a well-defined trace functional on the algebra. We discuss how this is realised within the AdS/CFT correspondence for the eternal black hole. On the other hand, a non-vanishing geometric phase indicates missing information for a local observer, associated to reference frames covering only parts of the quantum system considered. We illustrate this with several examples, ranging from a single spin in a magnetic field to Virasoro Berry phases and the geometric phase associated to the eternal black hole in AdS spacetime. For the latter, a non-vanishing geometric phase is tied to the presence of a centre in the associated von Neumann algebra.

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Quantum information metric and Berry curvature from a Lagrangian approach

Cited by 25 publications

References 39 publications

Geometry of the Parameter Space of a Quantum System: Classical Point of View

Geometry of the Parameter Space of a Quantum System: Classical Point of View

Berry phases on Virasoro orbits

Geometric phases characterise operator algebras and missing information

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