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Geometry and Topology in Condensed Matter Physics Geometry and Topology in Condensed Matter Physics Raffaele Resta Istituto Officina dei Materiali, CNR, Trieste, Italy Società Italiana di Fisica, 1050 Congresso Nazionale L’Aquila, September 25th, 2019 . Outline 1 Introduction: the polarization dilemma 2 What does it mean “geometrical”? 3 Bloch orbitals & geometry in k space 4 Two paradigmatic observables: P and M 5 Observables defined modulo 2π 6 Observables exempt from 2π ambiguity 7 Geometry in r space vs. k space . Outline 1 Introduction: the polarization dilemma 2 What does it mean “geometrical”? 3 Bloch orbitals & geometry in k space 4 Two paradigmatic observables: P and M 5 Observables defined modulo 2π 6 Observables exempt from 2π ambiguity 7 Geometry in r space vs. k space . Elementary textbook definition Z d 1 P = = dr r ρ(micro)(r) V V P appears as dominated by surface contributions Phenomenologically P is a bulk property . ?? . ation of solids – p. 24/ . How is polarizationDIELECTRIC retrieved? INSIDE A CAPACITOR silicon polarization density . min max The creative role of computationsto understand the polariz Silicon in a capacitor: induced charge density How is polarization retrieved? DIELECTRIC INSIDE A CAPACITOR silicon polarization density min max min max planar average Silicon (pseudo) charge density, unperturbed planar average. How a series of computationschanged our viewof the polarization of solids – p. 24/61 Silicon (pseudo)charge density, unperturbed . planar average . How a series of computationschanged our viewof the polarization of solids – p. 24/61 Dilemma’s solution Bulk macroscopic polarization P has nothing to do with the charge density in the bulk of the material (contrary to what many texbooks pretend!) Instead, P is a geometric phase (Berry’s phase) of the ground electronic wavefunction Breakthrough due to King-Smith & Vanderbilt, 1993 An early account (in Italian): R. Resta, Che cos’è la polarizzazione dielettrica macroscopica? Il Nuovo Saggiatore 9 (5/6), 79 (1993). Dilemma’s solution Bulk macroscopic polarization P has nothing to do with the charge density in the bulk of the material (contrary to what many texbooks pretend!) Instead, P is a geometric phase (Berry’s phase) of the ground electronic wavefunction Breakthrough due to King-Smith & Vanderbilt, 1993 An early account (in Italian): R. Resta, Che cos’è la polarizzazione dielettrica macroscopica? Il Nuovo Saggiatore 9 (5/6), 79 (1993). Outline 1 Introduction: the polarization dilemma 2 What does it mean “geometrical”? 3 Bloch orbitals & geometry in k space 4 Two paradigmatic observables: P and M 5 Observables defined modulo 2π 6 Observables exempt from 2π ambiguity 7 Geometry in r space vs. k space . The simplest geometrical property: Distance Two state vectors jΨ1i and jΨ2i in the same Hilbert space 2 − jh j ij2 D12 = log Ψ1 Ψ2 2 D12 = 0 if the two quantum states coincide apart for an irrelevant phase: gauge-invariant 2 1 D12 = if the two states are orthogonal . A second geometrical property: Connection 2 − jh j ij2 − h j i − h j i D12 = log Ψ1 Ψ2 = log Ψ1 Ψ2 log Ψ2 Ψ1 The two terms are not gauge-invariant Each of the two terms is a complex number What is the meaning of Im log hΨ1jΨ2i ? i'12 hΨ1jΨ2i = jhΨ1jΨ2ije −Im log hΨ1jΨ2i = '12;'21 = −'12 The connection fixes the phase difference The connection is arbitrary Given that it is arbitrary, why bother? . A second geometrical property: Connection 2 − jh j ij2 − h j i − h j i D12 = log Ψ1 Ψ2 = log Ψ1 Ψ2 log Ψ2 Ψ1 The two terms are not gauge-invariant Each of the two terms is a complex number What is the meaning of Im log hΨ1jΨ2i ? i'12 hΨ1jΨ2i = jhΨ1jΨ2ije −Im log hΨ1jΨ2i = '12;'21 = −'12 The connection fixes the phase difference The connection is arbitrary Given that it is arbitrary, why bother? . A second geometrical property: Connection 2 − jh j ij2 − h j i − h j i D12 = log Ψ1 Ψ2 = log Ψ1 Ψ2 log Ψ2 Ψ1 The two terms are not gauge-invariant Each of the two terms is a complex number What is the meaning of Im log hΨ1jΨ2i ? i'12 hΨ1jΨ2i = jhΨ1jΨ2ije −Im log hΨ1jΨ2i = '12;'21 = −'12 The connection fixes the phase difference The connection is arbitrary Given that it is arbitrary, why bother? . Change ofDownloaded paradigm from rspa.royalsocietypublishing.org due to onSir January Michael2, 2012 Berry Interview with Professor Sir Michael Ber… YouTube Visit Add to Collections Related images: Get help - Send feedback The connection by itself cannot have any physical meaning, but it can be used to build a gauge-invariant480 × 360 - Images may be subject to copyright. Find out more quantity Within QM, any gauge-invariant quantity is in principle Ford Colloquium - Nature's Optics and ... Plenary & Keynote Talks Sir Michael Berry Stock Photos an… Honorary Degrees | Cabot Institute ... Listen Free to Michael Berr… measurable physics.gatech.edu metaconferences.org gettyimages.co.uk bristol.ac.uk iheart.com Today, Berry’s geometric phase enters all modern quantum-mechanics texbooks . 1: Sir Michael Berry. | Dow… Sir Michael Berry - WM2014 ... 690 - Sir Michael Berry in the museum ... Sir Michael Victor Berry - … DTU Fotonik Colloquium - Sir Mic… researchgate.net youtube.com teylersmuseum.ning.com geni.com cachet.dk Michael Berry - Posts | Facebook Pictures of Famous Physic… UniKent SPS on Twitter: "Our second ... 2nd Stephen Gray Lecture:… News from ICTP 95 - Dateline — IC… facebook.com phys.bspu.by twitter.com blogs.kent.ac.uk portal.ictp.it Professor Sir Michael Berry ... How quantum physics democratised music Sir Michael Parkinson cooks up a treat ... Thank you for attending WEB 2016 | Wave ... youtube.com iop.org business-reporter.co.uk wave-engineering-bristol.uk The Royal Society is collaborating with JSTOR to digitize, preserve, and extend access to Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. ® www.jstor.org Change ofDownloaded paradigm from rspa.royalsocietypublishing.org due to onSir January Michael2, 2012 Berry Interview with Professor Sir Michael Ber… YouTube Visit Add to Collections Related images: Get help - Send feedback The connection by itself cannot have any physical meaning, but it can be used to build a gauge-invariant480 × 360 - Images may be subject to copyright. Find out more quantity Within QM, any gauge-invariant quantity is in principle Ford Colloquium - Nature's Optics and ... Plenary & Keynote Talks Sir Michael Berry Stock Photos an… Honorary Degrees | Cabot Institute ... Listen Free to Michael Berr… measurable physics.gatech.edu metaconferences.org gettyimages.co.uk bristol.ac.uk iheart.com Today, Berry’s geometric phase enters all modern quantum-mechanics texbooks . 1: Sir Michael Berry. | Dow… Sir Michael Berry - WM2014 ... 690 - Sir Michael Berry in the museum ... Sir Michael Victor Berry - … DTU Fotonik Colloquium - Sir Mic… researchgate.net youtube.com teylersmuseum.ning.com geni.com cachet.dk Michael Berry - Posts | Facebook Pictures of Famous Physic… UniKent SPS on Twitter: "Our second ... 2nd Stephen Gray Lecture:… News from ICTP 95 - Dateline — IC… facebook.com phys.bspu.by twitter.com blogs.kent.ac.uk portal.ictp.it Professor Sir Michael Berry ... How quantum physics democratised music Sir Michael Parkinson cooks up a treat ... Thank you for attending WEB 2016 | Wave ... youtube.com iop.org business-reporter.co.uk wave-engineering-bristol.uk The Royal Society is collaborating with JSTOR to digitize, preserve, and extend access to Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. ® www.jstor.org Change ofDownloaded paradigm from rspa.royalsocietypublishing.org due to onSir January Michael2, 2012 Berry Interview with Professor Sir Michael Ber… YouTube Visit Add to Collections Related images: Get help - Send feedback The connection by itself cannot have any physical meaning, but it can be used to build a gauge-invariant480 × 360 - Images may be subject to copyright. Find out more quantity Within QM, any gauge-invariant quantity is in principle Ford Colloquium - Nature's Optics and ... Plenary & Keynote Talks Sir Michael Berry Stock Photos an… Honorary Degrees | Cabot Institute ... Listen Free to Michael Berr… measurable physics.gatech.edu metaconferences.org gettyimages.co.uk bristol.ac.uk iheart.com Today, Berry’s geometric phase enters all modern quantum-mechanics texbooks . 1: Sir Michael Berry. | Dow… Sir Michael Berry - WM2014 ... 690 - Sir Michael Berry in the museum ... Sir Michael Victor Berry - … DTU Fotonik Colloquium - Sir Mic… researchgate.net youtube.com teylersmuseum.ning.com geni.com cachet.dk Michael Berry - Posts | Facebook Pictures of Famous Physic… UniKent SPS on Twitter: "Our second ... 2nd Stephen Gray Lecture:… News from ICTP 95 - Dateline — IC… facebook.com phys.bspu.by twitter.com blogs.kent.ac.uk portal.ictp.it Professor Sir Michael Berry ... How quantum physics democratised music Sir Michael Parkinson cooks up a treat ... Thank you for attending WEB 2016 | Wave ... youtube.com iop.org business-reporter.co.uk wave-engineering-bristol.uk The Royal Society is collaborating with JSTOR to digitize, preserve, and extend access to Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. ® www.jstor.org Differential forms in quantum geometry The state vector jΨκi depends on the continuous parameter κ Quantum metric gαβ: 2 2 d D = Dκ;κ+dκ = gαβdκαdκβ Berry connection Aα: d ' = Aαdκα A − A Berry curvature Ωαβ = @κα β @κβ α d × d' = Ωαβ dκαdκβ The above list is nonexahustive: other geometrical quantities can be defined. Differential forms in quantum geometry The state vector jΨκi depends on the continuous parameter κ Quantum metric : 2 2 d D = Dκ;κ+dκ = gαβdκαdκβ 2-form Berry connection : d ' = Aαdκα 1-form Berry curvature d × d' = Ωαβ dκαdκβ 2-form The above list is nonexahustive: other geometrical quantities can be defined. Differential forms in quantum geometry The state vector jΨκi depends on the continuous parameter κ Quantum metric : 2 2 d D = Dκ;κ+dκ = gαβdκαdκβ Berry connection : d ' = Aαdκα Berry curvature d × d' = Ωαβ dκαdκβ The above list is nonexahustive: other geometrical quantities can be defined.
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