IPMU Focus Week on Quantum Mechanical Systems at Large Quantum Number

Bulk-boundary
correspondence of
topologically trivial insulators
Haruki Watanabe
University of Tokyo
HW and S. Ono, Phys. Rev. B 102, 165120 (2020)
HW and H. C. Po, arXiv:2009.04845

Seishiro Ono
The University of Tokyo

Hoi Chun Po

 The Hong Kong Universityof Science and Technology

2

Variety of condensed matter
The Space Group List Project
P1 (#1)P1 (#2)P2 (#3)

P21 (#4)

C2 (#5)

Dioxo-((S)-N-salicylidene-3aminopyrrolidine)-vanadium(v)Gilmarite

Chancalthite

Thomasclarkite

Alloclasite

C2/c (#15)

P222 (#16)

P2221 (#17)

P21212 (#18)

JadeiteP212121 (#19)

Pm (#6)

Pc (#7)

Cm (#8)BaBe2Si2O7

Tashelgite

Gerstleyite

C2221 (#20)

C222 (#21)

F222 (#22)

I222 (#23)

Pottasium-L-alaninatodichloridoplatinate

Cs2O(B2O3)9La3InS6

NaAlCl4

K2AgS

Godlevskite

NaAg(NO2)2

NaFeS2

Pnc2 (#30)

Pmn21 (#31)

Pba2 (#32)

Pna21 (#33)

Pnn2 (#34)

Cmm2 (#35)

Cmc21 (#36)

Ccc2 (#37)

Cobaltite

Terskite

Enargite

Minyulite

Wakabayashilite

Li2TiTeO6

KNbW2O9

Imm2 (#44)

Iba2 (#45)

Ima2 (#46)

AgNO2

Banalsite

Batisite

Edenharterite

Pnnm (#58)

Pmmn (#59)

Pbcn (#60)

P43 (#78)

I4 (#79)

Na0.1WO3

Percleveite

Pinnoite

Sr2As2O7

WOBr4

P4122 (#91)

P41212 (#92)

P4222 (#93)

Mg2(TiO4)

Cristobalite[AsPh4][Ce(S2PMe2)4]

NbO2

P42212 (#94)

Rb5Nb3OF18

NbP

P4mm (#99)

Na2S

Li(AlSi2O6)

EkaniteCPF-1

BaTiO3

Sm2Cu4Sn5

Jasmundite

Ice XIIPbTiO3 (550 °C)

P42/ncm (#138)

I4/mmm (#139) I4/mcm (#140) I41/amd (#141) I41/acd (#142) P3 (#143)

Be(BH4)2

AuI

IndiumR32 (#155)

P3m1 (#156)

Al2Cu

Fencooperite

TeTe4

CaNa2As4O12

CeSiO4(Sr1.9Ba0.1)IrO4

P31m (#157)

RuCl3

LiNaSO4

QuartzTincalconite

P61 (#169)

P65 (#170)

P62 (#171)

GaleiteP64 (#172)

Al2S3

NaCoPO4

Sr(S2O6)(H2O)4

Ca (S2O6)(H2O)4

P6mm (#183) P6cc (#184) P63cm (#185) P63mc (#186)

KNiCl3AgI

P6m2 (#187)

KCaF(CO3)Pn3 (#201)

Fm3 (#202)

MgSn(OH)6

K2Pb(Cu(NO2)6)

Dodecasil

P43m (#215) F43m (#216) I43m (#217)

P43n (#218)

F43c (#219)

Sodalite

Fd3 (#203)

Mn3B7O13I

P63 (#173)

Nepheline

P6c2 (#188)

Ibca (#73)

Imma (#74)

VNi2

Leningradite

Chesnokovite

Weberite

Thenardite

P42/n (#86)

I4/m (#87)

I41/a (#88)

P422 (#89)

PCl4SbF6

NaSb(OH)6

Sr2Ni(WO6)

Na(AlH4)[Pt4(µ-OCOCH3)4(µ-OCOC5H4FeCp)4]

P4212 (#90)

Ba(VO)Cu4(PO4)4P4bm (#100)

P42cm (#101)

P42nm (#102)

P4cc (#103)

P4nc (#104)

Na0.5Bi0.5TiO3[Me4N][Cu2(NCS)3]

S4N2

VOSe2O5

HgTlBa2CuOx

P4c2 (#116)

P4b2 (#117)

P4n2 (#118)

I4m2 (#119)

I4c2 (#120)

CuFeSe2(NF4)(BF4)

Na3(PS4)

Hf2CuSb3

KNbF6

Pb3O4

Fe(NH3)6AgSbS4

KMnTe2

Be(SO4)(H2O)4

PbO (red)

Bi2CuO4

Sheldrickite

R3m (#160)

P3 (#147)

Mg(SO3)(H2O)6

K2TeO3R3c (#161)

Tourmaline

P6 (#174)

LiNaCO3

BaTi(Si3O9)

Li2ZrF6

P63/m (#176)

catena-[2,2'-(biphenyl-4,4'diyldiimino)dibenzene-1,3,5-triol]

P62m (#189)

P31m (#162)

Proustite

P6/m (#175)

P42/mcm (#132)

CaPt2O4

R3 (#146)

P32 (#145)

Fluorapatite

P42/nbc (#133) P42/nnm (#134)P42/mbc (#135) P42/mnm (#136) P42/nmc (#137)

NiZn(CN)4

Mn2(Te2O5)2CuSn(OH)6

SeO2

Na2O2

SrBe3O4

Dolomite

P31c (#163)

Coquimbite

P622 (#177)

Berlinite

CrCl3

R3c (#167)

P6 (#168)

CalciteK2Ta4F4O9

RbGe(IO6) Sr3(TaGa3Si2O14) Muscovite 3T

P3m1 (#164) P3c1 (#165) R3m (#166)

Ia3 (#206)

P432 (#207)

Na1-xWO3

PyriteYttriaBIF-9-Cu

Portlandite

Fluocerite-(La)PCN-6

P6522 (#179)

P6222 (#180) P6422 (#181) P6322 (#182)

AgF3

LaBTB

Beryl

ZrI3

Rhabdophane-(Ce)PCN-20Ba(AuF6)2Graphite

P4232 (#208) F432 (#209) F4132 (#210)

Be3P2

β-Eucryptite

Te(OH)6

tetrakis((18-Crown-6)thallium) MnCl4 (TlCl4)2

I432 (#211)

P4332 (#212)

NiHg4

LiFe5O8

I43d (#220) Pm3m (#221) Pn3n (#222) Pm3n (#223) Pn3m (#224) Fm3m (#225) Fm3c (#226) Fd3m (#227) Fd3c (#228)

Co-Squarate

V6SnSi(NH4)[(Mo12O36)(AsO4)Mo(MoO)]

NaCl

HgI2

P3212 (#153)

P6122 (#178)

tetracosakis(µ2-Methoxo)dodecakis(µ2-proline)-dodecairon(iii) dodecaperchlorate

AlB2

Pa3 (#205)

ZIF-71-RHO

RutileR3 (#148) P312 (#149) P321 (#150) P3112 (#151) P3121 (#152)

P62c (#190) P6/mmm (#191) P6/mcc (#192) P63/mcm (#193) P63/mmc (#194) P23 (#195) F23 (#196)

Im3 (#204)

Katoitehydrogarnet

BaTiOF4

Ibam (#72)

P4m2 (#115)

Stillwellite-Ce

P31c (#159)

Valentinite

Immm (#71)

P421c (#114)

P31 (#144)

Simpsonite

Reinerite

Pbcm (#57)

P421m (#113)

Phosgenite Fluorapophyllite

P3c1 (#158)

AgClO2

CH2I2

Pccn (#56)

P42c (#112)

Cu2(WS4)

BaAl2Se4

Pcca (#54) Pbam (#55)

P4/n (#85)

P4/mcc (#124) P4/nbm (#125) P4/nnc (#126) P4/mbm (#127) P4/mnc (#128) P4/nmm (#129) P4/ncc (#130) P42/mmc (#131)

P3221 (#154)

Zeolite Rho

BaLa4Cu5O13 Sr2Fe0.97Mo0.94O5.81

P42/m (#84)

I4122 (#98)

P4/mmm (#123)

Fe3O4

P4/m (#83)

In(PS4)NaZn(OH)3

Co3L2(tpt)2

Si2CN4

Johachidolite Magnesiocarpholite La2NiO4.15

I422 (#97)

I42d (#122)

Sr3C60

LiBH4(at 2.4 Gpa)

CordieriteZr(P2S6)

BaGe2P2

K2Pb2O32,5-bis(4-Bromobenzylidene)cyclopentanone

I4 (#82)

P43212 (#96)

I42m (#121)

AlPO-5

Ca2Na2(CO3)3

Fmmm (#69) Fddd (#70)

P4322 (#95)

Zinc(II) 4,4‘-Bipyridine2,6,2‘,6‘-tetracarboxylate

Krennerite

Fmm2 (#42)

FeNbTe2

MgVO3

Carbocernaite

B8S16

Pma2 (#28)

Abc2 (#41)

SnWO4

P4 (#81)Ferberite

Pcc2 (#27)

2,5-Bis(5-bromo-2thienyl)-thiophene

BaThBr6

I41 (#80)

Augelite

Ama2 (#40)

RetzianFerruccite

CaSb2

Pmc21 (#26)

P21/c (#14)

Abm2 (#39)(µ2-Oxo)-bis(Cl2- bis(1-H3Cimidazole)-oxo-rhenium(v)CsPr(MoO4)2P2/c (#13)

GaAs

Pmna (#53)

Ta4O

C2/m (#12)

Amm2 (#38)

Pnna (#52)

P42mc (#105) P42bc (#106) I4mm (#107) I4cm (#108) I41md (#109) I41cd (#110) P42m (#111)

AuCN

ZP-4

Tuhualite

CuNb2O6

P42 (#77)(1R,2R)-1,2-Diaminocyclohexane(2R,3R)-2,3-butanediol

by Frank Hoffmann

P21/m (#11)

Pmm2 (#25)

Pmma (#51)

Hambergite Avogadrite

Pasavaite

P41 (#76)

Muthmannite

Pban (#50)

Cmca (#64) Cmmm (#65) Cccm (#66) Cmma (#67) Ccca (#68)

P4 (#75)

FeMo2S4

Pmmm (#47) Pnnn (#48) Pccm (#49)

Pbca (#61) Pnma (#62) Cmcm (#63)

Cu(NH3)4(NO3)2

I213 (#199) Pm3 (#200)

Spertiniite

P2/m (#10)

I212121 (#24)

Pca21 (#29)

Fdd2 (#43)

Cc (#9)LTA

Spinel(Cr(NH3)6)(CuCl5)

Ca(Ta,Nb)4O11

I23 (#197)

P213 (#198)

Bi2O3

Langbeinite

P4132 (#213)

I4132 (#214)

C(NH2)3)2(SO4)

Gd3Cl3C

Im3m (#229)

Ia3d (#230)

BaCuO2Ba3(Al(OH)6)2

More information at

crystalsymmetry.wordpress.com

Crystal structures drawn with VESTA
K. Momma and F. Izumi, J. Appl. Crystallogr. 2011, 44, 1272-1276

2Variety of condensed matter
The Space Group List Project
P1 (#1)P1 (#2)P2 (#3)

P21 (#4)

C2 (#5)

Dioxo-((S)-N-salicylidene-3aminopyrrolidine)-vanadium(v)Gilmarite

Chancalthite

Thomasclarkite

Alloclasite

C2/c (#15)

P222 (#16)

P2221 (#17)

P21212 (#18)

JadeiteP212121 (#19)

Pm (#6)

Pc (#7)

Cm (#8)BaBe2Si2O7

Tashelgite

Gerstleyite

C2221 (#20)

C222 (#21)

I222 (#23)

Pottasium-L-alaninatodichloridoplatinate

Cs2O(B2O3)9La3InS6

NaAlCl4

K2AgS

Godlevskite

NaAg(NO2)2

NaFeS2

Pnc2 (#30)

Pmn21 (#31)

Pba2 (#32)

Pna21 (#33)

Pnn2 (#34)

Cmm2 (#35)

Cmc21 (#36)

Ccc2 (#37)

Cobaltite

Terskite

Enargite

Minyulite

Wakabayashilite

Li2TiTeO6

KNbW2O9

Imm2 (#44)

Iba2 (#45)

Ima2 (#46)

AgNO2

Banalsite

Batisite

Edenharterite

Pnnm (#58)

Pmmn (#59)

Pbcn (#60)

P43 (#78)

I4 (#79)

Na0.1WO3

Percleveite

Pinnoite

Sr2As2O7

WOBr4

P4122 (#91)

P41212 (#92)

P4222 (#93)

Mg2(TiO4)

Cristobalite[AsPh4][Ce(S2PMe2)4]

NbO2

P42212 (#94)

Rb5Nb3OF18

NbP

P4mm (#99)

Na2S

Li(AlSi2O6)

EkaniteCPF-1

BaTiO3

Sm2Cu4Sn5

Jasmundite

Ice XIIPbTiO3 (550 °C)

P42/ncm (#138)

I4/mmm (#139) I4/mcm (#140) I41/amd (#141) I41/acd (#142) P3 (#143)

Be(BH4)2

AuI

IndiumR32 (#155)

P3m1 (#156)

Al2Cu

Fencooperite

TeTe4

CaNa2As4O12

CeSiO4(Sr1.9Ba0.1)IrO4

P31m (#157)

RuCl3

LiNaSO4

QuartzTincalconite

P61 (#169)

P65 (#170)

P62 (#171)

GaleiteP64 (#172)

Al2S3

NaCoPO4

Sr(S2O6)(H2O)4

Ca (S2O6)(H2O)4

P6mm (#183) P6cc (#184) P63cm (#185) P63mc (#186)

KNiCl3AgI

P6m2 (#187)

KCaF(CO3)Pn3 (#201)

Fm3 (#202)

MgSn(OH)6

K2Pb(Cu(NO2)6)

Dodecasil

P43m (#215) F43m (#216) I43m (#217)

P43n (#218)

F43c (#219)

Sodalite

Fd3 (#203)

Mn3B7O13I

P63 (#173)

Nepheline

P6c2 (#188)

Ibca (#73)

Imma (#74)

VNi2

Leningradite

Chesnokovite

Weberite

Thenardite

P42/n (#86)

I4/m (#87)

I41/a (#88)

P422 (#89)

PCl4SbF6

NaSb(OH)6

Sr2Ni(WO6)

Na(AlH4)[Pt4(µ-OCOCH3)4(µ-OCOC5H4FeCp)4]

P4212 (#90)

Ba(VO)Cu4(PO4)4P4bm (#100)

P42cm (#101)

P42nm (#102)

P4cc (#103)

P4nc (#104)

Na0.5Bi0.5TiO3[Me4N][Cu2(NCS)3]

S4N2

VOSe2O5

HgTlBa2CuOx

P4c2 (#116)

P4b2 (#117)

P4n2 (#118)

I4m2 (#119)

I4c2 (#120)

CuFeSe2(NF4)(BF4)

Na3(PS4)

Hf2CuSb3

KNbF6

Pb3O4

Fe(NH3)6AgSbS4

KMnTe2

Be(SO4)(H2O)4

PbO (red)

Bi2CuO4

Sheldrickite

R3m (#160)

P3 (#147)

Mg(SO3)(H2O)6

K2TeO3R3c (#161)

Tourmaline

P6 (#174)

LiNaCO3

BaTi(Si3O9)

Li2ZrF6

P63/m (#176)

catena-[2,2'-(biphenyl-4,4'diyldiimino)dibenzene-1,3,5-triol]

P62m (#189)

P31m (#162)

Proustite

P6/m (#175)

P42/mcm (#132)

CaPt2O4

R3 (#146)

P32 (#145)

Fluorapatite

P42/nbc (#133) P42/nnm (#134)P42/mbc (#135) P42/mnm (#136) P42/nmc (#137)

NiZn(CN)4

Mn2(Te2O5)2CuSn(OH)6

SeO2

Na2O2

SrBe3O4

Dolomite

P31c (#163)

Coquimbite

P622 (#177)

Berlinite

CrCl3

R3c (#167)

P6 (#168)

CalciteK2Ta4F4O9

RbGe(IO6) Sr3(TaGa3Si2O14) Muscovite 3T

P3m1 (#164) P3c1 (#165) R3m (#166)

Ia3 (#206)

P432 (#207)

Na1-xWO3

PyriteYttriaBIF-9-Cu

Portlandite

Fluocerite-(La)PCN-6

P6522 (#179)

P6222 (#180) P6422 (#181) P6322 (#182)

AgF3

LaBTB

Beryl

ZrI3

Rhabdophane-(Ce)PCN-20Ba(AuF6)2Graphite

P4232 (#208) F432 (#209) F4132 (#210)

Be3P2

β-Eucryptite

Te(OH)6

tetrakis((18-Crown-6)thallium) MnCl4 (TlCl4)2

I432 (#211)

P4332 (#212)

NiHg4

LiFe5O8

I43d (#220) Pm3m (#221) Pn3n (#222) Pm3n (#223) Pn3m (#224) Fm3m (#225) Fm3c (#226) Fd3m (#227) Fd3c (#228)

Co-Squarate

V6SnSi(NH4)[(Mo12O36)(AsO4)Mo(MoO)]

NaCl

HgI2

P3212 (#153)

P6122 (#178)

tetracosakis(µ2-Methoxo)dodecakis(µ2-proline)-dodecairon(iii) dodecaperchlorate

AlB2

Pa3 (#205)

ZIF-71-RHO

RutileR3 (#148) P312 (#149) P321 (#150) P3112 (#151) P3121 (#152)

P62c (#190) P6/mmm (#191) P6/mcc (#192) P63/mcm (#193) P63/mmc (#194) P23 (#195) F23 (#196)

Im3 (#204)

Katoitehydrogarnet

BaTiOF4

Ibam (#72)

P4m2 (#115)

Stillwellite-Ce

P31c (#159)

Valentinite

Immm (#71)

P421c (#114)

P31 (#144)

Simpsonite

Reinerite

Pbcm (#57)

P421m (#113)

Phosgenite Fluorapophyllite

P3c1 (#158)

AgClO2

CH2I2

Pccn (#56)

P42c (#112)

Cu2(WS4)

BaAl2Se4

Pcca (#54) Pbam (#55)

P4/n (#85)

P4/mcc (#124) P4/nbm (#125) P4/nnc (#126) P4/mbm (#127) P4/mnc (#128) P4/nmm (#129) P4/ncc (#130) P42/mmc (#131)

P3221 (#154)

Zeolite Rho

BaLa4Cu5O13 Sr2Fe0.97Mo0.94O5.81

P42/m (#84)

I4122 (#98)

P4/mmm (#123)

Fe3O4

P4/m (#83)

In(PS4)NaZn(OH)3

Co3L2(tpt)2

Si2CN4

Johachidolite Magnesiocarpholite La2NiO4.15

I422 (#97)

I42d (#122)

Sr3C60

LiBH4(at 2.4 Gpa)

CordieriteZr(P2S6)

BaGe2P2

K2Pb2O32,5-bis(4-Bromobenzylidene)cyclopentanone

I4 (#82)

P43212 (#96)

I42m (#121)

AlPO-5

Ca2Na2(CO3)3

Fmmm (#69) Fddd (#70)

P4322 (#95)

Zinc(II) 4,4‘-Bipyridine2,6,2‘,6‘-tetracarboxylate

Krennerite

Fmm2 (#42)

FeNbTe2

MgVO3

Carbocernaite

B8S16

Pma2 (#28)

Abc2 (#41)

SnWO4

P4 (#81)Ferberite

Pcc2 (#27)

2,5-Bis(5-bromo-2thienyl)-thiophene

BaThBr6

I41 (#80)

Augelite

Ama2 (#40)

RetzianFerruccite

CaSb2

Pmc21 (#26)

P21/c (#14)

Abm2 (#39)(µ2-Oxo)-bis(Cl2- bis(1-H3Cimidazole)-oxo-rhenium(v)CsPr(MoO4)2P2/c (#13)

GaAs

Pmna (#53)

Ta4O

C2/m (#12)

Amm2 (#38)

Pnna (#52)

P42mc (#105) P42bc (#106) I4mm (#107) I4cm (#108) I41md (#109) I41cd (#110) P42m (#111)

AuCN

ZP-4

Tuhualite

CuNb2O6

P42 (#77)(1R,2R)-1,2-Diaminocyclohexane(2R,3R)-2,3-butanediol

by Frank Hoffmann

P21/m (#11)

Pmm2 (#25)

Pmma (#51)

Hambergite Avogadrite

Pasavaite

P41 (#76)

Muthmannite

Pban (#50)

Cmca (#64) Cmmm (#65) Cccm (#66) Cmma (#67) Ccca (#68)

P4 (#75)

FeMo2S4

Pmmm (#47) Pnnn (#48) Pccm (#49)

Pbca (#61) Pnma (#62) Cmcm (#63)

Cu(NH3)4(NO3)2

I213 (#199) Pm3 (#200)

Spertiniite

P2/m (#10)

I212121 (#24)

Pca21 (#29)

Fdd2 (#43)“Quantum Mechanical Systems”
at “Large Quantum Number”

F222 (#22)

Cc (#9)LTA

Spinel(Cr(NH3)6)(CuCl5)

Ca(Ta,Nb)4O11

I23 (#197)

P213 (#198)

Bi2O3

Langbeinite

P4132 (#213)

I4132 (#214)

C(NH2)3)2(SO4)

Gd3Cl3C

Im3m (#229)

Ia3d (#230)

BaCuO2Ba3(Al(OH)6)2

More information at

crystalsymmetry.wordpress.com

Crystal structures drawn with VESTA
K. Momma and F. Izumi, J. Appl. Crystallogr. 2011, 44, 1272-1276

2Variety of condensed matter
The Space Group List Project
P1 (#1)P1 (#2)P2 (#3)

P21 (#4)

C2 (#5)

Dioxo-((S)-N-salicylidene-3aminopyrrolidine)-vanadium(v)Gilmarite

Chancalthite

Thomasclarkite

Alloclasite

C2/c (#15)

P222 (#16)

P2221 (#17)

P21212 (#18)

JadeiteP212121 (#19)

Pm (#6)

Pc (#7)

Cm (#8)BaBe2Si2O7

Tashelgite

Gerstleyite

C2221 (#20)

C222 (#21)

Cs2O(B2O3)9La3InS6

NaAlCl4

K2AgS

Godlevskite

NaAg(NO2)2

NaFeS2

Pnc2 (#30)

Pmn21 (#31)

Pba2 (#32)

Pna21 (#33)

Pnn2 (#34)

Cmm2 (#35)

Cmc21 (#36)

Ccc2 (#37)

Cobaltite

Terskite

Enargite

Minyulite

Wakabayashilite

Li2TiTeO6

KNbW2O9

Imm2 (#44)

Iba2 (#45)

Ima2 (#46)

AgNO2

Banalsite

Batisite

Edenharterite

Pmmn (#59)

Pbcn (#60)

P43 (#78)

I4 (#79)

Na0.1WO3

Percleveite

Pinnoite

Sr2As2O7

WOBr4

P4122 (#91)

P41212 (#92)

P4222 (#93)

Mg2(TiO4)

Cristobalite[AsPh4][Ce(S2PMe2)4]

NbO2

P42212 (#94)

Rb5Nb3OF18

NbP

P4mm (#99)

Na2S

Li(AlSi2O6)

EkaniteCPF-1

BaTiO3

Sm2Cu4Sn5

Jasmundite

Ice XIIPbTiO3 (550 °C)

P42/ncm (#138)

I4/mmm (#139) I4/mcm (#140) I41/amd (#141) I41/acd (#142) P3 (#143)

Be(BH4)2

AuI

IndiumR32 (#155)

P3m1 (#156)

Al2Cu

Fencooperite

TeTe4

CaNa2As4O12

CeSiO4(Sr1.9Ba0.1)IrO4

P31m (#157)

RuCl3

LiNaSO4

QuartzTincalconite

P61 (#169)

P65 (#170)

P62 (#171)

GaleiteP64 (#172)

Al2S3

NaCoPO4

Sr(S2O6)(H2O)4

Ca (S2O6)(H2O)4

P6mm (#183) P6cc (#184) P63cm (#185) P63mc (#186)

KNiCl3AgI

P6m2 (#187)

KCaF(CO3)Pn3 (#201)

Fm3 (#202)

MgSn(OH)6

K2Pb(Cu(NO2)6)

Dodecasil

P43m (#215) F43m (#216) I43m (#217)

P43n (#218)

F43c (#219)

Sodalite

Fd3 (#203)

Mn3B7O13I

P63 (#173)

Nepheline

P6c2 (#188)

Ibca (#73)

Imma (#74)

VNi2

Leningradite

Chesnokovite

Weberite

Thenardite

P42/n (#86)

I4/m (#87)

I41/a (#88)

P422 (#89)

PCl4SbF6

NaSb(OH)6

Sr2Ni(WO6)

Na(AlH4)[Pt4(µ-OCOCH3)4(µ-OCOC5H4FeCp)4]

P4212 (#90)

Ba(VO)Cu4(PO4)4P4bm (#100)

P42cm (#101)

P42nm (#102)

P4cc (#103)

P4nc (#104)

Na0.5Bi0.5TiO3[Me4N][Cu2(NCS)3]

S4N2

VOSe2O5

HgTlBa2CuOx

P4c2 (#116)

P4b2 (#117)

P4n2 (#118)

I4m2 (#119)

I4c2 (#120)

CuFeSe2(NF4)(BF4)

Na3(PS4)

Hf2CuSb3

KNbF6

Pb3O4

Fe(NH3)6AgSbS4

KMnTe2

Be(SO4)(H2O)4

PbO (red)

Bi2CuO4

Sheldrickite

R3m (#160)

P3 (#147)

Mg(SO3)(H2O)6

K2TeO3R3c (#161)

Tourmaline

P6 (#174)

LiNaCO3

BaTi(Si3O9)

Li2ZrF6

P63/m (#176)

catena-[2,2'-(biphenyl-4,4'diyldiimino)dibenzene-1,3,5-triol]

P62m (#189)

P31m (#162)

Proustite

P6/m (#175)

P42/mcm (#132)

CaPt2O4

R3 (#146)

P32 (#145)

Fluorapatite

P42/nbc (#133) P42/nnm (#134)P42/mbc (#135) P42/mnm (#136) P42/nmc (#137)

NiZn(CN)4

Mn2(Te2O5)2CuSn(OH)6

SeO2

Na2O2

SrBe3O4

Dolomite

P31c (#163)

Coquimbite

P622 (#177)

Berlinite

CrCl3

R3c (#167)

P6 (#168)

CalciteK2Ta4F4O9

RbGe(IO6) Sr3(TaGa3Si2O14) Muscovite 3T

P3m1 (#164) P3c1 (#165) R3m (#166)

Ia3 (#206)

P432 (#207)

Na1-xWO3

PyriteYttriaBIF-9-Cu

Portlandite

Fluocerite-(La)PCN-6

P6522 (#179)

P6222 (#180) P6422 (#181) P6322 (#182)

AgF3

LaBTB

Beryl

ZrI3

Rhabdophane-(Ce)PCN-20Ba(AuF6)2Graphite

P4232 (#208) F432 (#209) F4132 (#210)

Be3P2

β-Eucryptite

Te(OH)6

tetrakis((18-Crown-6)thallium) MnCl4 (TlCl4)2

I432 (#211)

P4332 (#212)

NiHg4

LiFe5O8

I43d (#220) Pm3m (#221) Pn3n (#222) Pm3n (#223) Pn3m (#224) Fm3m (#225) Fm3c (#226) Fd3m (#227) Fd3c (#228)

Co-Squarate

V6SnSi(NH4)[(Mo12O36)(AsO4)Mo(MoO)]

NaCl

HgI2

P3212 (#153)

P6122 (#178)

tetracosakis(µ2-Methoxo)dodecakis(µ2-proline)-dodecairon(iii) dodecaperchlorate

AlB2

Pa3 (#205)

ZIF-71-RHO

RutileR3 (#148) P312 (#149) P321 (#150) P3112 (#151) P3121 (#152)

P62c (#190) P6/mmm (#191) P6/mcc (#192) P63/mcm (#193) P63/mmc (#194) P23 (#195) F23 (#196)

Im3 (#204)

Katoitehydrogarnet

BaTiOF4

Ibam (#72)

P4m2 (#115)

Stillwellite-Ce

P31c (#159)

Valentinite

Immm (#71)

P421c (#114)

P31 (#144)

Simpsonite

Reinerite

Pbcm (#57)

P421m (#113)

Phosgenite Fluorapophyllite

P3c1 (#158)

AgClO2

CH2I2

Pccn (#56)

P42c (#112)

Cu2(WS4)

BaAl2Se4

Pcca (#54) Pbam (#55)

P4/n (#85)

P4/mcc (#124) P4/nbm (#125) P4/nnc (#126) P4/mbm (#127) P4/mnc (#128) P4/nmm (#129) P4/ncc (#130) P42/mmc (#131)

P3221 (#154)

Zeolite Rho

BaLa4Cu5O13 Sr2Fe0.97Mo0.94O5.81

P42/m (#84)

I4122 (#98)

P4/mmm (#123)

Fe3O4

P4/m (#83)

In(PS4)NaZn(OH)3

Co3L2(tpt)2

Si2CN4

Johachidolite Magnesiocarpholite La2NiO4.15

I422 (#97)

I42d (#122)

Sr3C60

LiBH4(at 2.4 Gpa)

CordieriteZr(P2S6)

BaGe2P2

K2Pb2O32,5-bis(4-Bromobenzylidene)cyclopentanone

I4 (#82)

P43212 (#96)

I42m (#121)

AlPO-5

Ca2Na2(CO3)3

Fmmm (#69) Fddd (#70)

P4322 (#95)

Zinc(II) 4,4‘-Bipyridine2,6,2‘,6‘-tetracarboxylate

Krennerite

Fmm2 (#42)

FeNbTe2

MgVO3

Carbocernaite

B8S16

Pma2 (#28)

Abc2 (#41)

SnWO4

P4 (#81)Ferberite

Pcc2 (#27)

2,5-Bis(5-bromo-2thienyl)-thiophene

BaThBr6

I41 (#80)

Augelite

Ama2 (#40)

RetzianFerruccite

CaSb2

Pmc21 (#26)

P21/c (#14)

Abm2 (#39)(µ2-Oxo)-bis(Cl2- bis(1-H3Cimidazole)-oxo-rhenium(v)CsPr(MoO4)2P2/c (#13)

GaAs

Pmna (#53)

Ta4O

C2/m (#12)

Amm2 (#38)

Pnna (#52)

P42mc (#105) P42bc (#106) I4mm (#107) I4cm (#108) I41md (#109) I41cd (#110) P42m (#111)

AuCN

ZP-4

Tuhualite

CuNb2O6

P42 (#77)(1R,2R)-1,2-Diaminocyclohexane(2R,3R)-2,3-butanediol

by Frank Hoffmann

P21/m (#11)

Pmm2 (#25)

Pmma (#51)

Hambergite Avogadrite

Pasavaite

P41 (#76)

Muthmannite

Pban (#50)

Cmca (#64) Cmmm (#65) Cccm (#66) Cmma (#67) Ccca (#68)

P4 (#75)

FeMo2S4

Pmmm (#47) Pnnn (#48) Pccm (#49)

Pbca (#61) Pnma (#62) Cmcm (#63)

Cu(NH3)4(NO3)2

I213 (#199) Pm3 (#200)

Spertiniite

P2/m (#10)

I212121 (#24)

Pottasium-L-alaninatodichloridoplatinate

Pnnm (#58)(The topic today is unfortunately
not really quantum or with large quantum number)

I222 (#23)

Pca21 (#29)

Fdd2 (#43)“Quantum Mechanical Systems”
at “Large Quantum Number”

F222 (#22)

Cc (#9)LTA

Spinel(Cr(NH3)6)(CuCl5)

Ca(Ta,Nb)4O11

I23 (#197)

P213 (#198)

Bi2O3

Langbeinite

P4132 (#213)

I4132 (#214)

C(NH2)3)2(SO4)

Gd3Cl3C

Im3m (#229)

Ia3d (#230)

BaCuO2Ba3(Al(OH)6)2

More information at

crystalsymmetry.wordpress.com

Crystal structures drawn with VESTA
K. Momma and F. Izumi, J. Appl. Crystallogr. 2011, 44, 1272-1276

3Classi cation of T=0 phases
Systems described by
Hamiltonian with symmetry G
Yes

Spontaneous symmetry breaking?

Ordered phase (G→H)
Yes

Disordered phase

No

Yes

Gapless
Ordered phase

GappedDisordered phase

Excitation gap?

GappedOrdered phase

No

Excitation gap?

• Continuous symmetry breaking• …
Yes

Degeneracy
more than |G/H|?

No

Long-range entangled
phasesTopologically ordered phases (FQHE, QSL, …)Fracton topological orders

Yes

Degeneracy? No

Short-range entangled
phases• Trivial phases• Topological insulators• Symmetry-protectedfitopological phases

No

Gapless
Disordered phase• Fermi liquids• Critical phases• Gapless spin liquids• …

3

Classi cation of T=0 phases
Systems described by
Hamiltonian with symmetry G
Yes

Spontaneous symmetry breaking?

Ordered phase (G→H)
Yes

Disordered phase

No

Yes

Gapless
Ordered phase

GappedDisordered phase

Excitation gap?

GappedOrdered phase

No

Excitation gap?

• Continuous symmetry breaking• …
Yes

Degeneracy
more than |G/H|?

No

Long-range entangled
phasesTopologically ordered phases (FQHE, QSL, …)Fracton topological orders

Yes

Degeneracy? No

Short-range entangled
phases• Trivial phases• Topological insulators• Symmetry-protectedfitopological phases

No

Gapless
Disordered phase• Fermi liquids• Critical phases• Gapless spin liquids• …

3

Classi cation of T=0 phases
Systems described by
Hamiltonian with symmetry G
Yes

Spontaneous symmetry breaking?

Ordered phase (G→H)
Yes

Disordered phase

No

Yes

Gapless
Ordered phase

GappedDisordered phase

Excitation gap?

GappedOrdered phase

No

Excitation gap?

• Continuous symmetry breaking• …
Yes

Degeneracy
more than |G/H|?

No

Long-range entangled
phasesTopologically ordered phases (FQHE, QSL, …)Fracton topological orders

Yes

Degeneracy? No

Short-range entangled
phases• Trivial phases• Topological insulators• Symmetry-protectedfitopological phases

No

Gapless
Disordered phase• Fermi liquids• Critical phases• Gapless spin liquids• …

Nambu-Goldstone modes inrelativistic systems
Upon spontaneous breaking of G into H

E(k) Energy dispersion

Counting rule:
Gap Δ

ff

low-E e ective Lagrangian

Maurer–Cartan one-formk4Nambu-Goldstone modes inrelativistic systems
Upon spontaneous breaking of G into H

E(k) Energy dispersion

Counting rule:
Gap Δ

ff

low-E e ective Lagrangian

Maurer–Cartan one-formk4Nambu-Goldstone modes in
non-relativistic systems

5

HW, H. Murayama, PRL (2012), HW, Annual Review of Condensed Matter Physics (2020)

E(k)E(k) quadratic

linearkk

Nambu-Goldstone modes in
non-relativistic systems

5

HW, H. Murayama, PRL (2012), HW, Annual Review of Condensed Matter Physics (2020)

E(k)E(k) quadratic

linearkk

6

Quantum Time Crystals
F. Wilczek, PRL (2012)(conventional) crystal

Quantum Time Crystal<ρ(x)> : ground-state expectation
value of density of ions, atoms, …

<O(t)> : ground-state expectation
value of an observable

position x

We showed f(t) is time-independent → absence of QTCtime t

HW, M. Oshikawa, PRL (2015)→ Recent realization of Discrete Time Crystals (in driven systems) D. V. Else, et al, ARCMP (2020)

6

Quantum Time Crystals
F. Wilczek, PRL (2012)(conventional) crystal

Quantum Time Crystal<ρ(x)> : ground-state expectation
value of density of ions, atoms, …

<O(t)> : ground-state expectation
value of an observable

position x

We showed f(t) is time-independent → absence of QTCtime t

HW, M. Oshikawa, PRL (2015)→ Recent realization of Discrete Time Crystals (in driven systems) D. V. Else, et al, ARCMP (2020)

6

Quantum Time Crystals
F. Wilczek, PRL (2012)(conventional) crystal

Quantum Time Crystal<ρ(x)> : ground-state expectation
value of density of ions, atoms, …

<O(t)> : ground-state expectation
value of an observable

position x

We showed f(t) is time-independent → absence of QTCtime t

HW, M. Oshikawa, PRL (2015)→ Recent realization of Discrete Time Crystals (in driven systems) D. V. Else, et al, ARCMP (2020)

7

Classi cation of T=0 phases
Systems described by
Hamiltonian with symmetry G
Yes

Spontaneous symmetry breaking?

Ordered phase (G→H)
Yes

Disordered phase

No

Yes

Gapless
Ordered phase

GappedDisordered phase

Excitation gap?

GappedOrdered phase

No

Excitation gap?

• Continuous symmetry breaking• …
Yes

Degeneracy
more than |G/H|?

No

Long-range entangled
phasesTopologically ordered phases (FQHE, QSL, …)Fracton topological orders

Yes

Degeneracy? No

Short-range entangled
phases• Trivial phases• Topological insulators• Symmetry-protectedfitopological phases

No

Gapless
Disordered phase• Fermi liquids• Critical phases• Gapless spin liquids• …

Classi cation ofshort-range entangled phases• Assume a symmetry G and excitation gap Δ• H1 ~ H2 if H1 and H2 are connected without breaking symmetry G
or closing gap Δ (with or without ancillas)
H3
H1
H2
product state 1

product state 2• Trivial phases are connected to a real-space product state.
fi• Topological phases contain irremovable quantum entanglement.

8

Classi cation ofshort-range entangled phases• Assume a symmetry G and excitation gap Δ• H1 ~ H2 if H1 and H2 are connected without breaking symmetry G
or closing gap Δ (with or without ancillas)
H3
H1
H2
product state 1

product state 2• Trivial phases are connected to a real-space product state.
fi• Topological phases contain irremovable quantum entanglement.

8

Bulk-boundary correspondenceof topological phases• 2D topological insulator

C. L. Kane and E. J. Mele, PRL (2005)

S. Oh, Science (2013)• Haldane phase
S = 1 Heisenberg model<latexit sha1_base64="nTIGcnQyuqQxy8Vc0I6LL7mPGkU=">AAACJHicbZDLSgMxFIYzXmu9VV26GSyCoJQZUexGKLgprirYC3SGIZOmbWiSGZIzQhnmVVz6JC7VjXThxmcxnXahrT8E/nznHJLzhzFnGhzny1pZXVvf2CxsFbd3dvf2SweHLR0litAmiXikOiHWlDNJm8CA006sKBYhp+1wdDett5+o0iySjzCOqS/wQLI+IxgMCkpVb4ghrWe3955ORJDKLAdeKFKdZfmd9CJYgOduFpTKTsXJZS8bd27KaK5GUJp4vYgkgkogHGvddZ0Y/BQrYITTrOglmsaYjPCAdo2VWFDtp/mGmX1qSM/uR8ocCXZOf0+kWGg9FuFFKEyzwDDUi+Up/K/WTaBf9VMm4wSoJLO3+gm3IbKnidk9pigBPjYGE8XMd20yxAoTMLkWTQ7u4tbLpnVZca8rzsNVuVadJ1JAx+gEnSEX3aAaqqMGaiKCntErekcf1ov1Zn1ak1nrijWfOUJ/ZH3/AD4MpvI=</latexit>

Ĥ = JXn

S = 1/2 edge spinŝn · ŝn+1

Bulk topology implies nontrivial boundary.
Boundary states(=degrees of freedom) / surface topological order9Bulk-boundary correspondence— anomaly in ow —1+1 dimedge

2+1 dim bulk:
Chern Simons theory

1+1 dim edge
Chiral edge mode…

2+1 dimbulk

gauge dependence
cancels between
bulk and edge
Chiral anomaly
robust against
decoration / perturbationof surfacefl- Gauge invariant if no boundary- Gauge dependent with boundary10Bulk-boundary correspondence— anomaly in ow —1+1 dimedge

2+1 dim bulk:
Chern Simons theory

1+1 dim edge
Chiral edge mode…

2+1 dimbulk

gauge dependence
cancels between
bulk and edge
Chiral anomaly
robust against
decoration / perturbationof surfacefl- Gauge invariant if no boundary- Gauge dependent with boundary10Bulk-boundary correspondence— anomaly in ow —1+1 dimedge

2+1 dim bulk:
Chern Simons theory

1+1 dim edge
Chiral edge mode…

2+1 dimbulk

gauge dependence
cancels between
bulk and edge
Chiral anomaly
robust against
decoration / perturbationof surfacefl- Gauge invariant if no boundary- Gauge dependent with boundary10Bulk-boundary correspondence— anomaly in ow —1+1 dimedge

2+1 dim bulk:
Chern Simons theory

1+1 dim edge
Chiral edge mode…

2+1 dimbulk

gauge dependence
cancels between
bulk and edge
Chiral anomaly
robust against
decoration / perturbationof surfacefl- Gauge invariant if no boundary- Gauge dependent with boundary10Bulk-boundary correspondence
for higher-order topology
1D gaplessmode

2D gaplessmode

Inversion symmetric 3D topologicalinsulator under magnetic eld

0D zeromode
1D chiral mode
0D zeromode

1D gapless
mode

Higher order topology in Bismuth

0D zeromode
M. Fruchart et al, Nature (2018)
F. Schindler et al, Nature Physics (2018)
F. Schindler et al, Science Advances (2018)fi

Boundary states can be localized to corners and hinges
depending on the bulk topology.

11Symmetry indicators ofband topology• A handy diagnosis of topology• Similar relation holds in the band structure of electrons in solids

A paradigm shift
in material search• Investigation of topological properties for all materials listed on database
Material
DatahHigher-order topology in Bismut
Schindler et al, Nat. Phys. (2018)

Fist principles calculations+ symmetry indicator method

Topological propertyof materials

A paradigm shift
in material search• Investigation of topological properties for all materials listed on database
Material
DatahHigher-order topology in Bismut
Schindler et al, Nat. Phys. (2018)

Fist principles calculations+ symmetry indicator method

Topological propertyof materials

Bulk-boundary correspondenceof trivial insulators

Fractionally quantized→ something topological
W. A. Benalcazar, B. A. Bernevig, T. L. Hughes,
Science (2017) / PRB (2017)

W. A. Benalcazar, T. Li, T. L. Hughes,
PRB (2019)

Bulk multipole moment implies boundary charges.
Charges are “frozen” i.e., not degrees of freedom

14

15

Polarization and Edge charge• Assume U(1) symmetry and translation symmetry<latexit sha1_base64="GWjewyERa53FsQpgIukf7NpORTg=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBiyURxR4LXjxWsR/QhrLZTtqlm03Y3Qgl9B948aCIV/+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzSRBP6JDyUPOqLHSwwX2yxW36s5BVomXkwrkaPTLX71BzNIIpWGCat313MT4GVWGM4HTUi/VmFA2pkPsWipphNrP5pdOyZlVBiSMlS1pyFz9PZHRSOtJFNjOiJqRXvZm4n9eNzVhzc+4TFKDki0WhakgJiazt8mAK2RGTCyhTHF7K2EjqigzNpySDcFbfnmVtC6r3nXVvb+q1Gt5HEU4gVM4Bw9uoA530IAmMAjhGV7hzRk7L86787FoLTj5zDH8gfP5AzERjRo=</latexit>e+e<latexit sha1_base64="+kQmhFHqx6WICfqKYe2mDqxw48s=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBEEoiij0WvHisYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjlo5TxbDJYhGrTkA1Ci6xabgR2EkU0igQ2A7GtzO//YRK81g+mkmCfkSHkoecUWOlhwvslytu1Z2DrBIvJxXI0eiXv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtVTSCLWfzS+dkjOrDEgYK1vSkLn6eyKjkdaTKLCdETUjvezNxP+8bmrCmp9xmaQGJVssClNBTExmb5MBV8iMmFhCmeL2VsJGVFFmbDglG4K3/PIqaV1Wveuqe39VqdfyOIpwAqdwDh7cQB3uoAFNYBDCM7zCmzN2Xpx352PRWnDymWP4A+fzBy4HjRg=</latexit>x<latexit sha1_base64="nlOYUdOcLB+k6EYW3cewfF0yWss=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Koko9ljw4rEF+wFtKJvtpF272YTdjVhCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8O/Pbj6g0j+W9mSToR3QoecgZNVZqPPVLZbfizkFWiZeTMuSo90tfvUHM0gilYYJq3fXcxPgZVYYzgdNiL9WYUDamQ+xaKmmE2s/mh07JuVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NWHVz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m6INwVt+eZW0LivedcVtXJVr1TyOApzCGVyABzdQgzuoQxMYIDzDK7w5D86L8+58LFrXnHzmBP7A+fwB5ImM9g==</latexit>• For classical point charges<latexit sha1_base64="dHTjY3EXNJEpnRS40SWJuaqiVpY=">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</latexit>

1(classical)
Px=<latexit sha1_base64="9kkvNJLWhcSba9Ey/kG8p0S4p4g=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsceCF48t2A9oQ9lspu3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5EPOqLFSMxyUK27VXYCsEy8nFcjRGJS/+mHM0gilYYJq3fPcxPgZVYYzgbNSP9WYUDahI+xZKmmE2s8Wh87IhVVCMoyVLWnIQv09kdFI62kU2M6ImrFe9ebif14vNcOan3GZpAYlWy4apoKYmMy/JiFXyIyYWkKZ4vZWwsZUUWZsNiUbgrf68jppX1W9m6rbvK7Ua3kcRTiDc7gED26hDvfQgBYwQHiGV3hzHp0X5935WLYWnHzmFP7A+fwBxjmM4g==</latexit>dX( e)(x0 ) + (+e)(x0 + d)dQi xi ==eaaai2unit cell
Z
Qedge ⌘
dx @x P (x) = Px
edge

Polarization

Polarization charge<latexit sha1_base64="hPb6D15tAGu6KTZ+RoR4UyFnSaA=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRahXsquKHosePFYwX5Au5Rsmm1js8mSZKVl6X/w4kERr/4fb/4bs+0etPXBwOO9GWbmBTFn2rjut1NYW9/Y3Cpul3Z29/YPyodHLS0TRWiTSC5VJ8CaciZo0zDDaSdWFEcBp+1gfJv57SeqNJPiwUxj6kd4KFjICDZWajX6k+rkvF+uuDV3DrRKvJxUIEejX/7qDSRJIioM4VjrrufGxk+xMoxwOiv1Ek1jTMZ4SLuWChxR7afza2fozCoDFEplSxg0V39PpDjSehoFtjPCZqSXvUz8z+smJrzxUybixFBBFovChCMjUfY6GjBFieFTSzBRzN6KyAgrTIwNqGRD8JZfXiWti5p3VXPvLyv1eh5HEU7gFKrgwTXU4Q4a0AQCj/AMr/DmSOfFeXc+Fq0FJ585hj9wPn8A3sKOqA==</latexit>

Px (x)<latexit sha1_base64="YncSotVauE/pTTVebMbOn//Xyns=">AAACAnicbVDLSgMxFM3UV62vUVfiJliEurDMiKIboeDGZQX7gM4wZNJMG5pJhiQjLaW48VfcuFDErV/hzr8x085CWw9cOJxzb3LvCRNGlXacb6uwtLyyulZcL21sbm3v2Lt7TSVSiUkDCyZkO0SKMMpJQ1PNSDuRBMUhI61wcJP5rQciFRX8Xo8S4seox2lEMdJGCuwDT/ZFZXhyfeolSGqKWDCsB0OjBHbZqTpTwEXi5qQMctQD+8vrCpzGhGvMkFId10m0P85exYxMSl6qSILwAPVIx1COYqL88fSECTw2ShdGQpriGk7V3xNjFCs1ikPTGSPdV/NeJv7ndVIdXfljypNUE45nH0Upg1rALA/YpZJgzUaGICyp2RXiPpIIa5NayYTgzp+8SJpnVfei6tydl2u1PI4iOARHoAJccAlq4BbUQQNg8AiewSt4s56sF+vd+pi1Fqx8Zh/8gfX5Aw0vlo8=</latexit>⇢(x) =

Px<latexit sha1_base64="yekXo91nr6xxMVs1QdJ2WmEQFQA=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsceCF48V7Qe0oWy2k3bpZhN2N2IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4Zua3H1FpHssHM0nQj+hQ8pAzaqx03+g/9csVt+rOQVaJl5MK5Gj0y1+9QczSCKVhgmrd9dzE+BlVhjOB01Iv1ZhQNqZD7FoqaYTaz+anTsmZVQYkjJUtachc/T2R0UjrSRTYzoiakV72ZuJ/Xjc1Yc3PuExSg5ItFoWpICYms7/JgCtkRkwsoUxxeythI6ooMzadkg3BW355lbQuqt5V1b27rNRreRxFOIFTOAcPrqEOt9CAJjAYwjO8wpsjnBfn3flYtBacfOYY/sD5/AE4rI25</latexit>@x Px (x)x<latexit sha1_base64="nlOYUdOcLB+k6EYW3cewfF0yWss=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Koko9ljw4rEF+wFtKJvtpF272YTdjVhCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8O/Pbj6g0j+W9mSToR3QoecgZNVZqPPVLZbfizkFWiZeTMuSo90tfvUHM0gilYYJq3fXcxPgZVYYzgdNiL9WYUDamQ+xaKmmE2s/mh07JuVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NWHVz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m6INwVt+eZW0LivedcVtXJVr1TyOApzCGVyABzdQgzuoQxMYIDzDK7w5D86L8+58LFrXnHzmBP7A+fwB5ImM9g==</latexit>

Edge charge is given by bulk polarization!

x<latexit sha1_base64="nlOYUdOcLB+k6EYW3cewfF0yWss=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Koko9ljw4rEF+wFtKJvtpF272YTdjVhCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8O/Pbj6g0j+W9mSToR3QoecgZNVZqPPVLZbfizkFWiZeTMuSo90tfvUHM0gilYYJq3fXcxPgZVYYzgdNiL9WYUDamQ+xaKmmE2s/mh07JuVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NWHVz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m6INwVt+eZW0LivedcVtXJVr1TyOApzCGVyABzdQgzuoQxMYIDzDK7w5D86L8+58LFrXnHzmBP7A+fwB5ImM9g==</latexit>

16

Solitons in polyacetylene•

R. Jackiw and C. Rebbi, “Solitons with fermion number 1/2,” PRD (1976).

•

W. P. Su, J. R. Schrie er, and A. J. Heeger, “Solitons in polyacetylene,” PRL (1979).

•

A. J. Heeger, S. Kivelson, J. R. Schrie er, and W. P. Su, “Solitons in conducting polymers,” RMP (1988).

q/2

q = 2eff

ff

from: wikipedia

Formula of bulk polarization“Modern theory”•††
Periodic Boundary Condition (PBC): ĉx+L = ĉx<latexit sha1_base64="9ijwJFhx3spgoJ55H4ymOF0iyXs=">AAACEnicbVDLSsNAFJ3UV62vqEs3g0VQhJKIYjdCwY0LFxXsA5o0TCaTdOjkwcxEWkK+wY2/4saFIm5dufNvnLZZaOuBC4dz7uXee9yEUSEN41srLS2vrK6V1ysbm1vbO/ruXlvEKcekhWMW866LBGE0Ii1JJSPdhBMUuox03OH1xO88EC5oHN3LcULsEAUR9SlGUkmOfmINkMxw3s8sDwUB4bmTjU5v86tCd0b9wnD0qlEzpoCLxCxIFRRoOvqX5cU4DUkkMUNC9EwjkXaGuKSYkbxipYIkCA9RQHqKRigkws6mL+XwSCke9GOuKpJwqv6eyFAoxDh0VWeI5EDMexPxP6+XSr9uZzRKUkkiPFvkpwzKGE7ygR7lBEs2VgRhTtWtEA8QR1iqFCsqBHP+5UXSPquZFzXj7rzaqBdxlMEBOATHwASXoAFuQBO0AAaP4Bm8gjftSXvR3rWPWWtJK2b2wR9onz/F9Z7B</latexit>x⇠x+L<latexit sha1_base64="mt+kSwfxhq8Rb4jgF/WW5iLoOIk=">AAAB83icbVBNSwMxEJ2tX7V+VT16CRZBUMquKPZY8OLBQwX7Ie1Ssmm2DU2yS5KVlqW/wqN6Ea/+HA/+G9N2D9r6YODx3gwz84KYM21c99vJrayurW/kNwtb2zu7e8X9g4aOEkVonUQ8Uq0Aa8qZpHXDDKetWFEsAk6bwfBm6jefqNIskg9mHFNf4L5kISPYWOlx1NFMoNHZXbdYcsvuDGiZeBkpQYZat/jV6UUkEVQawrHWbc+NjZ9iZRjhdFLoJJrGmAxxn7YtlVhQ7aezgyfoxCo9FEbKljRopv6eSLHQeiyC80DYZoHNQC/aU/E/r52YsOKnTMaJoZLMd4UJRyZC0wBQjylKDB9bgoli9lxEBlhhYmxMBZuDt/j1MmlclL2rsnt/WapWskTycATHcAoeXEMVbqEGdSAg4Ble4c1JnBfn3fmYt+acbOYQ/sD5/AEhNJEH</latexit>• Conserved U(1) charge and polarization operator<latexit sha1_base64="NMyBksZ+kXE6AVUHMTv9T3oopWY=">AAACD3icbVBNS8NAEN34WetX1KOXYFEqSElEsceCFw8eWrAf0MSy2W7bpZtN3J2UltB/4MW/4sWDIl69evPfuG1z0NYHA4/3ZpiZ50ecKbDtb2NpeWV1bT2zkd3c2t7ZNff2ayqMJaFVEvJQNnysKGeCVoEBp41IUhz4nNb9/vXErw+oVCwUdzCKqBfgrmAdRjBoqWWeuD0MSWXs0oeYDVwmoGXf37aH7tnUcGUvHOeHpy0zZxfsKaxF4qQkh1KUW+aX2w5JHFABhGOlmo4dgZdgCYxwOs66saIRJn3cpU1NBQ6o8pLpP2PrWCttqxNKXQKsqfp7IsGBUqPA150Bhp6a9ybif14zhk7RS5iIYqCCzBZ1Ym5BaE3CsdpMUgJ8pAkmkulbLdLDEhPQEWZ1CM78y4ukdl5wLgt25SJXKqZxZNAhOkJ55KArVEI3qIyqiKBH9Ixe0ZvxZLwY78bHrHXJSGcO0B8Ynz9W55zM</latexit>

Q̂ ⌘

Z L

<latexit sha1_base64="Hkwyakr+jfF5kDVlRVdwr8fbd1o=">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</latexit>dx ⇢ˆ(x)0• Berry phase formula

1
P̂x ⌘
L
<latexit sha1_base64="kDn+yn4r9hi2CCc95jRgo5Svn/8=">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</latexit>0

X Z 2⇡an2occ<latexit sha1_base64="PRCBNoYzARIT7qYv6kNNR5lDOEE=">AAACGnicbVDLSgNBEJz1bXxFPXoZDIIghF1R9CIIXjxGMFHIhqV30psMzswuM7NiWPY7vPgrXjwo4k28+DdOHgdfBQ1FVTfdXXEmuLG+/+lNTc/Mzs0vLFaWlldW16rrGy2T5pphk6Ui1dcxGBRcYdNyK/A60wgyFngV35wN/atb1Ian6tIOMuxI6CmecAbWSVE1aER3J6EA1RNIwz7YolFGd6EeCXthooEVWBb7YcbLsAdSQlSt+XV/BPqXBBNSIxM0oup72E1ZLlFZJsCYduBntlOAtpwJLCthbjADdgM9bDuqQKLpFKPXSrrjlC5NUu1KWTpSv08UII0ZyNh1SrB989sbiv957dwmx52Cqyy3qNh4UZILalM6zIl2uUZmxcARYJq7Wynrg4vDujQrLoTg98t/SWu/HhzW/YuD2unxJI4FskW2yS4JyBE5JeekQZqEkXvySJ7Ji/fgPXmv3tu4dcqbzGySH/A+vgBcsqGt</latexit>

Px̂ is inconsistent with PBC.
⟨Px̂ ⟩ does not work in general!

dx xˆ
⇢(x)

Band insulators=e
Px = hP̂x i +
2⇡

Z L

0

R. D. King-Smith, D. VanderbiltPRB (1993)

Interacting systems<latexit sha1_base64="K0n/wccHev/wIcSre23ZleV5K98=">AAACK3icbVDLSgMxFM34tr6qLt0MFsGFlhlRdCOIblwq2Ac0dbiTpm0wyYQkI5Sx/+PGX3GhCx+49T/M1Fmo9UDg3HPuTXJPrDgzNgjevInJqemZ2bn50sLi0vJKeXWtbpJUE1ojCU90MwZDOZO0ZpnltKk0BRFz2ohvznK/cUu1YYm8sgNF2wJ6knUZAeukqHyKeyAEHO9iJm0UXGd7WLFhB6s+wzsMc5A9TvPqDivQlgGPiso16JEblStBNRjBHydhQSqowEVUfsKdhKSCSks4GNMKA2XbWX494XRYwqmhCsgN9GjLUQmCmnY22nXobzml43cT7Y60/kj9OZGBMGYgYtcpwPbNXy8X//Naqe0etTMmVWqpJN8PdVPu28TPg/M7TFNi+cARIJq5v/qkDxqIdfGWXAjh35XHSX2vGh5Ug8v9yslREccc2kCbaBuF6BCdoHN0gWqIoHv0iF7Qq/fgPXvv3sd364RXzKyjX/A+vwBhs6kC</latexit>

=

Z 2⇡
0

• Resta’s formula⌦ i2⇡P̂x /e ↵ePx =Im log e
2⇡<latexit sha1_base64="LlNqeAKxYRzWpm9vJrbdz99ciVg=">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</latexit>

R. Resta, PRL (1998)

un k+ 2⇡= unkai∂k ∼ x<latexit sha1_base64="2KN1DIdvqmi/VoTRpr6hVBgEvy8=">AAACB3icbVDLSsNAFJ3UV62vqEtBBosgKCUpit0IBTcuK9gHNCFMppN2yGQSZiZCCdm58VfcuFDErb/gzr9x0mah1QMXDufcy733+AmjUlnWl1FZWl5ZXauu1zY2t7Z3zN29noxTgUkXxywWAx9JwignXUUVI4NEEBT5jPT98Lrw+/dESBrzOzVNiBuhMacBxUhpyTMPUy/jzll46gQC4azpJDTPUJ5fFXqYe2bdalgzwL/ELkkdlOh45qczinEaEa4wQ1IObStRboaEopiRvOakkiQIh2hMhppyFBHpZrM/cnislREMYqGLKzhTf05kKJJyGvm6M0JqIhe9QvzPG6YqaLkZ5UmqCMfzRUHKoIphEQocUUGwYlNNEBZU3wrxBOlAlI6upkOwF1/+S3rNhn3RsG7P6+1WGUcVHIAjcAJscAna4AZ0QBdg8ACewAt4NR6NZ+PNeJ+3VoxyZh/8gvHxDY/ambs=</latexit>

dk u⇤nk i@k unk

d ih |@ | i

G. Ortiz, R. M. Martin, PRB (1994)
HW, M. Oshikawa, PRX (2017)
Twisted Boundary Conditionĉ†x+L = e i<latexit sha1_base64="xtUP53V9aGHnJO6FyPizvfFcxcM=">AAACHHicbVDLSsNAFJ34rPUVdekmWARBLIkP7EYouHHhooJ9QJOGyfSmHTp5MDORlpAPceOvuHGhiBsXgn/jtM1CWw8MHM45lzv3eDGjQprmt7awuLS8slpYK65vbG5t6zu7DRElnECdRCziLQ8LYDSEuqSSQSvmgAOPQdMbXI/95gNwQaPwXo5icALcC6lPCZZKcvUzu49lSrJOandxrwc8c9Ph8W12BZ30hNpxn2Z5wh128oirl8yyOYExT6yclFCOmqt/2t2IJAGEkjAsRNsyY+mkmEtKGGRFOxEQYzLAPWgrGuIAhJNOjsuMQ6V0DT/i6oXSmKi/J1IcCDEKPJUMsOyLWW8s/ue1E+lXnJSGcSIhJNNFfsIMGRnjpowu5UAkGymCCafqrwbpY46JVH0WVQnW7MnzpHFati7K5t15qVrJ6yigfXSAjpCFLlEV3aAaqiOCHtEzekVv2pP2or1rH9PogpbP7KE/0L5+AOQ3oxM=</latexit>ĉ†x| + 2⇡i = | i<latexit sha1_base64="m4xLhiIkRfJiSrLG1BAV+x8pJjg=">AAACCnicbVDLSgMxFM3UV62vUZeCBIsgKGWmKHYjFNy4rGAf0BlKJs20oZlMSDJCGbtz6Ze4VDfi1l9w4d+YTmehrQcunJxzL7n3BIJRpR3n2yosLa+srhXXSxubW9s79u5eS8WJxKSJYxbLToAUYZSTpqaakY6QBEUBI+1gdD312/dEKhrzOz0WxI/QgNOQYqSN1LMPHzwxpKdVT1BPIj5g5CpT8kfPLjsVJwNcJG5OyiBHo2d/ef0YJxHhGjOkVNd1hPZTJDXFjExKXqKIQHiEBqRrKEcRUX6a3TGBx0bpwzCWpriGmfp7IkWRUuMoOAsi0xwhPVTz9lT8z+smOqz5KeUi0YTj2V9hwqCO4TQX2KeSYM3GhiAsqVkX4iGSCGuTXsnk4M5fvUha1Yp7UXFuz8v1Wp5IERyAI3ACXHAJ6uAGNEATYPAInsEreLOerBfr3fqYtRasfGYf/IH1+QM3z5qJ</latexit>

We know how to compute the bulk Pxeven for interacting systems!

17Formula of bulk
higher-order multipoles?
• Di cultiesZ in generalizing the BerryZ phase formula<latexit sha1_base64="YRVtu8QieL9WGAh4eT2fwqhQwAg=">AAAC03iclVLLThsxFPVMaUvTB6Esuxk1qkS7iGZGoNIdgk2XVGoANU5GHseTWOPHyL5DiazZILZ8XVd8Qv+iniSVgLDhSpbOPec+5GPnleAW4vg2CJ9tPH/xcvNV5/Wbt++2utvvT62uDWUDqoU25zmxTHDFBsBBsPPKMCJzwc7y8rjVzy6YsVyrnzCv2EiSqeIFpwQ8lXX/YFvLzCnMFQZ2CU5T2jQ+g8wtiaNfPi8MoW5SZpeNS3HFm9q3tNn4C8cVMcCJyFxL/Bew4dMZEGP0785TVozTsnG77Y7P43QxDefSlc36pnvp/G5p1u3F/XgR0TpIVqCHVnGSdf/iiaa1ZAqoINYOk7iCkWvHU8GaDq4tqwgtyZQNPVREMjtyC/Ob6JNnJlGhjT8KogV7t8MRae1c5r5SEpjZh1pLPqrl8jF6WENxMHJcVTUwRZf7i1pEoKP2gaMJN4yCmHtAqOH+ChGdEW8u+G/Q8d4kD51YB6dpP9nvxz/2eofpyqVN9AF9RLsoQV/RIfqOTtAA0eBbkAWzgIeD0IVX4fWyNAxWPTvoXoQ3/wD5JewZ</latexit>

X

n2occ

X
dkx ⇤
unkx i@kx unkx !
BZ 2⇡
n2occ

d2 k ⇤
unk i@kx i@ky unk
2
BZ (2⇡)→gauge dependent
i✓k
u
!
e
unk
nk
i∂kxi∂ky ∼ xy

• Attemps in generalizing Resta’s formula
B. Kang, K. Shiozaki, G. Y. Cho, PRB (2019)
W. A. Wheeler, L. K. Wagner, T. L. Hughes, PRB (2019)
S. Ono, L. Trifunovic, HW, PRB (2019)• Nested Wilson loop approach for band insulators (gives only the edge polarization.)
W. A. Benalcazar, B. A. Bernevig, T. L. Hughes, Science (2017) / PRB (2017)• More direct generalization of the “modern theory”?

ffi

HW, S. Ono, PRB (2020)
L. Trifunovic, PRR (2020)
S. H. Kooi, G. van Miert, C. Ortix, npj QM (2021)
S. Ren, I. Souza, D. Vanderbilt, PRB (2021)

18

19

Our goal
1. Give a general formulation of the bulk multipole moments→ Characterization of trivial phases of matter
2. Clarify the relation to corner / hinges charges
3. Propose materials / experiments

Images from http://www.atomsinmotion.com/book/chapter4/salts

Polarizationin 1D systems

21

Polarization and Edge charge• Assume U(1) symmetry and translation symmetry<latexit sha1_base64="GWjewyERa53FsQpgIukf7NpORTg=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBiyURxR4LXjxWsR/QhrLZTtqlm03Y3Qgl9B948aCIV/+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzSRBP6JDyUPOqLHSwwX2yxW36s5BVomXkwrkaPTLX71BzNIIpWGCat313MT4GVWGM4HTUi/VmFA2pkPsWipphNrP5pdOyZlVBiSMlS1pyFz9PZHRSOtJFNjOiJqRXvZm4n9eNzVhzc+4TFKDki0WhakgJiazt8mAK2RGTCyhTHF7K2EjqigzNpySDcFbfnmVtC6r3nXVvb+q1Gt5HEU4gVM4Bw9uoA530IAmMAjhGV7hzRk7L86787FoLTj5zDH8gfP5AzERjRo=</latexit>e+e<latexit sha1_base64="+kQmhFHqx6WICfqKYe2mDqxw48s=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBEEoiij0WvHisYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjlo5TxbDJYhGrTkA1Ci6xabgR2EkU0igQ2A7GtzO//YRK81g+mkmCfkSHkoecUWOlhwvslytu1Z2DrBIvJxXI0eiXv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtVTSCLWfzS+dkjOrDEgYK1vSkLn6eyKjkdaTKLCdETUjvezNxP+8bmrCmp9xmaQGJVssClNBTExmb5MBV8iMmFhCmeL2VsJGVFFmbDglG4K3/PIqaV1Wveuqe39VqdfyOIpwAqdwDh7cQB3uoAFNYBDCM7zCmzN2Xpx352PRWnDymWP4A+fzBy4HjRg=</latexit>x<latexit sha1_base64="nlOYUdOcLB+k6EYW3cewfF0yWss=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Koko9ljw4rEF+wFtKJvtpF272YTdjVhCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8O/Pbj6g0j+W9mSToR3QoecgZNVZqPPVLZbfizkFWiZeTMuSo90tfvUHM0gilYYJq3fXcxPgZVYYzgdNiL9WYUDamQ+xaKmmE2s/mh07JuVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NWHVz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m6INwVt+eZW0LivedcVtXJVr1TyOApzCGVyABzdQgzuoQxMYIDzDK7w5D86L8+58LFrXnHzmBP7A+fwB5ImM9g==</latexit>• For classical point charges<latexit sha1_base64="dHTjY3EXNJEpnRS40SWJuaqiVpY=">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</latexit>

1(classical)
Px=<latexit sha1_base64="9kkvNJLWhcSba9Ey/kG8p0S4p4g=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsceCF48t2A9oQ9lspu3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5EPOqLFSMxyUK27VXYCsEy8nFcjRGJS/+mHM0gilYYJq3fPcxPgZVYYzgbNSP9WYUDahI+xZKmmE2s8Wh87IhVVCMoyVLWnIQv09kdFI62kU2M6ImrFe9ebif14vNcOan3GZpAYlWy4apoKYmMy/JiFXyIyYWkKZ4vZWwsZUUWZsNiUbgrf68jppX1W9m6rbvK7Ua3kcRTiDc7gED26hDvfQgBYwQHiGV3hzHp0X5935WLYWnHzmFP7A+fwBxjmM4g==</latexit>dX( e)(x0 ) + (+e)(x0 + d)dQi xi ==eaaai2unit cell
Z
Qedge ⌘
dx @x P (x) = Px
edge

Polarization

Polarization charge<latexit sha1_base64="hPb6D15tAGu6KTZ+RoR4UyFnSaA=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRahXsquKHosePFYwX5Au5Rsmm1js8mSZKVl6X/w4kERr/4fb/4bs+0etPXBwOO9GWbmBTFn2rjut1NYW9/Y3Cpul3Z29/YPyodHLS0TRWiTSC5VJ8CaciZo0zDDaSdWFEcBp+1gfJv57SeqNJPiwUxj6kd4KFjICDZWajX6k+rkvF+uuDV3DrRKvJxUIEejX/7qDSRJIioM4VjrrufGxk+xMoxwOiv1Ek1jTMZ4SLuWChxR7afza2fozCoDFEplSxg0V39PpDjSehoFtjPCZqSXvUz8z+smJrzxUybixFBBFovChCMjUfY6GjBFieFTSzBRzN6KyAgrTIwNqGRD8JZfXiWti5p3VXPvLyv1eh5HEU7gFKrgwTXU4Q4a0AQCj/AMr/DmSOfFeXc+Fq0FJ585hj9wPn8A3sKOqA==</latexit>

Px (x)<latexit sha1_base64="YncSotVauE/pTTVebMbOn//Xyns=">AAACAnicbVDLSgMxFM3UV62vUVfiJliEurDMiKIboeDGZQX7gM4wZNJMG5pJhiQjLaW48VfcuFDErV/hzr8x085CWw9cOJxzb3LvCRNGlXacb6uwtLyyulZcL21sbm3v2Lt7TSVSiUkDCyZkO0SKMMpJQ1PNSDuRBMUhI61wcJP5rQciFRX8Xo8S4seox2lEMdJGCuwDT/ZFZXhyfeolSGqKWDCsB0OjBHbZqTpTwEXi5qQMctQD+8vrCpzGhGvMkFId10m0P85exYxMSl6qSILwAPVIx1COYqL88fSECTw2ShdGQpriGk7V3xNjFCs1ikPTGSPdV/NeJv7ndVIdXfljypNUE45nH0Upg1rALA/YpZJgzUaGICyp2RXiPpIIa5NayYTgzp+8SJpnVfei6tydl2u1PI4iOARHoAJccAlq4BbUQQNg8AiewSt4s56sF+vd+pi1Fqx8Zh/8gfX5Aw0vlo8=</latexit>⇢(x) =

Px<latexit sha1_base64="yekXo91nr6xxMVs1QdJ2WmEQFQA=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsceCF48V7Qe0oWy2k3bpZhN2N2IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4Zua3H1FpHssHM0nQj+hQ8pAzaqx03+g/9csVt+rOQVaJl5MK5Gj0y1+9QczSCKVhgmrd9dzE+BlVhjOB01Iv1ZhQNqZD7FoqaYTaz+anTsmZVQYkjJUtachc/T2R0UjrSRTYzoiakV72ZuJ/Xjc1Yc3PuExSg5ItFoWpICYms7/JgCtkRkwsoUxxeythI6ooMzadkg3BW355lbQuqt5V1b27rNRreRxFOIFTOAcPrqEOt9CAJjAYwjO8wpsjnBfn3flYtBacfOYY/sD5/AE4rI25</latexit>@x Px (x)x<latexit sha1_base64="nlOYUdOcLB+k6EYW3cewfF0yWss=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Koko9ljw4rEF+wFtKJvtpF272YTdjVhCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8O/Pbj6g0j+W9mSToR3QoecgZNVZqPPVLZbfizkFWiZeTMuSo90tfvUHM0gilYYJq3fXcxPgZVYYzgdNiL9WYUDamQ+xaKmmE2s/mh07JuVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NWHVz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m6INwVt+eZW0LivedcVtXJVr1TyOApzCGVyABzdQgzuoQxMYIDzDK7w5D86L8+58LFrXnHzmBP7A+fwB5ImM9g==</latexit>

Edge charge is given by bulk polarization!

x<latexit sha1_base64="nlOYUdOcLB+k6EYW3cewfF0yWss=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Koko9ljw4rEF+wFtKJvtpF272YTdjVhCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8O/Pbj6g0j+W9mSToR3QoecgZNVZqPPVLZbfizkFWiZeTMuSo90tfvUHM0gilYYJq3fXcxPgZVYYzgdNiL9WYUDamQ+xaKmmE2s/mh07JuVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NWHVz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m6INwVt+eZW0LivedcVtXJVr1TyOApzCGVyABzdQgzuoQxMYIDzDK7w5D86L8+58LFrXnHzmBP7A+fwB5ImM9g==</latexit>

22

Ambiguity of bulk polarization<latexit sha1_base64="GWjewyERa53FsQpgIukf7NpORTg=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBiyURxR4LXjxWsR/QhrLZTtqlm03Y3Qgl9B948aCIV/+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzSRBP6JDyUPOqLHSwwX2yxW36s5BVomXkwrkaPTLX71BzNIIpWGCat313MT4GVWGM4HTUi/VmFA2pkPsWipphNrP5pdOyZlVBiSMlS1pyFz9PZHRSOtJFNjOiJqRXvZm4n9eNzVhzc+4TFKDki0WhakgJiazt8mAK2RGTCyhTHF7K2EjqigzNpySDcFbfnmVtC6r3nXVvb+q1Gt5HEU4gVM4Bw9uoA530IAmMAjhGV7hzRk7L86787FoLTj5zDH8gfP5AzERjRo=</latexit>e+e<latexit sha1_base64="+kQmhFHqx6WICfqKYe2mDqxw48s=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBEEoiij0WvHisYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjlo5TxbDJYhGrTkA1Ci6xabgR2EkU0igQ2A7GtzO//YRK81g+mkmCfkSHkoecUWOlhwvslytu1Z2DrBIvJxXI0eiXv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtVTSCLWfzS+dkjOrDEgYK1vSkLn6eyKjkdaTKLCdETUjvezNxP+8bmrCmp9xmaQGJVssClNBTExmb5MBV8iMmFhCmeL2VsJGVFFmbDglG4K3/PIqaV1Wveuqe39VqdfyOIpwAqdwDh7cQB3uoAFNYBDCM7zCmzN2Xpx352PRWnDymWP4A+fzBy4HjRg=</latexit>x<latexit sha1_base64="nlOYUdOcLB+k6EYW3cewfF0yWss=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Koko9ljw4rEF+wFtKJvtpF272YTdjVhCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8O/Pbj6g0j+W9mSToR3QoecgZNVZqPPVLZbfizkFWiZeTMuSo90tfvUHM0gilYYJq3fXcxPgZVYYzgdNiL9WYUDamQ+xaKmmE2s/mh07JuVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NWHVz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m6INwVt+eZW0LivedcVtXJVr1TyOApzCGVyABzdQgzuoQxMYIDzDK7w5D86L8+58LFrXnHzmBP7A+fwB5ImM9g==</latexit><latexit sha1_base64="9kkvNJLWhcSba9Ey/kG8p0S4p4g=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsceCF48t2A9oQ9lspu3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5EPOqLFSMxyUK27VXYCsEy8nFcjRGJS/+mHM0gilYYJq3fPcxPgZVYYzgbNSP9WYUDahI+xZKmmE2s8Wh87IhVVCMoyVLWnIQv09kdFI62kU2M6ImrFe9ebif14vNcOan3GZpAYlWy4apoKYmMy/JiFXyIyYWkKZ4vZWwsZUUWZsNiUbgrf68jppX1W9m6rbvK7Ua3kcRTiDc7gED26hDvfQgBYwQHiGV3hzHp0X5935WLYWnHzmFP7A+fwBxjmM4g==</latexit><latexit sha1_base64="dHTjY3EXNJEpnRS40SWJuaqiVpY=">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</latexit>

1(classical)Px=a<latexit sha1_base64="fSWXUWjnvZ4aOu7MyLWn/EiPj1U=">AAACX3icbVFNa9wwFJTdNEk3H3XaU+nFZCl4WXaxS0tyCQR66XED3SSw3hhZfk5EZNlIz2EX4T/ZWyGX/JPKax/aJA8Eo5l50tMorQTXGIZ/HPfN1tvtnd13g739g8P33tGHS13WisGclaJU1ynVILiEOXIUcF0poEUq4Cq9/9HqVw+gNC/lL1xXsCzoreQ5ZxQtlXgPs2R1Y4IYYYWGCaq1lUQzas7iXFFmosbQJtZ1kRgec9n5asnRZyBE01wkfJXw3hxMYBSsknA0DsYdGmcTOmqPOIPOkrWbCSTeMJyGm/JfgqgHQ9LXLPF+x1nJ6gIkbqZcRGGFS0MVciagGcS1hoqye3oLCwslLUAvzSafxv9imczPS2WXtJO37L8dhhZar4vUOguKd/q51pKvaYsa89Ol4bKqESTrLspr4WPpt2H7GVfAUKwtoExxO6vP7qjNAe2XDGwI0fMnvwSXX6fR92l48W14ftrHsUs+k2MSkIickHPyk8zInDDy6LjOnrPvPLk77qHrdVbX6Xs+kv/K/fQXA4+2Tg==</latexit>

1(classical)
Px
=

a

d

a
<latexit sha1_base64="M4JPRLxFvI7lUzxNlMzjWaERXS0=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBiyURxR4LXjxWtB/QhjLZbNqlm03Y3Qil9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSq4Nq777RTW1jc2t4rbpZ3dvf2D8uFRSyeZoqxJE5GoToCaCS5Z03AjWCdVDONAsHYwup357SemNE/koxmnzI9xIHnEKRorPeBF2C9X3Ko7B1klXk4qkKPRL3/1woRmMZOGCtS667mp8SeoDKeCTUu9TLMU6QgHrGupxJhpfzI/dUrOrBKSKFG2pCFz9ffEBGOtx3FgO2M0Q73szcT/vG5mopo/4TLNDJN0sSjKBDEJmf1NQq4YNWJsCVLF7a2EDlEhNTadkg3BW355lbQuq9511b2/qtRreRxFOIFTOAcPbqAOd9CAJlAYwDO8wpsjnBfn3flYtBacfOYY/sD5/AHoOY2E</latexit>dX( e)(x0 ) + (+e)(x0 + d)dQi xi ==eaaX( e)(x0 ) + (+e)(x0 + d
Qi xi =
ai2unit celli2unit cell

a)

d
=e
a

Bulk polarization has an integer ambiguity.
(Choice of unit cell)e23

Ambiguity of edge charge<latexit sha1_base64="GWjewyERa53FsQpgIukf7NpORTg=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBiyURxR4LXjxWsR/QhrLZTtqlm03Y3Qgl9B948aCIV/+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzSRBP6JDyUPOqLHSwwX2yxW36s5BVomXkwrkaPTLX71BzNIIpWGCat313MT4GVWGM4HTUi/VmFA2pkPsWipphNrP5pdOyZlVBiSMlS1pyFz9PZHRSOtJFNjOiJqRXvZm4n9eNzVhzc+4TFKDki0WhakgJiazt8mAK2RGTCyhTHF7K2EjqigzNpySDcFbfnmVtC6r3nXVvb+q1Gt5HEU4gVM4Bw9uoA530IAmMAjhGV7hzRk7L86787FoLTj5zDH8gfP5AzERjRo=</latexit>e+e<latexit sha1_base64="+kQmhFHqx6WICfqKYe2mDqxw48s=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBEEoiij0WvHisYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjlo5TxbDJYhGrTkA1Ci6xabgR2EkU0igQ2A7GtzO//YRK81g+mkmCfkSHkoecUWOlhwvslytu1Z2DrBIvJxXI0eiXv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtVTSCLWfzS+dkjOrDEgYK1vSkLn6eyKjkdaTKLCdETUjvezNxP+8bmrCmp9xmaQGJVssClNBTExmb5MBV8iMmFhCmeL2VsJGVFFmbDglG4K3/PIqaV1Wveuqe39VqdfyOIpwAqdwDh7cQB3uoAFNYBDCM7zCmzN2Xpx352PRWnDymWP4A+fzBy4HjRg=</latexit>x<latexit sha1_base64="nlOYUdOcLB+k6EYW3cewfF0yWss=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Koko9ljw4rEF+wFtKJvtpF272YTdjVhCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8O/Pbj6g0j+W9mSToR3QoecgZNVZqPPVLZbfizkFWiZeTMuSo90tfvUHM0gilYYJq3fXcxPgZVYYzgdNiL9WYUDamQ+xaKmmE2s/mh07JuVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NWHVz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m6INwVt+eZW0LivedcVtXJVr1TyOApzCGVyABzdQgzuoQxMYIDzDK7w5D86L8+58LFrXnHzmBP7A+fwB5ImM9g==</latexit><latexit sha1_base64="9kkvNJLWhcSba9Ey/kG8p0S4p4g=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsceCF48t2A9oQ9lspu3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5EPOqLFSMxyUK27VXYCsEy8nFcjRGJS/+mHM0gilYYJq3fPcxPgZVYYzgbNSP9WYUDahI+xZKmmE2s8Wh87IhVVCMoyVLWnIQv09kdFI62kU2M6ImrFe9ebif14vNcOan3GZpAYlWy4apoKYmMy/JiFXyIyYWkKZ4vZWwsZUUWZsNiUbgrf68jppX1W9m6rbvK7Ua3kcRTiDc7gED26hDvfQgBYwQHiGV3hzHp0X5935WLYWnHzmFP7A+fwBxjmM4g==</latexit>

+e

d

a
<latexit sha1_base64="M4JPRLxFvI7lUzxNlMzjWaERXS0=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBiyURxR4LXjxWtB/QhjLZbNqlm03Y3Qil9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSq4Nq777RTW1jc2t4rbpZ3dvf2D8uFRSyeZoqxJE5GoToCaCS5Z03AjWCdVDONAsHYwup357SemNE/koxmnzI9xIHnEKRorPeBF2C9X3Ko7B1klXk4qkKPRL3/1woRmMZOGCtS667mp8SeoDKeCTUu9TLMU6QgHrGupxJhpfzI/dUrOrBKSKFG2pCFz9ffEBGOtx3FgO2M0Q73szcT/vG5mopo/4TLNDJN0sSjKBDEJmf1NQq4YNWJsCVLF7a2EDlEhNTadkg3BW355lbQuq9511b2/qtRreRxFOIFTOAcPbqAOd9CAJlAYwDO8wpsjnBfn3flYtBacfOYY/sD5/AHoOY2E</latexit>d<latexit sha1_base64="+kQmhFHqx6WICfqKYe2mDqxw48s=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBEEoiij0WvHisYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjlo5TxbDJYhGrTkA1Ci6xabgR2EkU0igQ2A7GtzO//YRK81g+mkmCfkSHkoecUWOlhwvslytu1Z2DrBIvJxXI0eiXv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtVTSCLWfzS+dkjOrDEgYK1vSkLn6eyKjkdaTKLCdETUjvezNxP+8bmrCmp9xmaQGJVssClNBTExmb5MBV8iMmFhCmeL2VsJGVFFmbDglG4K3/PIqaV1Wveuqe39VqdfyOIpwAqdwDh7cQB3uoAFNYBDCM7zCmzN2Xpx352PRWnDymWP4A+fzBy4HjRg=</latexit><latexit sha1_base64="GWjewyERa53FsQpgIukf7NpORTg=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBiyURxR4LXjxWsR/QhrLZTtqlm03Y3Qgl9B948aCIV/+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzSRBP6JDyUPOqLHSwwX2yxW36s5BVomXkwrkaPTLX71BzNIIpWGCat313MT4GVWGM4HTUi/VmFA2pkPsWipphNrP5pdOyZlVBiSMlS1pyFz9PZHRSOtJFNjOiJqRXvZm4n9eNzVhzc+4TFKDki0WhakgJiazt8mAK2RGTCyhTHF7K2EjqigzNpySDcFbfnmVtC6r3nXVvb+q1Gt5HEU4gVM4Bw9uoA530IAmMAjhGV7hzRk7L86787FoLTj5zDH8gfP5AzERjRo=</latexit>ex<latexit sha1_base64="nlOYUdOcLB+k6EYW3cewfF0yWss=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Koko9ljw4rEF+wFtKJvtpF272YTdjVhCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8O/Pbj6g0j+W9mSToR3QoecgZNVZqPPVLZbfizkFWiZeTMuSo90tfvUHM0gilYYJq3fXcxPgZVYYzgdNiL9WYUDamQ+xaKmmE2s/mh07JuVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NWHVz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m6INwVt+eZW0LivedcVtXJVr1TyOApzCGVyABzdQgzuoQxMYIDzDK7w5D86L8+58LFrXnHzmBP7A+fwB5ImM9g==</latexit>x<latexit sha1_base64="nlOYUdOcLB+k6EYW3cewfF0yWss=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Koko9ljw4rEF+wFtKJvtpF272YTdjVhCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8O/Pbj6g0j+W9mSToR3QoecgZNVZqPPVLZbfizkFWiZeTMuSo90tfvUHM0gilYYJq3fXcxPgZVYYzgdNiL9WYUDamQ+xaKmmE2s/mh07JuVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NWHVz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m6INwVt+eZW0LivedcVtXJVr1TyOApzCGVyABzdQgzuoQxMYIDzDK7w5D86L8+58LFrXnHzmBP7A+fwB5ImM9g==</latexit>

Edge charge also has an integer ambiguity.
(Choice of edge termination & decoration)

24

Quantization under symmetry• Assume inversion symmetry (x→−x) in addition to U(1) and translation.
• Inversion symmetry ips the sign of polarization → quantization to 0 or e/2.
1
Px = 0 or e2<latexit sha1_base64="uCzK32gA0xQDNjRCxFu//fd7yXY=">AAACC3icbVC7SsRAFJ34XNfXqqXN4CJYLYkobiMINpYruA/YhDCZvdHBSSbM3IhLSG/jr9hYKGLrD9j5N84+Cl091eGce7nnniiTwqDrfjlz8wuLS8uVlerq2vrGZm1ru2NUrjm0uZJK9yJmQIoU2ihQQi/TwJJIQje6PR/53TvQRqj0CocZBAm7TkUsOEMrhbW9Vnh/6voI91hQpWnpx5rxwiuLwxL8RA0ohLW623DHoH+JNyV1MkUrrH36A8XzBFLkkhnT99wMg4JpFFxCWfVzAxnjt+wa+pamLAETFONfSrpvlQGNbZRYpUjH6s+NgiXGDJPITiYMb8ysNxL/8/o5xs2gEGmWI6R8cijOJUVFR8XQgdDAUQ4tYVwLm5XyG2bLQFtf1Zbgzb78l3QOG95xw708qp81p3VUyC7ZIwfEIyfkjFyQFmkTTh7IE3khr86j8+y8Oe+T0TlnurNDfsH5+AZfsZqY</latexit><latexit sha1_base64="LJsUVMjY/h1G0gppEwFLxbPpAh8=">AAAB+XicbVDLSgMxFM3UV62vUZdugkVwY5kRxW6EghuXFewD2mHIZDJtaB5DkimWoX/ixoUibv0Td/6NaTsLbT1wL4dz7iU3J0oZ1cbzvp3S2vrG5lZ5u7Kzu7d/4B4etbXMFCYtLJlU3QhpwqggLUMNI91UEcQjRjrR6G7md8ZEaSrFo5mkJOBoIGhCMTJWCl33ohk+wVtoe5/LGJLQrXo1bw64SvyCVEGBZuh+9WOJM06EwQxp3fO91AQ5UoZiRqaVfqZJivAIDUjPUoE40UE+v3wKz6wSw0QqW8LAufp7I0dc6wmP7CRHZqiXvZn4n9fLTFIPcirSzBCBFw8lGYNGwlkMMKaKYMMmliCsqL0V4iFSCBsbVsWG4C9/eZW0L2v+dc17uKo26kUcZXACTsE58MENaIB70AQtgMEYPINX8Obkzovz7nwsRktOsXMM/sD5/AGvG5Je</latexit>

Px = P x

mod e

mod e

• There are at least two phases → trivial phases are not unique!
Y. Fuji, F. Pollmann, M. Oshikawa, PRL (2015)
H. C. Po, A. Vishwanath, HW, Nature Comm (2017)+ many other works in the context of TCI<latexit sha1_base64="oz1+3enmm18/8/dwuc9wnWzFSd4=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoseCF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1Fip6Q7KFbfqLkDWiZeTCuRoDMpf/WHM0gilYYJq3fPcxPgZVYYzgbNSP9WYUDahI+xZKmmE2s8Wh87IhVWGJIyVLWnIQv09kdFI62kU2M6ImrFe9ebif14vNeGtn3GZpAYlWy4KU0FMTOZfkyFXyIyYWkKZ4vZWwsZUUWZsNiUbgrf68jpp16reddVtXlXqtTyOIpzBOVyCBzdQh3toQAsYIDzDK7w5j86L8+58LFsLTj5zCn/gfP4AdZuMqA==</latexit>

0 0<latexit sha1_base64="oz1+3enmm18/8/dwuc9wnWzFSd4=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoseCF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1Fip6Q7KFbfqLkDWiZeTCuRoDMpf/WHM0gilYYJq3fPcxPgZVYYzgbNSP9WYUDahI+xZKmmE2s8Wh87IhVWGJIyVLWnIQv09kdFI62kU2M6ImrFe9ebif14vNeGtn3GZpAYlWy4KU0FMTOZfkyFXyIyYWkKZ4vZWwsZUUWZsNiUbgrf68jpp16reddVtXlXqtTyOIpzBOVyCBzdQh3toQAsYIDzDK7w5j86L8+58LFsLTj5zCn/gfP4AdZuMqA==</latexit><latexit sha1_base64="9zJ1U5o5+JSaWBS08OAnB1GeiX0=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBZBEEpSFD0WvHisaD+gDWWznbRLN5uwuxFK6E/w4kERr/4ib/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjlo5TxbDJYhGrTkA1Ci6xabgR2EkU0igQ2A7GtzO//YRK81g+mkmCfkSHkoecUWOlh4sa9ssVt+rOQVaJl5MK5Gj0y1+9QczSCKVhgmrd9dzE+BlVhjOB01Iv1ZhQNqZD7FoqaYTaz+anTsmZVQYkjJUtachc/T2R0UjrSRTYzoiakV72ZuJ/Xjc14Y2fcZmkBiVbLApTQUxMZn+TAVfIjJhYQpni9lbCRlRRZmw6JRuCt/zyKmnVqt5V1b2/rNRreRxFOIFTOAcPrqEOd9CAJjAYwjO8wpsjnBfn3flYtBacfOYY/sD5/AGdRI1O</latexit>

e +e
2e +2e<latexit sha1_base64="9zJ1U5o5+JSaWBS08OAnB1GeiX0=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBZBEEpSFD0WvHisaD+gDWWznbRLN5uwuxFK6E/w4kERr/4ib/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjlo5TxbDJYhGrTkA1Ci6xabgR2EkU0igQ2A7GtzO//YRK81g+mkmCfkSHkoecUWOlh4sa9ssVt+rOQVaJl5MK5Gj0y1+9QczSCKVhgmrd9dzE+BlVhjOB01Iv1ZhQNqZD7FoqaYTaz+anTsmZVQYkjJUtachc/T2R0UjrSRTYzoiakV72ZuJ/Xjc14Y2fcZmkBiVbLApTQUxMZn+TAVfIjJhYQpni9lbCRlRRZmw6JRuCt/zyKmnVqt5V1b2/rNRreRxFOIFTOAcPrqEOd9CAJjAYwjO8wpsjnBfn3flYtBacfOYY/sD5/AGdRI1O</latexit><latexit sha1_base64="zndZINlBSHNLddtsi2QSxs8R2X4=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBiyUpih4LXjxWtB/QhrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzSRBP6JDyUPOqLHSw0UN++WKW3XnIKvEy0kFcjT65a/eIGZphNIwQbXuem5i/Iwqw5nAaamXakwoG9Mhdi2VNELtZ/NTp+TMKgMSxsqWNGSu/p7IaKT1JApsZ0TNSC97M/E/r5ua8MbPuExSg5ItFoWpICYms7/JgCtkRkwsoUxxeythI6ooMzadkg3BW355lbRqVe+q6t5fVuq1PI4inMApnIMH11CHO2hAExgM4Rle4c0Rzovz7nwsWgtOPnMMf+B8/gCgUI1Q</latexit>

The U(1) charge Q mod 2e at the inversion center
is a topological invariant.
fl<latexit sha1_base64="ms9Mq06qduhg5ax+qUjOyBGuWGY=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBZBEEpSFD0WvHhswdZCG8pmO2nXbjZhdyOU0F/gxYMiXv1J3vw3btsctPXBwOO9GWbmBYng2rjut1NYW9/Y3Cpul3Z29/YPyodHbR2nimGLxSJWnYBqFFxiy3AjsJMopFEg8CEY3878hydUmsfy3kwS9CM6lDzkjBorNS/65Ypbdecgq8TLSQVyNPrlr94gZmmE0jBBte56bmL8jCrDmcBpqZdqTCgb0yF2LZU0Qu1n80On5MwqAxLGypY0ZK7+nshopPUkCmxnRM1IL3sz8T+vm5rwxs+4TFKDki0WhakgJiazr8mAK2RGTCyhTHF7K2EjqigzNpuSDcFbfnmVtGtV76rqNi8r9VoeRxFO4BTOwYNrqMMdNKAFDBCe4RXenEfnxXl3PhatBSefOYY/cD5/AG4HjKM=</latexit><latexit sha1_base64="P66qSVZe3WXL2/3HV3RyX0gyXug=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoheh4MVjC7YW2lA220m7drMJuxuhhP4CLx4U8epP8ua/cdvmoK0PBh7vzTAzL0gE18Z1v53C2vrG5lZxu7Szu7d/UD48aus4VQxbLBax6gRUo+ASW4YbgZ1EIY0CgQ/B+HbmPzyh0jyW92aSoB/RoeQhZ9RYqXnTL1fcqjsHWSVeTiqQo9Evf/UGMUsjlIYJqnXXcxPjZ1QZzgROS71UY0LZmA6xa6mkEWo/mx86JWdWGZAwVrakIXP190RGI60nUWA7I2pGetmbif953dSE137GZZIalGyxKEwFMTGZfU0GXCEzYmIJZYrbWwkbUUWZsdmUbAje8surpF2repdVt3lRqdfyOIpwAqdwDh5cQR3uoAEtYIDwDK/w5jw6L86787FoLTj5zDH8gfP5A4lPjLU=</latexit>+2e<latexit sha1_base64="SWue4KGKpwZI7TnixNGYhX2IkNw=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBEEpSFD0WvHisYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjlo5TxbDJYhGrTkA1Ci6xabgR2EkU0igQ2A7GtzO//YRK81g+mkmCfkSHkoecUWOlhwvslytu1Z2DrBIvJxXI0eiXv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtVTSCLWfzS+dkjOrDEgYK1vSkLn6eyKjkdaTKLCdETUjvezNxP+8bmrCGz/jMkkNSrZYFKaCmJjM3iYDrpAZMbGEMsXtrYSNqKLM2HBKNgRv+eVV0qpVvauqe39ZqdfyOIpwAqdwDh5cQx3uoAFNYBDCM7zCmzN2Xpx352PRWnDymWP4A+fzByw5jRI=</latexit><latexit sha1_base64="Px+NR54fmuwgXc68NTRnaChwTsU=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBiyUpih4LXjxWsR/QhrLZTtqlm03Y3Qgl9B948aCIV/+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzSRBP6JDyUPOqLHSwwX2yxW36s5BVomXkwrkaPTLX71BzNIIpWGCat313MT4GVWGM4HTUi/VmFA2pkPsWipphNrP5pdOyZlVBiSMlS1pyFz9PZHRSOtJFNjOiJqRXvZm4n9eNzXhjZ9xmaQGJVssClNBTExmb5MBV8iMmFhCmeL2VsJGVFFmbDglG4K3/PIqadWq3lXVvb+s1Gt5HEU4gVM4Bw+uoQ530IAmMAjhGV7hzRk7L86787FoLTj5zDH8gfP5Ay9DjRQ=</latexit>+=+e<latexit sha1_base64="SWue4KGKpwZI7TnixNGYhX2IkNw=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBEEpSFD0WvHisYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjlo5TxbDJYhGrTkA1Ci6xabgR2EkU0igQ2A7GtzO//YRK81g+mkmCfkSHkoecUWOlhwvslytu1Z2DrBIvJxXI0eiXv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtVTSCLWfzS+dkjOrDEgYK1vSkLn6eyKjkdaTKLCdETUjvezNxP+8bmrCGz/jMkkNSrZYFKaCmJjM3iYDrpAZMbGEMsXtrYSNqKLM2HBKNgRv+eVV0qpVvauqe39ZqdfyOIpwAqdwDh5cQx3uoAFNYBDCM7zCmzN2Xpx352PRWnDymWP4A+fzByw5jRI=</latexit>+e<latexit sha1_base64="SWue4KGKpwZI7TnixNGYhX2IkNw=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBEEpSFD0WvHisYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjlo5TxbDJYhGrTkA1Ci6xabgR2EkU0igQ2A7GtzO//YRK81g+mkmCfkSHkoecUWOlhwvslytu1Z2DrBIvJxXI0eiXv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtVTSCLWfzS+dkjOrDEgYK1vSkLn6eyKjkdaTKLCdETUjvezNxP+8bmrCGz/jMkkNSrZYFKaCmJjM3iYDrpAZMbGEMsXtrYSNqKLM2HBKNgRv+eVV0qpVvauqe39ZqdfyOIpwAqdwDh5cQx3uoAFNYBDCM7zCmzN2Xpx352PRWnDymWP4A+fzByw5jRI=</latexit>25“Filling anomaly”• Sometimes charge neutrality and point-group symmetry
cannot be simultaneously respected.
<latexit sha1_base64="GWjewyERa53FsQpgIukf7NpORTg=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBiyURxR4LXjxWsR/QhrLZTtqlm03Y3Qgl9B948aCIV/+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzSRBP6JDyUPOqLHSwwX2yxW36s5BVomXkwrkaPTLX71BzNIIpWGCat313MT4GVWGM4HTUi/VmFA2pkPsWipphNrP5pdOyZlVBiSMlS1pyFz9PZHRSOtJFNjOiJqRXvZm4n9eNzVhzc+4TFKDki0WhakgJiazt8mAK2RGTCyhTHF7K2EjqigzNpySDcFbfnmVtC6r3nXVvb+q1Gt5HEU4gVM4Bw9uoA530IAmMAjhGV7hzRk7L86787FoLTj5zDH8gfP5AzERjRo=</latexit>eW. A. Benalcazar, T. Li, T. L. Hughes,
PRB (2019)+e<latexit sha1_base64="+kQmhFHqx6WICfqKYe2mDqxw48s=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBEEoiij0WvHisYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjlo5TxbDJYhGrTkA1Ci6xabgR2EkU0igQ2A7GtzO//YRK81g+mkmCfkSHkoecUWOlhwvslytu1Z2DrBIvJxXI0eiXv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtVTSCLWfzS+dkjOrDEgYK1vSkLn6eyKjkdaTKLCdETUjvezNxP+8bmrCmp9xmaQGJVssClNBTExmb5MBV8iMmFhCmeL2VsJGVFFmbDglG4K3/PIqaV1Wveuqe39VqdfyOIpwAqdwDh7cQB3uoAFNYBDCM7zCmzN2Xpx352PRWnDymWP4A+fzBy4HjRg=</latexit>

1
Qedge = Qtot
2<latexit sha1_base64="SqnIWjhzGg19HO9J3wgzVTpULY8=">AAACL3icbZDLSgMxFIYz3q23qks3wSK4qjOi6EYQ3LhUsBfolJLJnKnBZDIkZ8QyzGP4IK7d6jOIG3HbtzCtRbz9EPj5/3MI54syKSz6/qs3NT0zOze/sFhZWl5ZXauubzStzg2HBtdSm3bELEiRQgMFSmhnBpiKJLSim7NR37oFY4VOr3CQQVexfioSwRm6qFfdu+wVIcIdFhD3oSxPwsQwXgRlsV9+VaixLEOlYwq9as2v+2PRvyaYmBqZ6KJXHYax5rmCFLlk1nYCP8NuwQwKLqGshLmFjPEb1oeOsylTYLvF+LCS7rgkpok27qVIx+n3jYIpawcqcpOK4bX93Y3Cf7tI/Rd3ckyOu4VIsxwh5Z//J7mkqOkIHo2FAY5y4AzjRrgTKL9mDhg6xBXHJvhN4q9p7teDw7p/eVA7PZ5QWiBbZJvskoAckVNyTi5Ig3ByTx7JE3n2HrwX7817/xyd8iY7m+SHvOEHpqGqxA==</latexit>

Note:
1. This formula assumes charges are localized to edges.
2. Termination has to be designed carefully.

mod e

25

“Filling anomaly”• Sometimes charge neutrality and point-group symmetry
cannot be simultaneously respected.
<latexit sha1_base64="GWjewyERa53FsQpgIukf7NpORTg=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBiyURxR4LXjxWsR/QhrLZTtqlm03Y3Qgl9B948aCIV/+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzSRBP6JDyUPOqLHSwwX2yxW36s5BVomXkwrkaPTLX71BzNIIpWGCat313MT4GVWGM4HTUi/VmFA2pkPsWipphNrP5pdOyZlVBiSMlS1pyFz9PZHRSOtJFNjOiJqRXvZm4n9eNzVhzc+4TFKDki0WhakgJiazt8mAK2RGTCyhTHF7K2EjqigzNpySDcFbfnmVtC6r3nXVvb+q1Gt5HEU4gVM4Bw9uoA530IAmMAjhGV7hzRk7L86787FoLTj5zDH8gfP5AzERjRo=</latexit>eW. A. Benalcazar, T. Li, T. L. Hughes,
PRB (2019)+e<latexit sha1_base64="+kQmhFHqx6WICfqKYe2mDqxw48s=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBEEoiij0WvHisYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjlo5TxbDJYhGrTkA1Ci6xabgR2EkU0igQ2A7GtzO//YRK81g+mkmCfkSHkoecUWOlhwvslytu1Z2DrBIvJxXI0eiXv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtVTSCLWfzS+dkjOrDEgYK1vSkLn6eyKjkdaTKLCdETUjvezNxP+8bmrCmp9xmaQGJVssClNBTExmb5MBV8iMmFhCmeL2VsJGVFFmbDglG4K3/PIqaV1Wveuqe39VqdfyOIpwAqdwDh7cQB3uoAFNYBDCM7zCmzN2Xpx352PRWnDymWP4A+fzBy4HjRg=</latexit>

1
Qedge = Qtot
2<latexit sha1_base64="SqnIWjhzGg19HO9J3wgzVTpULY8=">AAACL3icbZDLSgMxFIYz3q23qks3wSK4qjOi6EYQ3LhUsBfolJLJnKnBZDIkZ8QyzGP4IK7d6jOIG3HbtzCtRbz9EPj5/3MI54syKSz6/qs3NT0zOze/sFhZWl5ZXauubzStzg2HBtdSm3bELEiRQgMFSmhnBpiKJLSim7NR37oFY4VOr3CQQVexfioSwRm6qFfdu+wVIcIdFhD3oSxPwsQwXgRlsV9+VaixLEOlYwq9as2v+2PRvyaYmBqZ6KJXHYax5rmCFLlk1nYCP8NuwQwKLqGshLmFjPEb1oeOsylTYLvF+LCS7rgkpok27qVIx+n3jYIpawcqcpOK4bX93Y3Cf7tI/Rd3ckyOu4VIsxwh5Z//J7mkqOkIHo2FAY5y4AzjRrgTKL9mDhg6xBXHJvhN4q9p7teDw7p/eVA7PZ5QWiBbZJvskoAckVNyTi5Ig3ByTx7JE3n2HrwX7817/xyd8iY7m+SHvOEHpqGqxA==</latexit>

mod e

Note:
1. This formula assumes charges are localized to edges.
2. Termination has to be designed carefully.
Qtot = 0ff

Corner charge is a local property.
Not a ected by the other edge

26

De nition of the edge charge• Conserved charge of the U(1) symmetry:<latexit sha1_base64="ZhkVTgfEBNEXbQdnbI4SV8mumIA=">AAACBXicbVDLSsNAFJ34rPUVdamLwSLUTUlEsRuh4MZlC/YBTSiTyaQdOsmEmYm0hGzc+CtuXCji1n9w5984TbPQ1gMDZ865l3vv8WJGpbKsb2NldW19Y7O0Vd7e2d3bNw8OO5InApM25oyLnockYTQibUUVI71YEBR6jHS98e3M7z4QISmP7tU0Jm6IhhENKEZKSwPzxBkhlbayG4dGCvqT/OuIEc+qk/OBWbFqVg64TOyCVECB5sD8cnyOk5BECjMkZd+2YuWmSCiKGcnKTiJJjPAYDUlf0wiFRLppfkUGz7Tiw4AL/fQuufq7I0WhlNPQ05UhUiO56M3E/7x+ooK6m9IoThSJ8HxQkDCoOJxFAn0qCFZsqgnCgupdIR4hgbDSwZV1CPbiycukc1Gzr2pW67LSqBdxlMAxOAVVYINr0AB3oAnaAINH8AxewZvxZLwY78bHvHTFKHqOwB8Ynz9wBJh+</latexit>

Q̂ =

Z

<latexit sha1_base64="ApLAgJ+JeVebO1jr9lNqnBV0ovs=">AAACHnicbVBNSwMxFMz6bf2qevQSLEJ7KbtisRdB8OJRwarQrSWbvrbBbLIkb6Vl6S/x4l/x4kERwZP+G7O1B7UOBIaZeby8iRIpLPr+pzczOze/sLi0XFhZXVvfKG5uXVqdGg4NrqU21xGzIIWCBgqUcJ0YYHEk4Sq6Pcn9qzswVmh1gcMEWjHrKdEVnKGT2sVaaPr6JgsRBpiVY8GNroxG5UHlKJRM9STQsM8wy1O5Gpqx2C6W/Ko/Bp0mwYSUyARn7eJ72NE8jUEhl8zaZuAn2MqYQcEljAphaiFh/Jb1oOmoYjHYVjY+b0T3nNKhXW3cU0jH6s+JjMXWDuPIJWOGffvXy8X/vGaK3XorEypJERT/XtRNJUVN865oRxjgKIeOMG6E+yvlfWYYR9dowZUQ/D15mlzuV4Na1T8/KB3XJ3UskR2yS8okIIfkmJySM9IgnNyTR/JMXrwH78l79d6+ozPeZGab/IL38QUHuKL/</latexit>⇢(micro) (x) = hˆ
⇢(x)i

dxˆ
⇢(x)

• Microscopic charge density:x<latexit sha1_base64="nlOYUdOcLB+k6EYW3cewfF0yWss=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Koko9ljw4rEF+wFtKJvtpF272YTdjVhCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8O/Pbj6g0j+W9mSToR3QoecgZNVZqPPVLZbfizkFWiZeTMuSo90tfvUHM0gilYYJq3fXcxPgZVYYzgdNiL9WYUDamQ+xaKmmE2s/mh07JuVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NWHVz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m6INwVt+eZW0LivedcVtXJVr1TyOApzCGVyABzdQgzuoQxMYIDzDK7w5D86L8+58LFrXnHzmBP7A+fwB5ImM9g==</latexit><latexit sha1_base64="ApLAgJ+JeVebO1jr9lNqnBV0ovs=">AAACHnicbVBNSwMxFMz6bf2qevQSLEJ7KbtisRdB8OJRwarQrSWbvrbBbLIkb6Vl6S/x4l/x4kERwZP+G7O1B7UOBIaZeby8iRIpLPr+pzczOze/sLi0XFhZXVvfKG5uXVqdGg4NrqU21xGzIIWCBgqUcJ0YYHEk4Sq6Pcn9qzswVmh1gcMEWjHrKdEVnKGT2sVaaPr6JgsRBpiVY8GNroxG5UHlKJRM9STQsM8wy1O5Gpqx2C6W/Ko/Bp0mwYSUyARn7eJ72NE8jUEhl8zaZuAn2MqYQcEljAphaiFh/Jb1oOmoYjHYVjY+b0T3nNKhXW3cU0jH6s+JjMXWDuPIJWOGffvXy8X/vGaK3XorEypJERT/XtRNJUVN865oRxjgKIeOMG6E+yvlfWYYR9dowZUQ/D15mlzuV4Na1T8/KB3XJ3UskR2yS8okIIfkmJySM9IgnNyTR/JMXrwH78l79d6+ozPeZGab/IL38QUHuKL/</latexit>⇢(micro) (x) = hˆ⇢(x)i

• Coarse-grained charge density:⇠a<latexit sha1_base64="cXDKXUtW0baLmw7B7nlz/WONAlA=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKwRwDXjxGMA9IljA7mU3GzGOZmRXCkn/w4kERr/6PN//GSbIHTSxoKKq66e6KEs6M9f1vr7CxubW9U9wt7e0fHB6Vj0/aRqWa0BZRXOluhA3lTNKWZZbTbqIpFhGnnWhyO/c7T1QbpuSDnSY0FHgkWcwItk5q9w0TCA/KFb/qL4DWSZCTCuRoDspf/aEiqaDSEo6N6QV+YsMMa8sIp7NSPzU0wWSCR7TnqMSCmjBbXDtDF04ZolhpV9Kihfp7IsPCmKmIXKfAdmxWvbn4n9dLbVwPMyaT1FJJlovilCOr0Px1NGSaEsunjmCimbsVkTHWmFgXUMmFEKy+vE7aV9WgVvXvryuNeh5HEc7gHC4hgBtowB00oQUEHuEZXuHNU96L9+59LFsLXj5zCn/gff4ALtKO1g==</latexit><latexit sha1_base64="ZNbQdCclvrqYHdvS5tZbPoOuuOA=">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</latexit><latexit sha1_base64="sNV6ySaOXAVB0Zo3bLiouy5jwtc=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0WoC0sSFLsRCm5cVrAPaNIymUzaoZOHMxNpCfkdN/6KGwVF3fojTh/4aD1w4XDOvXfmHjdmVEjDeNdyS8srq2v59cLG5tb2jr671xBRwjGp44hFvOUiQRgNSV1SyUgr5gQFLiNNd3A59pt3hAsahTdyFBMnQL2Q+hQjqaSuXu2VhscXts8RTs0stcUtl6llxxTaTG3xUMfKMtJJT75brB9jqKqrF42yMQFcJOaMFMEMta7+bHsRTgISSsyQEG3TiKWTIi4pZiQr2IkgMcID1CNtRUMUEOGkk0szeKQUD/oRVxVKOFF/T6QoEGIUuKozQLIv5r2x+J/XTqRfcVIaxokkIZ4+5CcMygiOY4Me5QRLNlIEYU7VXyHuIxWJVOEWVAjm/MmLpGGVzbOycX1arFZmceTBATgEJWCCc1AFV6AG6gCDe/AIXsCr9qA9aW/ax7Q1p81m9sEfaJ9fIDOnJQ==</latexit>

g(x) = p

1
2⇡

2

e

1

x
2 2

2

⇢(smooth) (x) =• De nition of edge charge:<latexit sha1_base64="ga31h6OUoa1jgJIamCb7Cyhb1O8=">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</latexit>fi

fi

Qedge ⌘Z W

1

dx0 g(xx0 )⇢(micro) (x)x<latexit sha1_base64="nlOYUdOcLB+k6EYW3cewfF0yWss=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Koko9ljw4rEF+wFtKJvtpF272YTdjVhCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8O/Pbj6g0j+W9mSToR3QoecgZNVZqPPVLZbfizkFWiZeTMuSo90tfvUHM0gilYYJq3fXcxPgZVYYzgdNiL9WYUDamQ+xaKmmE2s/mh07JuVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NWHVz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m6INwVt+eZW0LivedcVtXJVr1TyOApzCGVyABzdQgzuoQxMYIDzDK7w5D86L8+58LFrXnHzmBP7A+fwB5ImM9g==</latexit>⇠<latexit sha1_base64="+s6XH6TKzRy4PGQLDZIRGhOwEHs=">AAAB8nicbVDLSgMxFM3UV62vqks3wSK4KjOi2GXBjcsK9gEzQ8lkMm1oMhmSO0IZ+hluXCji1q9x59+YtrPQ1gOBwznnkntPlAluwHW/ncrG5tb2TnW3trd/cHhUPz7pGZVryrpUCaUHETFM8JR1gYNgg0wzIiPB+tHkbu73n5g2XKWPMM1YKMko5QmnBKzkB4bLQNh4TIb1htt0F8DrxCtJA5XoDOtfQaxoLlkKVBBjfM/NICyIBk4Fm9WC3LCM0AkZMd/SlEhmwmKx8gxfWCXGidL2pYAX6u+JgkhjpjKySUlgbFa9ufif5+eQtMKCp1kOLKXLj5JcYFB4fj+OuWYUxNQSQjW3u2I6JppQsC3VbAne6snrpHfV9G6a7sN1o90q66iiM3SOLpGHblEb3aMO6iKKFHpGr+jNAefFeXc+ltGKU86coj9wPn8AWwiRRA==</latexit>dx ⇢(smooth) (x)Z

27(De nition of the bulk
polarization)• Translation invariance→ a repetition unit of charge density+1X⇢(micro) (x) =⇢0 (x na)<latexit sha1_base64="ApLAgJ+JeVebO1jr9lNqnBV0ovs=">AAACHnicbVBNSwMxFMz6bf2qevQSLEJ7KbtisRdB8OJRwarQrSWbvrbBbLIkb6Vl6S/x4l/x4kERwZP+G7O1B7UOBIaZeby8iRIpLPr+pzczOze/sLi0XFhZXVvfKG5uXVqdGg4NrqU21xGzIIWCBgqUcJ0YYHEk4Sq6Pcn9qzswVmh1gcMEWjHrKdEVnKGT2sVaaPr6JgsRBpiVY8GNroxG5UHlKJRM9STQsM8wy1O5Gpqx2C6W/Ko/Bp0mwYSUyARn7eJ72NE8jUEhl8zaZuAn2MqYQcEljAphaiFh/Jb1oOmoYjHYVjY+b0T3nNKhXW3cU0jH6s+JjMXWDuPIJWOGffvXy8X/vGaK3XorEypJERT/XtRNJUVN865oRxjgKIeOMG6E+yvlfWYYR9dowZUQ/D15mlzuV4Na1T8/KB3XJ3UskR2yS8okIIfkmJySM9IgnNyTR/JMXrwH78l79d6+ozPeZGab/IL38QUHuKL/</latexit>⇢(micro) (x) = hˆ⇢(x)i<latexit sha1_base64="/fINFEmv1e29mI53CDG8oVq5Eyw=">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</latexit>

n= 1x<latexit sha1_base64="nlOYUdOcLB+k6EYW3cewfF0yWss=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Koko9ljw4rEF+wFtKJvtpF272YTdjVhCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8O/Pbj6g0j+W9mSToR3QoecgZNVZqPPVLZbfizkFWiZeTMuSo90tfvUHM0gilYYJq3fXcxPgZVYYzgdNiL9WYUDamQ+xaKmmE2s/mh07JuVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NWHVz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m6INwVt+eZW0LivedcVtXJVr1TyOApzCGVyABzdQgzuoQxMYIDzDK7w5D86L8+58LFrXnHzmBP7A+fwB5ImM9g==</latexit>⇢0 (x) ! 0<latexit sha1_base64="BbSBjDwpiXZXiwYCEUaBrZBpV0U=">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</latexit><latexit sha1_base64="P4iqPRKd/V+qEUwImrvnVOV881A=">AAACB3icbZDLSgMxGIX/qbdab1WXboJFqJsyI4pdFty4rGAv0g4lk2ba0GQyJBmxDH0A1271GdyJWx/DR/AtTNtZaNsDgY9z/p8kJ4g508Z1v53c2vrG5lZ+u7Czu7d/UDw8amqZKEIbRHKp2gHWlLOINgwznLZjRbEIOG0Fo5tp3nqkSjMZ3ZtxTH2BBxELGcHGWg9dNZQ9t/x03iuW3Io7E1oGL4MSZKr3ij/dviSJoJEhHGvd8dzY+ClWhhFOJ4VuommMyQgPaMdihAXVfjp78ASdWaePQqnsiQyauX83Uiy0HovATgpshnoxm5ors0CssjuJCat+yqI4MTQi8/vDhCMj0bQU1GeKEsPHFjBRzH4BkSFWmBhbXcF24y02sQzNi4p3VXHvLku1atZSHk7gFMrgwTXU4Bbq0AACAl7gFd6cZ+fd+XA+56M5J9s5hn9yvn4BHKaY/Q==</latexit>⇢0 (x) is not unique.

We require{<latexit sha1_base64="zkL+c3CS7YXS/yQAjPO1C7YuPu0=">AAACAHicbZDLSgMxGIX/eK31VnXpJlgEV2VGFLssuHFZxV6gHUomzbShSWZIMkIZunHtVp/Bnbj1TXwE38K0nYW2PRD4OOf/SXLCRHBjPe8bra1vbG5tF3aKu3v7B4elo+OmiVNNWYPGItbtkBgmuGINy61g7UQzIkPBWuHodpq3npg2PFaPdpywQJKB4hGnxDrroZv1SmWv4s2El8HPoQy56r3ST7cf01QyZakgxnR8L7FBRrTlVLBJsZsalhA6IgPWcaiIZCbIZi+d4HPn9HEUa3eUxTP370ZGpDFjGbpJSezQLGZTc2UWylV2J7VRNci4SlLLFJ3fH6UC2xhP28B9rhm1YuyAUM3dFzAdEk2odZ0VXTf+YhPL0Lys+NcV7/6qXKvmLRXgFM7gAny4gRrcQR0aQCGCF3iFN/SM3tEH+pyPrqF85wT+CX39ApkzlpE=</latexit>

when |x|a• De nition of bulk polarization<latexit sha1_base64="J2bweqgJ/VhN1gKyt1V/W48FvqU=">AAAB/3icbZDNTgIxFIXv4B/iH+rSTSMxYUVmjEaWJG5cQiI/CUxIp1ygoZ2ZtB0TMmHh2q0+gzvj1kfxEXwLC8xCgZM0+XLOvWl7glhwbVz328ltbe/s7uX3CweHR8cnxdOzlo4SxbDJIhGpTkA1Ch5i03AjsBMrpDIQ2A4m9/O8/YRK8yh8NNMYfUlHIR9yRo21GuV+seRW3IXIOngZlCBTvV/86Q0ilkgMDRNU667nxsZPqTKcCZwVeonGmLIJHWHXYkglaj9dPHRGrqwzIMNI2RMasnD/bqRUaj2VgZ2U1Iz1ajY3N2aB3GR3EzOs+ikP48RgyJb3DxNBTETmZZABV8iMmFqgTHH7BcLGVFFmbGUF24232sQ6tK4r3m3FbdyUatWspTxcwCWUwYM7qMED1KEJDBBe4BXenGfn3flwPpejOSfbOYd/cr5+AVrYldg=</latexit>

fi

fi

(

x

>tixetal/<=gdlYrVA+5rc/dYObfSOjepPwlf3nfGneXB4eBBDJEK1DEMq7MYwUWCwcxTpsWtaUydbF3m3r4Kt6Qs23242FUGbmFFVGLcB7HHTgqFmMi8VBAZZmTETBNxD3bJygR84Pki+sOzE3RG3Ba2N3Yja1zI1U2ZgV2jaURqb/DnsaMR2INMIzwqrGRHPd9jalgkYXHWHJILmGnoeVwZCcKTqPZsxn766UNRDMgkli0Q68/VvTBClZgnOIXI3WRes+VuG12oRy9RIHlUfYMNN8hy8KRY/8O/9m4A2QIDprMBsjA30i5hC1AkTpGhIJDbxS4olzOdxnc8RHewC3Xu7s/ebtl823zVbwhlg7lWvOLX+0MZgCx8CLwXExfk1jvzg+0q2hHmMT0BtZ2Zogy1pIxUC/IiQc5GJWaEjmVUYxMSTSr+Hi/B4vXIFxIgTNDZbci3/BAAA>"=UqvF84W/V1tyKg1NhV/Jgqewb2J"=46esab_1ahs tixetal<(x<latexit sha1_base64="nlOYUdOcLB+k6EYW3cewfF0yWss=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Koko9ljw4rEF+wFtKJvtpF272YTdjVhCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8O/Pbj6g0j+W9mSToR3QoecgZNVZqPPVLZbfizkFWiZeTMuSo90tfvUHM0gilYYJq3fXcxPgZVYYzgdNiL9WYUDamQ+xaKmmE2s/mh07JuVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NWHVz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m6INwVt+eZW0LivedcVtXJVr1TyOApzCGVyABzdQgzuoQxMYIDzDK7w5D86L8+58LFrXnHzmBP7A+fwB5ImM9g==</latexit>⇠a<latexit sha1_base64="cXDKXUtW0baLmw7B7nlz/WONAlA=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKwRwDXjxGMA9IljA7mU3GzGOZmRXCkn/w4kERr/6PN//GSbIHTSxoKKq66e6KEs6M9f1vr7CxubW9U9wt7e0fHB6Vj0/aRqWa0BZRXOluhA3lTNKWZZbTbqIpFhGnnWhyO/c7T1QbpuSDnSY0FHgkWcwItk5q9w0TCA/KFb/qL4DWSZCTCuRoDspf/aEiqaDSEo6N6QV+YsMMa8sIp7NSPzU0wWSCR7TnqMSCmjBbXDtDF04ZolhpV9Kihfp7IsPCmKmIXKfAdmxWvbn4n9dLbVwPMyaT1FJJlovilCOr0Px1NGSaEsunjmCimbsVkTHWmFgXUMmFEKy+vE7aV9WgVvXvryuNeh5HEc7gHC4hgBtowB00oQUEHuEZXuHNU96L9+59LFsLXj5zCn/gff4ALtKO1g==</latexit><latexit sha1_base64="nlOYUdOcLB+k6EYW3cewfF0yWss=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Koko9ljw4rEF+wFtKJvtpF272YTdjVhCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8O/Pbj6g0j+W9mSToR3QoecgZNVZqPPVLZbfizkFWiZeTMuSo90tfvUHM0gilYYJq3fXcxPgZVYYzgdNiL9WYUDamQ+xaKmmE2s/mh07JuVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NWHVz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m6INwVt+eZW0LivedcVtXJVr1TyOApzCGVyABzdQgzuoQxMYIDzDK7w5D86L8+58LFrXnHzmBP7A+fwB5ImM9g==</latexit>⇠<latexit sha1_base64="+s6XH6TKzRy4PGQLDZIRGhOwEHs=">AAAB8nicbVDLSgMxFM3UV62vqks3wSK4KjOi2GXBjcsK9gEzQ8lkMm1oMhmSO0IZ+hluXCji1q9x59+YtrPQ1gOBwznnkntPlAluwHW/ncrG5tb2TnW3trd/cHhUPz7pGZVryrpUCaUHETFM8JR1gYNgg0wzIiPB+tHkbu73n5g2XKWPMM1YKMko5QmnBKzkB4bLQNh4TIb1htt0F8DrxCtJA5XoDOtfQaxoLlkKVBBjfM/NICyIBk4Fm9WC3LCM0AkZMd/SlEhmwmKx8gxfWCXGidL2pYAX6u+JgkhjpjKySUlgbFa9ufif5+eQtMKCp1kOLKXLj5JcYFB4fj+OuWYUxNQSQjW3u2I6JppQsC3VbAne6snrpHfV9G6a7sN1o90q66iiM3SOLpGHblEb3aMO6iKKFHpGr+jNAefFeXc+ltGKU86coj9wPn8AWwiRRA==</latexit>

28(De nition of the bulk
polarization)• For band insulators, ρ0(x) is given by Wannier function<latexit sha1_base64="ApLAgJ+JeVebO1jr9lNqnBV0ovs=">AAACHnicbVBNSwMxFMz6bf2qevQSLEJ7KbtisRdB8OJRwarQrSWbvrbBbLIkb6Vl6S/x4l/x4kERwZP+G7O1B7UOBIaZeby8iRIpLPr+pzczOze/sLi0XFhZXVvfKG5uXVqdGg4NrqU21xGzIIWCBgqUcJ0YYHEk4Sq6Pcn9qzswVmh1gcMEWjHrKdEVnKGT2sVaaPr6JgsRBpiVY8GNroxG5UHlKJRM9STQsM8wy1O5Gpqx2C6W/Ko/Bp0mwYSUyARn7eJ72NE8jUEhl8zaZuAn2MqYQcEljAphaiFh/Jb1oOmoYjHYVjY+b0T3nNKhXW3cU0jH6s+JjMXWDuPIJWOGffvXy8X/vGaK3XorEypJERT/XtRNJUVN865oRxjgKIeOMG6E+yvlfWYYR9dowZUQ/D15mlzuV4Na1T8/KB3XJ3UskR2yS8okIIfkmJySM9IgnNyTR/JMXrwH78l79d6+ozPeZGab/IL38QUHuKL/</latexit><latexit sha1_base64="iUZblGsJMqCUHaUw571GdDLUGVo=">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</latexit>(el)⇢0 (x) =ea X X|wn,i (R)|2 (xL
n2occ⇢(micro) (x) = hˆ
⇢(x)i
R

xi )

R

x
<latexit sha1_base64="nlOYUdOcLB+k6EYW3cewfF0yWss=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Koko9ljw4rEF+wFtKJvtpF272YTdjVhCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8O/Pbj6g0j+W9mSToR3QoecgZNVZqPPVLZbfizkFWiZeTMuSo90tfvUHM0gilYYJq3fXcxPgZVYYzgdNiL9WYUDamQ+xaKmmE2s/mh07JuVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NWHVz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m6INwVt+eZW0LivedcVtXJVr1TyOApzCGVyABzdQgzuoQxMYIDzDK7w5D86L8+58LFrXnHzmBP7A+fwB5ImM9g==</latexit>• Our expression of Px reduces to the Berry phase formula<latexit sha1_base64="tfzXEYaT9KCJstJNKXbvqgMbeq4=">AAACIXicbVDLSsNAFJ34tr6iLt0MFqGClkQUuxTcuKxgW6GJYTKd2MHJJM7cSErIr7jxV9y4UMSd+DNOHwutHhju4Zx7uXNPmAquwXE+rZnZufmFxaXlysrq2vqGvbnV1kmmKGvRRCTqOiSaCS5ZCzgIdp0qRuJQsE54dz70Ow9MaZ7IKxikzI/JreQRpwSMFNiNZpB77D7jD16kCC3csiClxyUExaEpEQzKm3HFvdw7yD3VTwKnlu8HdtWpOyPgv8SdkCqaoBnYH14voVnMJFBBtO66Tgp+QRRwKlhZ8TLNUkLvyC3rGipJzLRfjC4s8Z5RejhKlHkS8Ej9OVGQWOtBHJrOmEBfT3tD8T+vm0HU8Asu0wyYpONFUSYwJHgYF+5xxSgIcz0nVHHzV0z7xCQFJtSKCcGdPvkvaR/V3ZO6c3lcPWtM4lhCO2gX1ZCLTtEZukBN1EIUPaJn9IrerCfrxXq3PsatM9ZkZhv9gvX1DfaXpIo=</latexit>

1
Px ⌘
a
<latexit sha1_base64="w0SLASdT7H1ae43fGZ9/l56Ecj8=">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</latexit>

Z 1

fi

Px = hP̂x i

dx x⇢0 (x)1e

X Z 2⇡
a dk

n2occ

0

2⇡

u⇤nk i@k unk

29(Proof of the bulk-edge correspondence)
In nite system<latexit sha1_base64="/fINFEmv1e29mI53CDG8oVq5Eyw=">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</latexit>⇢(micro) (x) =Semi-in nite system
+1
X

fi

fi

n= 1

⇢0 (x

na)

Higher order multipolesin higher D systems

2D: Quadrupole moment andcorner charge• Charge density<latexit sha1_base64="v2UJAKoumo3w5QiEbekJ4xBIkW0=">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</latexit>⇢(micro)

(r) =

X

ni 0

⇢0 (r

in the region R• The total chargeZPd

i=1 ni ai ).

<latexit sha1_base64="9dLUKACsd0dtRRvvA/HGBPzVqNw=">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</latexit>

dd r⇢(smooth) (r) = W1 2 + W2 1 + Qcorner

QR =

R

• Surface charge density<latexit sha1_base64="/mkMjMUIq7GPGkaLItfhiPIXyjA=">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</latexit>

i =

Z

Rd

dd r⇢0 (r)Pi (r)Pi (r) ⌘ bi · r• Corner charge<latexit sha1_base64="JCGJp1VJucMfyKETGSKjGg665MY=">AAADQnicbVLPb9MwFHbCr1F+dXDkYlExdRpUSQRiF6RJXDhuiG6T6iZyHKe1ljjBfplaRf7POPIvcODKEbggrhxw0kxqO55k6fu+5+89+9lxmQkNnvfNcW/cvHX7zs7d3r37Dx4+6u8+PtVFpRgfsyIr1HlMNc+E5GMQkPHzUnGaxxk/iy/eNfmzS660KORHWJZ8mtOZFKlgFKwU7Trjk6gmwBdQs0JJrozZe0uEBKvmFOZxXH8wYWJwEiaKqHkReUMS57Uy+9g6/cBcUUJ6m8Ie4Z8qcdnS2EQ+YUkBXe5KDNbFA5IqyuqG0fWUJb6pAxoFYVd9u1wYbHr9dW/Qev01b7DpJbIAOrPntzdvq9hu1mD7GZLxFIYrdREGL5dhU80QVmgCcw70AC/wkmghV5QoMZvD/ijqD7yR1wa+DvwODFAXx1H/K0kKVuVcAsuo1hPfK2FaUwWCZdz0SKV5SdkFnfGJhZLmXE/r9v0Nfm6VBKeFsksCbtV1R01zrZd5/CLO7ebmWfV2uhH/l5tUkB5OayHLCrhkq15plWEocPOfcCIUZ5AtLaBMCXtczObUzgvsr+vZOfjbt74OToOR/3rknbwaHB12E9lBT9EzNEQ+eoOO0Ht0jMaIOZ+d785P55f7xf3h/nb/rLa6Tud5gjbC/fsPBDoQcA==</latexit>

Qcorner =ZRdai · bj =<latexit sha1_base64="7f6rpYQsy/zarKVRlhrW9irY+VU=">AAACDXicbVDLSsNAFJ34rPUVdekmWgQXUhJR7EYouHFZwT6gKWEymbbTziRh5kYoIWuXfolLdSNu/QIX/o2TNgttPXDhzDn3MvceP+ZMgW1/G0vLK6tr66WN8ubW9s6uubffUlEiCW2SiEey42NFOQtpExhw2oklxcLntO2Pb3K//UClYlF4D5OY9gQehKzPCAYteeaR64sUZx5zSRBB/vAzb3TtBpQD9lI2yjyzYlftKaxF4hSkggo0PPPLDSKSCBoC4ViprmPH0EuxBEY4zcpuomiMyRgPaFfTEAuqeun0lMw60Upg9SOpKwRrqv6eSLFQaiL8M1/oZoFhqObtXPzP6ybQr/VSFsYJ0JDM/uon3ILIyqOxAiYpAT7RBBPJ9LoWGWKJCegAyzoHZ/7qRdI6rzqXVfvuolKvFYmU0CE6RqfIQVeojm5RAzURQY/oGb2iN+PJeDHejY9Z65JRzBygPzA+fwCGs5xp</latexit>ij

dd r⇢0 (r)Q12 (r)a2 · a1
a1 · a222
Q12 (r) ⌘ (b1 · r)(b2 · r) +
(b
·
r)
+
(b
·
r)
1
2
2a22
2a21
✓ 2
◆
2
1
x
y
a1 = a1 (1, 0)=
cos ✓ + xy sin ✓ .
a1 a2
2a2 = a2 (cos ✓, sin ✓)<latexit sha1_base64="7Ue3pVkGjQPspnuqYDuFsCvoaHA=">AAAB/XicbVDLSgMxFM3UV62vqS7dBItQoZSJKHYjFNy4rGAf0A5DJk3b0CQzJBmlDMUvcaluxK1f4sK/MW1noa0HLhzOuZd77wljzrTxvG8nt7a+sbmV3y7s7O7tH7jFw5aOEkVok0Q8Up0Qa8qZpE3DDKedWFEsQk7b4fhm5rcfqNIskvdmElNf4KFkA0awsVLgFnuhSPE0QNc4QGVU8c4Ct+RVvTngKkEZKYEMjcD96vUjkggqDeFY6y7yYuOnWBlGOJ0WeommMSZjPKRdSyUWVPvp/PQpPLVKHw4iZUsaOFd/T6RYaD0RYSUUtllgM9LL9kz8z+smZlDzUybjxFBJFrsGCYcmgrMoYJ8pSgyfWIKJYvZcSEZYYWJsYAWbA1r+epW0zqvosurdXZTqtSyRPDgGJ6AMELgCdXALGqAJCHgEz+AVvDlPzovz7nwsWnNONnME/sD5/AHfsJN8</latexit><latexit sha1_base64="ZYjfqUYsgJZcSoLBatazfkUmruc=">AAACD3icbVDLSgMxFM3Ud31VXboZLEKFUmaKohtBcONSwWqhMwx30rQNTTJDckcoQz/ApV/iUt2IWz/AhX9j+lho64HA4ZxzubknTgU36HnfTmFhcWl5ZXWtuL6xubVd2tm9M0mmKWvQRCS6GYNhgivWQI6CNVPNQMaC3cf9y5F//8C04Ym6xUHKQgldxTucAlopKpWDWOYwjOrnENUrAU1MgD2GUA0MVxN6ZFNezRvDnSf+lJTJFNdR6StoJzSTTCEVYEzL91IMc9DIqWDDYpAZlgLtQ5e1LFUgmQnz8TFD99AqbbeTaPsUumP190QO0piBjKuxtGEJ2DOz9kj8z2tl2DkLc67SDJmik12dTLiYuKNy3DbXjKIYWAJUc/tdl/ZAA0VbYdH24M9ePU/u6jX/pObdHJcvzqaNrJJ9ckAqxCen5IJckWvSIJQ8kmfySt6cJ+fFeXc+JtGCM53ZI3/gfP4AEPeb/A==</latexit>31

3D: Octupole moment andcorner/hinge charge• Charge density<latexit sha1_base64="v2UJAKoumo3w5QiEbekJ4xBIkW0=">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</latexit>⇢(micro)

(r) =

X

⇢0 (r

ni 0

Pd

i=1 ni ai ).

• The total
Z charge in the region3R<latexit sha1_base64="QLVIaCqwhA2J4OyeiDcm73veGOo=">AAACXXicbVHLahsxFNVMmpebJm6z6KKboabg0GJm+iDZGALddBmndRzw2EKjkW0RPabSnVAj9JkhdNUvKUR2ZpEmPSA4nHPv1dVRUQluIU1/R/HGs82t7Z3d1vO9F/sH7ZevLqyuDWVDqoU2lwWxTHDFhsBBsMvKMCILwUbF1deVP7pmxnKtfsCyYhNJ5orPOCUQJNz+OcDn/ZwrwO7cl9PS5Gahpy4H9gtc10qtYXHkfTcvpDP+qJ/bWmLH+5mffvqOeW75XBLM3z/UR0EXYYVyZQxwM4xqo5jxHrc7aS9dI3lKsoZ0UIMz3L7JS01ryRRQQawdZ2kFE0cMcCqYb+W1ZRWhV2TOxoEqIpmduHU0PnkXlDKZaROOgmStPuxwRFq7lMWHQoZiSWBhH9sr8X/euIbZycRxVdXAFL2/a1aLBHSyijopuWEUxDIQQg0P6yZ0QQyhED6kFXLIHr/6Kbn42Mu+9NLB587pSZPIDnqD3qIuytAxOkXf0BkaIopu0d9oK9qO/sSb8V68f18aR03PIfoH8es7hw233w==</latexit>

dd r⇢(smooth) (r) =

QR =

R

• Hinge charge density<latexit sha1_base64="hCZ3UfatgKaMpIuLlzVx1+EnUQs=">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</latexit>

3 =

Z

Rd

• Corner charge
Qcorner =Z

i=1

Si i +

3
X

Wi i + Qcorneri=1

dd r⇢0 (r)Q12 (r)Q12 (r) ⌘ (b1 · r)(b2 · r) +<latexit sha1_base64="W7AvoEZjwwatKqwx9XLJLV5rJII=">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</latexit>X

a1 · b1 ⇥ a3
a2 · b2 ⇥ a3
(b1 · r)2 +(b2 · r)22a2 · b1 ⇥ a3
2a1 · b2 ⇥ a3
<latexit sha1_base64="YD1pvj9Bac1IPM7EjJq226B5WIg=">AAACHHicbVDLSsNAFJ34rPUVdekmWAQXWpKi6LLgxmUF+4Amhslk0k47k8SZSaEM+Q2XfolLdSMuFfwbp20W2nrgwplz7mXuPUFKiZC2/W0sLa+srq2XNsqbW9s7u+befkskGUe4iRKa8E4ABaYkxk1JJMWdlGPIAorbwfB64rdHmAuSxHdynGKPwV5MIoKg1JJv2shXZJC7+CEjozM34hApN2AK5j5xUZjI4jHIFfTJfS33zYpdtaewFolTkAoo0PDNTzdMUMZwLBGFQnQdO5WeglwSRHFedjOBU4iGsIe7msaQYeGp6WW5dayV0IoSriuW1lT9PaEgE2LMgtOA6WYGZV/M2xPxP6+byejKUyROM4ljNPsryqglE2uSlBUSjpGkY00g4kSva6E+1AFJnWdZ5+DMX71IWrWqc1G1b88r9VqRSAkcgiNwAhxwCergBjRAEyDwCJ7BK3gznowX4934mLUuGcXMAfgD4+sH8l6jDA==</latexit>

Rd

dd r⇢0 (r)Q123 (r)cij ⌘ai · aja2i

ri ⌘ b i · r<latexit sha1_base64="bTlfdRriR4Zn8Rq77H/QfFGBzXo=">AAACCHicbVC7TsMwFHXKq5RXgBGGiAqJAVVJBYKxEgtjkehDaqLIcZzWqu0E26lURVkY+RJGYEGs/AMDf4PTZoCWI1k695x7dX1PkFAilW1/G5WV1bX1jepmbWt7Z3fP3D/oyjgVCHdQTGPRD6DElHDcUURR3E8EhiyguBeMbwq/N8FCkpjfq2mCPQaHnEQEQaUl3zwWPnHxQ0ombsCyINcVCmNVFCL3zbrdsGewlolTkjoo0fbNLzeMUcowV4hCKQeOnSgvg0IRRHFec1OJE4jGcIgHmnLIsPSy2RW5daqV0IpioR9X1kz9PZFBJuWUBecB080MqpFctAvxP2+QqujaywhPUoU5mu+KUmqp2CpSsUIiMFJ0qglEgujvWmgEBURKZ1fTOTiLVy+TbrPhXDbsu4t6q1kmUgVH4AScAQdcgRa4BW3QAQg8gmfwCt6MJ+PFeDc+5q0Vo5w5BH9gfP4A3F+abA==</latexit>

111(c23 r3 2 + c32 r2 2 )r1(c31 r1 2 + c13 r3 2 )r2(c12 r2 2 + c21 r1 2 )r32
22
c23 c31 2 + 2c21 c31 + c32 c21 2 3 c31 c12 2 + 2c32 c12 + c13 c32 2 3 c12 c23 2 + 2c13 c23 + c21 c13 2 3
+
r1 +
r2 +
r3
6(1 c23 c32 )6(1 c31 c13 )6(1 c12 c21 )O123 (r) = r1 r2 r3

32

33

(Ambiguities inbulk multipole moment)•<latexit sha1_base64="P4iqPRKd/V+qEUwImrvnVOV881A=">AAACB3icbZDLSgMxGIX/qbdab1WXboJFqJsyI4pdFty4rGAv0g4lk2ba0GQyJBmxDH0A1271GdyJWx/DR/AtTNtZaNsDgY9z/p8kJ4g508Z1v53c2vrG5lZ+u7Czu7d/UDw8amqZKEIbRHKp2gHWlLOINgwznLZjRbEIOG0Fo5tp3nqkSjMZ3ZtxTH2BBxELGcHGWg9dNZQ9t/x03iuW3Io7E1oGL4MSZKr3ij/dviSJoJEhHGvd8dzY+ClWhhFOJ4VuommMyQgPaMdihAXVfjp78ASdWaePQqnsiQyauX83Uiy0HovATgpshnoxm5ors0CssjuJCat+yqI4MTQi8/vDhCMj0bQU1GeKEsPHFjBRzH4BkSFWmBhbXcF24y02sQzNi4p3VXHvLku1atZSHk7gFMrgwTXU4Bbq0AACAl7gFd6cZ+fd+XA+56M5J9s5hn9yvn4BHKaY/Q==</latexit>⇢0 (x) is not unique!
<latexit sha1_base64="fQlfYEX8K0gV2y4L+baqub8qJV0=">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</latexit>(micro)⇢(r) ⌘ hˆ⇢(r)i =

<latexit sha1_base64="iUZblGsJMqCUHaUw571GdDLUGVo=">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</latexit>(el)⇢0 (x) =Z<latexit sha1_base64="ML4qaB+cQhjLHoGsbETLMWUXOA4=">AAACFHicbVDLSgMxFM34rPVVdekmWIQKUmZEsRuh4MZlFfuATjskmbQNTTJDkhHKML/g0i9xqW7ErTsX/o3pY6GtBy4czrmXe+/BMWfauO63s7S8srq2ntvIb25t7+wW9vYbOkoUoXUS8Ui1MNKUM0nrhhlOW7GiSGBOm3h4PfabD1RpFsl7M4ppR6C+ZD1GkLFSUCj5TJog9QUyA4zTu6wbZjDshspXgyhwSz4WqcpOrtygUHTL7gRwkXgzUgQz1ILClx9GJBFUGsKR1m3PjU0nRcowwmmW9xNNY0SGqE/blkokqO6kk48yeGyVEPYiZUsaOFF/T6RIaD0S+BQL2zw+Xs/bY/E/r52YXqWTMhknhkoy3dVLODQRHCcEQ6YoMXxkCSKK2XMhGSCFiLE55m0O3vzXi6RxVvYuyu7tebFamSWSA4fgCJSABy5BFdyAGqgDAh7BM3gFb86T8+K8Ox/T1iVnNnMA/sD5/AFvRJ5V</latexit>{<latexit sha1_base64="zkL+c3CS7YXS/yQAjPO1C7YuPu0=">AAACAHicbZDLSgMxGIX/eK31VnXpJlgEV2VGFLssuHFZxV6gHUomzbShSWZIMkIZunHtVp/Bnbj1TXwE38K0nYW2PRD4OOf/SXLCRHBjPe8bra1vbG5tF3aKu3v7B4elo+OmiVNNWYPGItbtkBgmuGINy61g7UQzIkPBWuHodpq3npg2PFaPdpywQJKB4hGnxDrroZv1SmWv4s2El8HPoQy56r3ST7cf01QyZakgxnR8L7FBRrTlVLBJsZsalhA6IgPWcaiIZCbIZi+d4HPn9HEUa3eUxTP370ZGpDFjGbpJSezQLGZTc2UWylV2J7VRNci4SlLLFJ3fH6UC2xhP28B9rhm1YuyAUM3dFzAdEk2odZ0VXTf+YhPL0Lys+NcV7/6qXKvmLRXgFM7gAny4gRrcQR0aQCGCF3iFN/SM3tEH+pyPrqF85wT+CX39ApkzlpE=</latexit>

X

Pd

⇢0 (r

i=1 ni ai ).

ni 2Z

ea X X|wn,i (R)|2 (x
L
n2occ

R

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Rd

dd r⇢0 (r) = 0⇢0 (r) ! 0

when |x|xi )R• Wannier center has only integer ambiguity→ fractional part of polarization is well-de ned.
But spread of Wannier function is gauge-dependent.

• Higher moments depend on the speci c choice of ρ0(r)Z<latexit sha1_base64="gaQgNckBi96oN7jpOez9PmTNZvI=">AAACGXicbVDLSgMxFM3UV62vUZdugkWooGVGFF0KblxWsVXojEOSSdvYJDMkGaEM8xUu/RKX6kbcigv/xvSx0OqBwOGce7k5B6ecaeN5X05pZnZufqG8WFlaXlldc9c3WjrJFKFNkvBE3WCkKWeSNg0znN6kiiKBOb3G/bOhf31PlWaJvDKDlIYCdSXrMIKMlSJ3P2DSRHkgkOlhnF8Wt3EB49tYBaqXRF4twCJXxS5UEVPRXeRWvbo3AvxL/AmpggkakfsZxAnJBJWGcKR12/dSE+ZIGUY4LSpBpmmKSB91adtSiQTVYT6KVcAdq8Swkyj7pIEj9edGjoTWA4H3sLDDwwR62h6K/3ntzHROwpzJNDNUkvGtTsahSeCwJhgzRYnhA0sQUcx+F5IeUogYW2bF9uBPp/5LWgd1/6juXRxWT08mjZTBFtgGNeCDY3AKzkEDNAEBD+AJvIBX59F5dt6c9/FoyZnsbIJfcD6+AcVFoK8=</latexit>

Rd

dd r⇢0 (r)riZ<latexit sha1_base64="KM3hwbWl+5PV3CeBLr1C69aXYZE=">AAACHHicbVDLSgMxFM34rPVVdekmWIQKUmZEscuCG5dV7AM67ZBk0jY2yQxJRijD/IZLv8SluhGXCv6N6WOhrQcCh3Pu5eYcHHOmjet+O0vLK6tr67mN/ObW9s5uYW+/oaNEEVonEY9UCyNNOZO0bpjhtBUrigTmtImHV2O/+UCVZpG8M6OYdgTqS9ZjBBkrBQXXZ9IEqS+QGWCc3mbdMINhN1S+GkSBW/KxSFV2AlXAVHCvgmFQKLpldwK4SLwZKYIZakHh0w8jkggqDeFI67bnxqaTImUY4TTL+4mmMSJD1KdtSyUSVHfSSbIMHlslhL1I2ScNnKi/N1IktB4JfIqFHR6H0PP2WPzPayemV+mkTMaJoZJMb/USDk0Ex03BkClKDB9Zgohi9ruQDJBCxNg+87YHbz71Immclb2LsntzXqxWZo3kwCE4AiXggUtQBdegBuqAgEfwDF7Bm/PkvDjvzsd0dMmZ7RyAP3C+fgBNKqIJ</latexit>

Rd

dd r⇢0 (r)ri rj

fi<latexit sha1_base64="4Y9kcPFD/sbveL+/S9cglCDBw5o=">AAACFnicbVDLSgMxFM3UV62vqks3wSJUkTIjil0W3LisYh/QaYckk7ahSWZIMkIZ5h9c+iUu1Y24dePCvzF9LLR64MLhnHu59x4cc6aN6345uaXlldW1/HphY3Nre6e4u9fUUaIIbZCIR6qNkaacSdowzHDajhVFAnPawqOrid+6p0qzSN6ZcUy7Ag0k6zOCjJWC4onPpAlSXyAzxDi9zXphBsNeqHw1jAK37GORquwYqoAFxZJbcaeAf4k3JyUwRz0ofvphRBJBpSEcad3x3Nh0U6QMI5xmBT/RNEZkhAa0Y6lEgupuOv0pg0dWCWE/UrakgVP150SKhNZjgU+xsM2T8/WiPRH/8zqJ6Ve7KZNxYqgks139hEMTwUlGMGSKEsPHliCimD0XkiFSiBibZMHm4C1+/Zc0zyreRcW9OS/VqvNE8uAAHIIy8MAlqIFrUAcNQMADeAIv4NV5dJ6dN+d91ppz5jP74Becj29CuZ9W</latexit>

fi

Z

Rd

dd r⇢0 (r)ri rj rkaAmbiguities incorner/hinger charge
Qcorner ! Qcorner + Px + Py

Qcorner ! Qcorner + Px + Py + Pz + Qxy + · · ·<latexit sha1_base64="JxZKrQghCTMwWTl86wfBGXSdTW4=">AAACInicbVDLSgNBEJz1bXxFPXoZDIKghF3xdRS8eIxgNJCEZXbSSYbMziwzvZqw5E88+iUe1YuIB8GPcfI4aGJBQ1HVTXdXlEhh0fe/vJnZufmFxaXl3Mrq2vpGfnPr1urUcChzLbWpRMyCFArKKFBCJTHA4kjCXdS5HPh392Cs0OoGewnUY9ZSoik4QyeF+dPrMKshdDHj2igw/X7NiFYbmTH6gU6ZB6Ww66oX5gt+0R+CTpNgTApkjFKY/6w1NE9jUMgls7Ya+AnWM2ZQcAn9XC21kDDeYS2oOqpYDLaeDf/r0z2nNGhTG1cK6VD9PZGx2NpeHB1GsWuOGbbtpD0Q//OqKTbP65lQSYqg+GhXM5UUNR3kRRvCAEfZc4RxI9y5lLeZYRxdqjmXQzD59TS5PSoGJ0X/+rhwcT5OZInskF2yTwJyRi7IFSmRMuHkkTyTV/LmPXkv3rv3MWqd8cYz2+QPvO8fKyKlzA==</latexit><latexit sha1_base64="TEnFAd+TKFQ6asQ93zG/CsgVN7c=">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</latexit>

+Px

+Py
+Py

Qyz
+Pz

Qxy
+Px
Qzx

-Py

Corner charge not well-de ned.
(Choice of termination & decoration)
Point group symmetry is required!!
fi

-Px

34

35

2D system under C4 rotation
Z

1
1
Qcorner =d r⇢0 (r)Q12 (r) = qa = qb
4
4
Rd<latexit sha1_base64="aZIec8NvuI1TCrYewwauLJIu6AY=">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</latexit>

fffiClassi cation ofhigh-symmetry points:(special) Wycko position

d

mod e

The U(1) charge
on high-symmetry points

36

2D system under C3 or C6 rotation<latexit sha1_base64="Kf8DdQrvpFn1I7/PDUsFoDI/1DQ=">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</latexit>

Qcorner =

Z

d

Rd

d r⇢0 (r)Q12 (r) =

(

1
2
1
q
=
q
+
a
b
6
3
2 qc
1
1
1
q
=
q
=
a
b
3
3
3 qc

mod e
mod e

(n = 6)(n = 3)

The U(1) charge Q mod ne at a Cn rotation axis is topological invariant.
Our results implies Qcorner = Q/n mod e

37

3D system under Oh symmetryZ

1
1
Qcorner =d r⇢0 (r)Q123 (r) = qa = qb
88Rd<latexit sha1_base64="aO0Uh9rftkubBYXpYb6F6EM9Ffw=">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</latexit>d1
mod e.
4

The U(1) charge Q mod 2e at a Oh center is topological invariant.
Our formula is Qcorner = Q/8 mod e/4.

38Filling anomaly• Simple formula for the corner charge
W. A. Benalcazar, T. Li, T. L. Hughes,
PRB (2019)<latexit sha1_base64="uLwUdFpJ1i4AszT5xEQxG5iq0PE=">AAACMnicbZDNSsNAFIUn/tb6F3XpJlgEVzUpim6EghtXomBtoQ1lMr3VoZOZMHMjlpD38EFcu9VX0J24EnwIpzWIVg8MHL5zL8M9USK4Qd9/dqamZ2bn5ksL5cWl5ZVVd2390qhUM2gwJZRuRdSA4BIayFFAK9FA40hAMxocj/LmDWjDlbzAYQJhTK8k73NG0aKuWzvvZh2EW8yY0hJ0nh99E1SY57unkwNdt+JX/bG8vyYoTIUUOuu6752eYmkMEpmgxrQDP8Ewoxo5E5CXO6mBhLIBvYK2tZLGYMJsfFvubVvS8/pK2yfRG9OfGxmNjRnGkZ2MKV6byWwE/82i+D/cTrF/GGZcJimCZF//91PhofJG/Xk9roGhGFpDmeb2BI9dU00Z2pbLtptgsom/5rJWDfar/vlepV4vWiqRTbJFdkhADkidnJAz0iCM3JEH8kienHvnxXl13r5Gp5xiZ4P8kvPxCZMQrOg=</latexit>

Qcorner = Qtot /Ncorner

Qtot = The total U(1) charge in the system
Ncorner = The number of point-group related corners<latexit sha1_base64="UpSMTgUe0ATSUwXXpqVWfWphG/w=">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</latexit>

Qtot = +e,

Qcorner = n1 e

Qtot = +e, Qcorner = 18 e (mod 14 e)<latexit sha1_base64="j0gmiQPDB84szmvcSz/6Z60GQnk=">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</latexit>(mod e)W. A. Benalcazar, T. Li, T. L. Hughes,
PRB (2019)

HW, S. Ono, PRB (2020)

39

Coupled-wire/layer argument

40

Similarity to HOTI

Matsugatani-Watanabe PRB (2018)

Traditional measurement of
the bulk polarization• Change a parameter and measure the electric current during
the process.

2Px =

Z T
0

Figure from https://www.globalsino.com/EM/

dtJx (t)41Possible direct measurement
via atomic force microscope
Robustness
against impurities

e2
10
2
F0 ⌘
=
7.25
⇥
10
kg
m
s
= 725 pN2
4⇡✏0 a<latexit sha1_base64="Usmzmdw2NvX8f/l82O5mUdDNVzQ=">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</latexit>

for a = 5.64 Å

42

Corrections from surface
dipoles & hinge polarizations

E(r) = (0, 0, 1)

Qc
d2

1
Qc
p
E(r) =(1, 1, 1) 2
d
3

+

+

43

44

Summary

M. Fruchart et al, Nature (2018)• Topological vs Trivial• Higher-order topological insulators• Gapless modes on hinges / corners• Irremovable quantum entanglement• Higher-order trivial insulators• Fractional charges on hinges / corners• Connected to product state• We bridged “multipole moments” and “ lling anomaly”
xy
d r⇢0 (r) 2
a
d

Z
1
1
1
d
d 1
Qcornerd r⇢0 (r)Q
Qcornerq(r)Qmode = qa = qb
Filling=anomaly
12 (r) = qa d=r⇢0b
12 (r)
4Rd
4
4
4
Rd<latexit sha1_base64="aZIec8NvuI1TCrYewwauLJIu6AY=">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</latexit>

fi<latexit sha1_base64="aZIec8NvuI1TCrYewwauLJIu6AY=">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</latexit>

Z=

Quadrupole moment

bulk
Qcorner = q12=Z

mod e