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.