《講義ノート》

物性研究・電子版 Vol. 8, No. 1, 081202（2020年2月号）

ରশੑࢦඪͷ‫͔ૅج‬Βߴ࣍τϙϩδΧϧઈԑମ΁ͷԠ༻·Ͱ͕ɺx, y ࠲ඪͷ৘ใ͔͠‫͍ͳ·ؚ‬ਅʹ 2 ࣍‫ݩ‬ͷ৔߹ͷนࢴ౦‫ژ‬େֶ ෺ཧ޻ֶઐ߈‫( ܈‬wallpaper group) ͱͷ૬ҧ఺Ͱ͋Δɻಉ༷ʹɺ1 ࣍‫ݩ‬౉ล ༔थస΍‫ڸ‬өͳͲʹΑͬͯ z ͷූ߸͕มΘΔ͜ͱ΋ߟྀ͢Δ఺ํ޲ʢx ࣠ͱ͢Δ2 ʣͷ཭ࢄฒਐରশੑ͔࣋ͨ͠ͳ͍৔߹͸༩͑ΒΕͨରশੑͷ΋ͱͰɺͲͷΑ͏ͳछྨͷτϙϩδ

ϩου‫( ܈‬rod group) ͱ‫ݺ‬͹Εɺશ 75 छྨ͋ΔɻຊߘͰ͸Χϧ૬͕͋Γ͑Δͷ͔Λ෼ྨͨ͠ΓɺͦΕΒͷτϙϩδΧ

ओʹۭؒ‫܈‬Λٞ࿦ͭͭ͠ɺྫͱͯ͠ϨΠϠʔ‫܈‬΍ϩου‫܈‬ϧ૬Λ࣮‫͢ݱ‬Δ۩ମతͳ෺࣭Λ୳ࡧ͢Δ͜ͱ͸ɺۙ೥ͷ෺΋ѻ͏͜ͱʹ͢Δ3 ɻੑ෺ཧֶͷओཁͳ‫ڀݚ‬ςʔϚͷҰͭʹͳ͍ͬͯΔɻ࣌ؒ൓సରশੑ΍ཻࢠਖ਼޸ରশɺΧΠϥϧରশੑͷ૊Έ߹Θ͔ͤΒͳΔ ʮ಺෦ରশੑʯʹՃ͑ͯɺ࠷ۙͰ͸ɺ཭ࢄฒਐରশ
ੑɺ൓సରশੑɺճసରশੑͳͲͷ૊Έ߹Θ͔ͤΒͳΔʮۭ

1.1.2. ఺‫܈‬
ۭؒ R3 ͷ߹ಉૢ࡞͔ΒͳΔ‫܈‬ͷ͏ͪগͳ͘ͱ΋͋ΔҰؒ‫܈‬ͷରশੑʯΛ‫ؚ‬Ίͨ෼ྨ͕੝Μʹ‫͞ڀݚ‬Ε͍ͯΔɻຊ఺Λಈ͔͞ͳ͍΋ͷΛ఺‫( ܈‬point group) ͱ͍͏ɻۭؒ‫܈‬ߨٛͰ͸ɺόϯυߏ଄ʹ‫ݱ‬ΕΔۭؒ‫܈‬ͷରশੑͷද‫ݱ‬Λ༻ͷٞ࿦ʹ‫ݱ‬ΕΔ఺‫ʹ܈‬͸શ෦Ͱ 32 छྨ͋ΓɺͦͷҐ਺ʢ‫ݩ‬͍ͯɺόϯυߏ଄ͷ΋ͭτϙϩδʔΛܾఆ͢Δʮରশੑࢦ

ͷ਺ʣ͸༗‫ Ͱݶ‬1, 2, 3, 4, 6, 8, 12, 16, 24, 48 ͷ͍ͣΕ͔ඪʯͷํ๏ʹ͍ͭͯৄ͘͠঺հ͢Δɻ

x0 ∈ R3 ͱॻ͚͹ɺP ͷ‫ ݩ‬g ͷ
Ͱ͋Δɻ఺‫ ܈‬P ͷෆಈ఺Λ ⃗1ຊߨٛϊʔτͷߏ੒͸ҎԼͷΑ͏ʹͳ͍ͬͯΔ ɻୈ 1 અͰ͸ɺ݁থ͕࣋ͭରশੑͰ͋Δʮۭؒ‫܈‬ʯͷఆٛ΍ͦͷද⃗x ∈ R3 ΁ͷ࡞༻͸g(⃗x) = pg (⃗x − ⃗x0 ) + ⃗x0 ∈ R3

‫ͱ·͍ͯͭʹݱ‬ΊΔɻୈ 2 અͰ͸ɺిࢠͷόϯυߏ଄Λ‫ه‬ड़͢Δ༗ޮ໛‫͋Ͱܕ‬Δʮ‫ڧ‬ଋറ໛‫ܕ‬ʯʹ͓͍ͯɺۭؒ‫͕܈‬ͲͷΑ͏ʹද‫͞ݱ‬ΕΔ͔Λٞ࿦͢Δɻ͜ΕΒͷ४උΛ‫ʹݩ‬ɺ
ୈ 3,4 અͰ͸όϯυߏ଄ʹ‫ݱ‬ΕΔۭؒ‫܈‬ͷରশੑͷද‫ͱݱ‬ɺͦͷઈԑମ͕࣋ͭඇࣗ໌ͳτϙϩδʔͷؔ܎Λৄࡉʹٞ࿦
͢Δɻ

(1)

ͱॻ͘͜ͱ͕Ͱ͖Δɻ͜͜ʹ pg ∈ O(3) ͸ 3 ࣍‫ݩ‬௚ަߦྻͰ͋Δɻpg ͱͯ͠͸ɺ߃౳ม‫ʹ׵‬ରԠ͢Δ୯Ґߦྻͷଞʹ (1) ۭؒ൓సʹରԠ͢Δ − ɺ(2) ͋Δ࣠पΓͷ֯౓ 2πmn

ͷճసʢn = 2, 3, 4, 6ʣߦྻɺ͓Αͼ (1) ͱ (2) ͷੵ͕ొ৔
͢Δɻྫ͑͹‫ڸ‬ө͸ۭؒ൓సͱ֯౓ π ͷճసͷੵͰ͋Δɻ

1. ۭؒ‫܈‬ɾ఺‫܈‬ͷ‫ط‬໿ද‫ݱ‬·ۭͣؒ‫ͦͱ܈‬ͷද‫ؔ͢ʹݱ‬Δ‫߲ࣄૅج‬Λ͓͞Β͍͠Α͏ɻಛʹిࢠ‫ܥ‬ͷόϯυߏ଄ͷද‫ݱ‬࿦ʹ͔ܽͤͳ͍ʮࣹӨද‫ݱ‬ʯͷৄࡉ΍ʮϊϯγϯϞʔϑΟοΫۭؒ‫܈‬ʯͷऔΓѻ͍ʹ͍ͭͯจ‫ݙ‬1, 2) Λิ଍͠ͳ͕Β·ͱΊΔɻ

1.1. ۭؒ‫܈‬ͷ‫߲ࣄૅج‬1.1.3. ฒਐ‫ͱ܈‬୯Ґ๔ฒਐ‫ ܈‬T ͸‫ج‬ຊฒਐϕΫτϧ ⃗aI (I = 1, . . . , d) Ͱಛ௃෇͚ΒΕΔɻҎԼɺd = 3 ͸ۭؒ‫܈‬ɺd = 2 ͸ϨΠϠʔ‫܈‬ɺ

d = 1 ͸ϩου‫ʹ܈‬ରԠ͢ΔɻT ͷ‫ ݩ‬g ͷ ⃗x ∈ R3 ΁ͷ࡞༻͸⃗ ∈ R3 ,g(⃗x) = ⃗x + R

1.1.1. ۭؒ‫܈‬ͷछྨզʑͷॅΉۭؒࣗମ͸Ұ༷͔ͭ౳ํతͰ͋Γɺ3 ࣍‫ݩ‬ϢʔΫϦου‫܈‬ʢ࿈ଓతฒਐɺ࿈ଓతճసɺۭؒ൓స͓ΑͼͦΕΒͷ૊Έ߹Θͤʣͷ΋ͱͰʢগͳ͘ͱ΋೔ৗੜ‫׆‬ͷΤωϧΪʔεέʔϧͰ͸ʣෆมͰ͋Δɻ෺࣭தͰ͸͜ΕΒͷରশੑ͕෦෼‫ʹ܈‬ഁΕ͍ͯΔɻෆ७෺ͷଘࡏΛແࢹ͢Ε͹ɺ

⃗ =
R

!

nI ⃗aI(2)

I⃗ Λฒਐϕͷ‫͚ॻʹܗ‬Δɻͨͩ͠ nI ͸੔਺ͱ͢Δɻ͜ͷ R⃗ ͚ͩฒਐ͢Δૢ࡞Ϋτϧͱ‫Ϳݺ‬ɻҎԼͰ͸ฒਐϕΫτϧ R
Λ TR⃗ ∈ T ͱॻ͘͜ͱʹ͢Δɻ

ฒਐ‫͕܈‬༩͑ΒΕΔͱɺͦΕʹରԠͯ͠୯Ґ๔ (unit cell)ΛఆٛͰ͖Δɻ୯Ґ๔ͱ͸ฒਐͷ࡞༻ʹΑͬͯ Rd શମΛ‫͕ࢠݪ‬प‫ظ‬తʹฒͿ݁থߏ଄Λ࣋ͭ෺࣭͸཭ࢄతฒਐରশ

ॏෳͳ͘ຒΊਚ͘͢ྖҬͷ͜ͱͰ͋Δɻ͜Εʹ͸༷ʑͳબੑΛ࣋ͭͱͯ͠ྑ͍ɻۭؒ‫܈‬ʢspace groupʣͱ͸Ұൠʹɺͼํ͕͋Δ͕ɺຊߘͰ͸‫ج‬ຊฒਐϕΫτϧͰுΒΕΔฏߦ"ଟ໘ମ { I xI ⃗aI |0 ≤ xI < 1} ͱ͢ΔɻҎԼͰ͸͜ΕΛ UC཭ࢄతฒਐ෦෼‫܈‬Λ࣋ͭΑ͏ͳ 3 ࣍‫ݩ‬ϢʔΫϦου‫܈‬ͷ෦෼‫܈‬ͷ͜ͱͰ͋Δɻ
શ 3 ࣍‫΁޲ํݩ‬ͷ཭ࢄฒਐରশੑΛۭ࣋ͭؒ‫܈‬͸શ෦Ͱͱॻ͘ɻ

230 छྨ͋Γɺ୯ʹۭؒ‫ͨͬݴͱ܈‬৔߹ʹ͸͜ͷҙຯʹཧղ͞ΕΔɻҰํɺάϥϑΣϯͷରশੑͷΑ͏ʹ 2 ࣍‫޲ํݩ‬ʢ͜ΕΛ xy ฏ໘ͱ͢Δʣͷ཭ࢄฒਐରশੑ͔࣋ͨ͠ͳ͍
ۭؒ‫܈‬͸ɺಛʹϨΠϠʔ‫( ܈‬layer group) ͱ‫ݺ‬͹Εɺશ 80

1.1.4. ۭؒ‫܈‬ͷ࣮ۭؒ΁ͷ࡞༻ۭؒ‫ ܈‬G ͷ‫ ݩ‬g ͸ۭؒͷ఺ ⃗x ∈ R3 Λछྨ͋Δɻz ࣠ํ޲΁ͷฒਐରশੑ͸࣋ͨͳ͍͕ɺۭؒ൓

g(⃗x) ≡ pg ⃗x + ⃗tg ∈ R3

1 ຊߘ͸ୈ 64 ճ෺ੑएखՆͷֶߍʢ2019 ೥ 8 ݄։࠵ʣͷߨٛͰ࢖༻͢ΔςΩετͰ͋Δɻ͜Ε͸ࡶࢽʮ‫ݻ‬ମ෺ཧʯʹ࿈ࡌதͷࢽ্ηϛφʔͷ‫ߘݪ‬Λ‫ڐ‬ՄΛಘ্ͨͰ·ͱΊ௚ͨ͠΋ͷͰ͋Δɻ

2 z ͕࣠༻͍ΒΕΔ͜ͱ͕ଟ͍͕ɺ‫ه‬๏ͷ౷ҰͷͨΊ x ʹͨ͠ɻ

(3)

3 ͳͥนࢴ‫Ͱ܈‬͸ͳ͘ϨΠϠʔ‫܈‬Λѻ͏ͷ͔ʹ͍ͭͯ͸ 1.3.2 અɻ《講義ノート》

物性研究・電子版 Vol. 8, No. 1, 081202（2020年2月号）

΁ͱม‫͢׵‬Δɻ͜͜ʹ pg ∈ O(3) ͸ (1) ࣜʹ‫ݱ‬Εͨ 3 ࣍‫ݩ‬௚ަߦྻͰ͋Δ͕ɺ⃗tg ∈ R3 ͸‫ج‬ຊฒਐϕΫτϧ΍ͦͷ࿨

(a)

(b)

ͷ‫ͱܗ‬͸‫ݶ‬Βͳ͍ɻG ͷ 2 ͭͷ‫ ݩ‬g, g ′ ͷੵ gg ′ ͸ɺ͜ͷม‫׵‬Λ߹੒͢Δͱ෼͔ΔΑ͏ʹ

pgg′ = pg pg′ ,⃗tgg′ = pg⃗tg′ + ⃗tg(4)Ͱಛ௃෇͚ΒΕΔɻ֤ۭؒ‫܈‬ͷͦΕͧΕͷ‫͍ͯͭʹݩ‬ͷ pg ,⃗tg ͷ৘ใ͸จ‫ݙ‬3) ʹ·ͱΊΒΕ͍ͯΔɻۭؒ‫ ܈‬G ͷฒਐ෦෼‫ ܈‬T ͸ɺಛʹ G ͷਖ਼‫ن‬෦෼‫͋Ͱ܈‬

Δ4 ɻͦͷͨΊ G ͷཁૉͷ͏ͪฒਐͷҧ͍Λແࢹͨ͠΋ͷɺ͢ͳΘͪ঎ू߹ G/T ͸࠶ͼ‫ͳͱ܈‬ΔɻG/T ͷ֤‫ݩ‬ͷ௚ަx = ⃗0 Λෆߦྻ pg ͷ෦෼ͷΈʹண໨͢Δ͜ͱͰɺG/T ͸ ⃗ਤ 1 (a) ϩου‫ ܈‬p1̄ Ͱෆมͳ֨ࢠͷྫɻ੺ɺ੨ɺ྘ͷ఺͸ͦΕͧΕຊจͷ Λ⃗xa ɺΛ⃗xb ɺΛ⃗xc ʹରԠ͢Δɻ (b) ϨΠϠʔ‫ ܈‬p6/mmm Ͱෆมͳ֨ࢠͷྫɻ੺ɺ੨ɺ྘ͷ఺͸ͦΕͧΕࡾ֯֨ࢠɺϋχΧϜ֨ࢠɺ᝷໨֨ࢠ
Λͳ͢ɻ͜ͷଞʹɺΑΓରশੑͷ௿͍఺͕ෳ਺ଘࡏ͢Δɻ(a),(b) ͱ΋ʹࠇ఺͸ۭؒͷ‫఺ݪ‬ɺ໼ҹ͸‫ج‬ຊฒਐϕΫτϧɺ੺৭ͷྖҬ͸୯Ґ๔ UC Λද͢ɻಈ఺ͱ͢Δ఺‫ͱ܈‬Έͳ͢͜ͱ͕Ͱ͖Δɻ(−1)m ͓Αͼ ⃗tg = n1⃗a1 Ͱಛ௃෇͚ΒΕɺ୯Ґ‫ ݩ‬e ͸
1.1.5. γϯϞʔϑΟοΫۭؒ‫ͱ܈‬ϊϯγϯϞʔϑΟοΫۭؒ‫܈‬n1 = m = 0 ͷ৔߹ʹରԠ͢Δɻ఺‫ ܈‬P = G/T ͸ pe =͓Αͼ pI = − ͔ΒͳΔɻ⃗tg = n1⃗a1 ͕ฒਐϕΫτϧͷ‫ܗ‬Λ͍ͯ͠ΔͷͰ p1̄ ͸γϯϞʔϑΟοΫͰ͋Δɻ′఺‫ ܈‬P ͱฒਐ‫ ܈‬T ͷੵू߹ P ×T ͷ‫( ݩ‬p, TR⃗ ), (p , TR⃗′) Λߟ͑Δɻ఺‫ ܈‬P ͷੵ͸ p∗p′ = pp′ ∈ P ɺฒਐ‫ ܈‬T ͷੵ͸ TR⃗∗

TR⃗ ′ = TR+⃗ R⃗ ′ ͱఆٛ͞Ε͍ͯΔɻ௚ੵ‫ ܈‬P ×T Ͱ͸ P , T ͦ′′ΕͧΕಠཱʹੵΛͱΓ (p, TR⃗ )∗(p , TR⃗ ′ ) = (pp , TR+⃗ R⃗ ′ ) ͱఆΊΔ͕ɺ͜ΕͰ͸ (4) ࣜͷੵͷఆٛͱ૬༰Εͳ͍ɻ୅ΘΓʹ
′′(p, TR⃗ ) ∗ (p , TR⃗ ′ ) = (pp , TR⃗ ∗ (pTR⃗ ′ p−1′)) = (pp , TR+p⃗⃗′)R

ͱ͢Δͱ (4) ࣜͱ੔߹͢Δ͜ͱ͕Θ͔Δɻ͜ΕΛ൒௚ੵ P !Tͱॻ͘ɻ͜ͷΑ͏ʹۭؒ‫ ܈‬G ͕ɺ͋Δ఺‫ ܈‬P ͱฒਐ‫ ܈‬T ͷ൒௚ੵG = P !T Ͱ͋Δͱ͖ɺG ͸γϯϞʔϑΟοΫ (symmorphic)

1.1.6. ࣮ۭؒʹ͓͚Δ‫ي‬ಓͱখ‫܈‬͋Δ ⃗x ∈ R3 ΛબΜͰ‫ݻ‬ఆͨ͠ͱ͖ɺू߹ G⃗x ≡ {g(⃗x) | g ∈G} ͸఺ ⃗x ͷ G ͷ࡞༻ͷ΋ͱͰͷ‫ي‬ಓ (orbit) ͱ‫ݺ‬͹Εɺ෺ཧతʹ͸ରশੑ G ͷ΋ͱͰෆมͳ֨ࢠΛද͢ɻ͜ͷ֨ࢠΛΛ⃗x ͱද͢͜ͱʹ͠Α͏ɻ·ͨ Λ⃗x ͷ఺ͷ͏ͪ୯Ґ๔ UC ʹ
UCଐ͢Δ΋ͷͷू߹Λ Λ⃗UCx ͸ Λ⃗x ͱॻ͜͏ɻ‫ ʹٯ‬Λ⃗x Λฒਐ

ରশੑ T ʹΑͬͯʮෑ͖٧ΊΔʯ͜ͱͰಘΒΕΔɻx ͷબͼํ࣍ୈͰ༷ʑͳ֨ࢠ͕ۭؒ‫ ܈‬G Λ‫ݻ‬ఆͯ͠΋ɺ⃗

Ͱ͋Δͱఆٛ͢Δɻ΋͏গ͠؆୯ʹ‫͑׵͍ݴ‬Δͱɺ͏·͘‫ݱ‬ΕΔɻྫ͑͹ϨΠϠʔ‫ ܈‬p6/mmm (No. 80) ͷ৔߹ʹ͸ɺ‫఺ݪ‬ΛબͿ͜ͱʹΑͬͯ͢΂ͯͷ‫ ݩ‬g ∈ G ʹରͯ͠ಉ࣌ʹ⃗ [(2) ࣜ] ͷ‫͖Ͱʹܗ‬Ε͹γϯ⃗tg [(3) ࣜ] ΛฒਐϕΫτϧ Rਤ 1(b) ʹࣔͨ͠Α͏ʹɺΛ⃗x ͸ࡾ֯֨ࢠɺϋχΧϜ֨ࢠɺ᝷ϞʔϑΟοΫͰ͋ΓɺͦΕҎ֎ͷͱ͖͸ϊϯγϯϞʔϑΟοΔ֨ࢠ͸ϫΠίϑҐஔʢ1.1.7 અʣʹΑͬͯ෼ྨ͞ΕΔɻΫ (nonsymmorphic) Ͱ͋Δɻ 3 ࣍‫ݩ‬ͷ 230 ‫ݸ‬ͷۭؒ‫܈‬ͷ

1.1.5 અͷγϯϞʔϑΟοΫۭؒ‫܈‬ͷఆٛ͸ɺ͜ͷ G ෆ໨֨ࢠͳͲͱͳΔɻۭؒ‫ ܈‬G ͷରশੑΛ࣋ͭຊ࣭తʹҟͳ͏ͪɺγϯϞʔϑΟοΫͳ΋ͷ͸ 73 ‫࢒Ͱ͚ͩݸ‬Γͷ 157 ‫ݸ‬x ΛมԽͤ͞
มͳ֨ࢠͷ‫ݴ‬༿Ͱ‫͑׵͍ݴ‬Δ͜ͱ͕Ͱ͖Δɻ⃗͸ϊϯγϯϞʔϑΟοΫͰ͋Δɻ

ͯಘΒΕΔछʑͷ Λ⃗x ͷதʹʮαΠτ਺͕୯Ґ๔͋ͨΓ 1ճసʹҾ͖ଓ͍ͯճస࣠ํ޲ʹ൒୺ͳ੒෼Λ࣋ͭฒਐΛͭʯ
ʢ|Λ⃗UCx | = 1ʣͰ͋Δ΋ͷ͕ଘࡏ͢Δͱ͖ɺG ͸γϯϞʔ͢ΔΒͤΜૢ࡞΍ɺ‫ڸ‬өΛͨ͠‫Ͱޙ‬໘಺ʹ൒୺ͳ੒෼Λ࣋ϑΟοΫͰ͋Δɻ‫ʹٯ‬ϊϯγϯϞʔϑΟοΫۭؒ‫܈‬ͷͱ͖ͭฒਐΛ͢ΔάϥΠυૢ࡞Λ‫܈ۭؒͭ࣋ʹݩ‬͸ඞͣϊϯγx ΛબΜͰ΋ Λ⃗UC
͸ɺͲΜͳʹ͏·͘ ⃗x ͸ඞͣ 2 ͭҎ্ͷ఺ϯϞʔϑΟοΫͰ͋Δɻଞํɺۭؒ‫ ܈‬I21 21 21 (No. 24) ΍Λ‫ؚ‬Ήɻઌͷ p6/mmm ͷྫͰ͸ɺࡾ֯֨ࢠ͸୯Ґ๔͋ͨ

I21 3 (No. 199) ͸ྫ֎తͰΒͤΜૢ࡞΍άϥΠυૢ࡞Λ࣋Γʹ 1 ͚͔ͭͩ͠αΠτΛ࣋ͨͳ͍ͷͰ p6/mmm ͸γϯͨͳ͍ɻ͔͠͠ɺ֤‫ʹݩ‬ରͯ͠͸ͦΕͧΕద౰ͳ‫͕఺ݪ‬ଘ⃗ ͷ‫ʹܗ‬औΕΔ͕ɺͦͷͱ͖ଞʹࡏ͠ ⃗tg ΛฒਐϕΫτϧ RϞʔϑΟοΫͰ͋Δɻ

൒୺ͳฒਐΛ࣋ͭ‫͕ݩ‬ଘࡏ͢ΔͨΊɺϊϯγϯϞʔϑΟοx Λෆมʹอͭ G ͷ෦෼‫͋Ͱ܈‬Γɺ͍͏΋ͷ͕͋Δɻ͜Ε͸ ⃗x) = ⃗x} ʹ౳͍͠5 ɻҎԼͰ͸͜Εू߹ͱͯ͠͸ {h ∈ G | h(⃗ΫͰ͋Δɻϩου‫ ܈‬p1̄ ͷྫ‫ي‬ಓͱີ઀ʹؔ࿈ͨ֓͠೦ͱͯ͠খ‫( ܈‬little group) ͱϩου‫ ܈‬G = p1̄ ͸ɺ⃗a1 ≡ (a, 0, 0) ͷ n1

ഒ (n1 ∈ Z) ͚ͩฒਐ͢Δฒਐૢ࡞ Tn1⃗a1 : (x, y, z) → (x +n1 a, y, z) ʹՃ͑ͯɺۭؒ൓స I : (x, y, z) → (−x, −y, −z)x Λෆಈ఺ͱ͢Δ఺‫͋Ͱ܈‬ΔɻΛ G⃗x ͱॻ͘ɻఆٛΑΓ G⃗x ͸ ⃗
UC
UC
Λ⃗UC
x
x ͷҐ਺ |G⃗x | ͷੵ |Λ⃗x | ͸ɺ⃗x ͷ‫ݩ‬ͷ਺ |Λ⃗x | ͱ G⃗
x ||G⃗

ʹΑΒͣৗʹ G/T (1.1.4 અ) ͷҐ਺ |G/T | ʹ౳͍͠ɻΛ࣋ͭɻҰൠͷ‫ ݩ‬g = Tn1⃗a1 I m ∈ G (m = 0, 1) ͸ pg =4 H Λ G ͷ෦෼‫͢ͱ܈‬Δɻશͯͷ g ∈ G, h ∈ H ʹରͯ͠ ghg −1 ∈ H
Ͱ͋Δͱ͖ɺH ͸ G ͷਖ਼‫ن‬෦෼‫ݴͱ܈‬ΘΕΔɻ

5 ‫ޙ‬ͷ౎߹Ͱখ‫܈‬ͷ‫ݩ‬͸ओʹ h ͱॻ͘ɻ

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1.1.7. ϫΠίϑҐஔ⃗ ͸ಉ͡ฒਐͷද‫ݱ‬Λ༩͑ΔͷͰ͢΂ͯͱॻ͘ͱɺ⃗k ͱ ⃗k + Gۭؒͷ 2 ఺ ⃗x, ⃗y ∈ R3 ʹର͢Δಉ஋ؔ܎ ⃗x ∼ ⃗y Λɺখ‫܈‬ಉҰࢹ͢Δɻ͢ͳΘͪ೾਺ۭؒ͸ͦͷ୯Ґ๔ͷΈΛߟ͑Ε͹े෼Ͱ͋Δɻ͜ΕΛୈ 1 ϒϦϧΞϯҬ (the first BrillouinG⃗x ͱ G⃗y ͕‫ڞ‬໾Ͱ͋Δ͜ͱɺ͢ͳΘͪʮG⃗y = gG⃗x g −1 ͳΔzone) ͱ‫ͼݺ‬ɺҎԼͰ͸ BZ ͱॻ͘ɻୈ 1 ϒϦϧΞϯҬͱ͠y = g(⃗x) (g ∈ G) ͱॻ͚ΔͳΒ͹ɺG⃗y = gG⃗x g −1্ʹ͋Γɺ⃗

ͯ͸΢ΟάφʔβΠπ๔ (Wigner-Seitz cell)1) ͕બ͹ΕΔ⃗ ΛಉҰࢹ
͜ͱ͕ଟ͍ͷͰຊߘͰ΋͜Εʹश͏ɻ⃗k ͱ ⃗k + G

g ∈ G ͕ଘࡏ͢Δ͜ͱʯͱఆΊΔɻྫ͑͹ɺ⃗x ͱ ⃗y ͕ಉ͡‫ي‬ಓͱͳΔͷͰ ⃗x ∼ ⃗y Ͱ͋Δɻ͔͠͠ಉ͡‫ي‬ಓ্ʹ͋Δඞཁ͸ͳ͘ɺྫ͑͹ࣗ໌ͳখ‫ ܈‬G⃗x = {e} Λ࣋ͭ఺͸͢΂ͯಉ஋͢Δ΋ͱͰ೾਺ۭؒ͸ d ࣍‫ݩ‬τʔϥεͱಉ૬ʹͳΔɻ

Ͱ͋Δɻۭؒ R3 ͷ͢΂ͯͷ఺Λ͜ͷಉ஋ؔ܎Ͱ෼ྨͨ͠
঎ू߹ʢR3 / ∼ʣ͕ϫΠίϑҐஔ (Wyckoﬀ position) Ͱ͋Δɻ͜ͷ৘ใ΋ 230 ‫ݸ‬ͷۭؒ‫ͦ܈‬ΕͧΕʹରͯ͠จ‫ݙ‬3) ʹ·ͱΊΒΕ͍ͯΔɻಉ͡‫ي‬ಓ্ʹ͋Δ৔߹ʹ͸ಉ͡ϫΠί

1.2.2. ۭؒ‫܈‬ͷ೾਺ۭؒ΁ͷ࡞༻(5) ࣜͷද‫ݱ‬ͷ‫ج‬ఈͱͳΔঢ়ଶΛ |⃗k⟩ ͱॻ͘ɻ͢ͳΘͪϑҐஔʹଐ͢Δ͜ͱ͔ΒɺҟͳΔϫΠίϑҐஔΛྻ‫͢ڍ‬Δ⃗ ⃗T̂R⃗ |⃗k⟩ = |⃗k⟩e−ik·R .

ͨΊʹ͸୯Ґ๔ UC ಺ͷ఺ʹண໨͢Ε͹े෼Ͱ͋Δɻྫ࠶ͼϩου‫ ܈‬p1̄ ͷྫΛ༻͍Δɻg = Tn1⃗a1 I͜ͷͱ͖ɺĝ|⃗k⟩ ͱ͍͏ঢ়ଶΛߟ͑ͯΈΔͱɺ(4) ࣜΛ༻͍ͯ
m

ʹର͠

⃗
⃗ −i(pg ⃗k)·R⃗
T̂R⃗ (ĝ|⃗k⟩) = ĝ(T̂p−1⃗ |k⟩) = (ĝ|k⟩)eg Rx) = ⃗x Λղ͍ͯΈΔͱɺm = 0 ͷͱ͖͸ n1 = 0ʢͭͯ g(⃗·Γ୯Ґ‫ݩ‬ʣͷͱ͖ͷΈղΛ࣋ͪɺͦͷͱ͖͸౰વ͢΂ͯx ͕ղͱͳΔɻҰํ m = 1 ͷͱ͖͸Ұൠͷ n1 ∈ Z ʹରͷ⃗

x = (n1 a/2, 0, 0) ͱ͍͏ղΛ࣋ͭ͜ͱ͕෼͔Δɻ͢ͳͯ͠ ⃗x ͕ (na/2, 0, 0) (n ∈ Z) ͷͱ͖ T(na,0,0) IΘͪখ‫ ܈‬G⃗x ͸ɺ⃗

(8)
⃗

⃗

−i(pg k)·R
ͱͳΔɻ͕ͨͬͯ͠ɺĝ|⃗k⟩ ͸ද‫ ݱ‬Upg ⃗k (TRͷ⃗) = e‫ج‬ఈͱͳΔ͜ͱ͕෼͔Δɻ͢ͳΘۭͪؒ‫܈‬ͷ‫ ݩ‬g ͸೾਺ ⃗kʹରͯ͠g(⃗k) = pg⃗kΛཁૉʹ࣋ͪɺ͜ͷଞͷ‫ܗ‬ͷͱ͖ʹ͸ࣗ໌Ͱ୯Ґ‫ ݩ‬e ͷΈ(9)ͱ࡞༻͢Δɻ⃗k ͷม‫ʹ׵‬͸ g ͷฒਐ෦෼ ⃗tg ͕ޮ͔ͳ͍͜ͱͱͳΔɻm·ͨɺG(na/2,0,0) = {e, T(na,0,0) I } ͕ T(n1 a,0,0) G(na/2,0,0)−1T(n1 a,0,0)(7)ʹಛʹ஫ҙ͍ͨ͠ɻ

= G((n1 +n/2)a,0,0) Λຬͨ͢͜ͱʹ஫ҙ͢Δͱɺ

⃗x = (na/2, 0, 0) ͱ ⃗y = ⃗x + n1⃗a ͸ಉ͡ϫΠίϑҐஔʹଐ͍ͯ͠Δ͜ͱ͕֬ೝͰ͖Δɻ͕ͨͬͯ͠ϫΠίϑҐஔ͸ 3 ͭଘ

xa = (0, 0, 0), ⃗xb = (a/2, 0, 0),ࡏ͠ɺͦͷ୅ද఺ͱͯ͠͸ ⃗⃗xc = (x, y, z)ʢͨͩ͠ 0 < x < a/2ʣΛબͿ͜ͱ͕Ͱ͖
Δɻਤ 1(a) ʹࣔͨ͠Α͏ʹɺͦΕͧΕ Λ⃗UC
xa = {(0, 0, 0)}ɺUC
Λ⃗UC
xb = {(a/2, 0, 0)}ɺΛ⃗
xc = {(x, y, z), (a − x, −y, −z)} ͱ͍͏֨ࢠʹରԠ͢Δɻ

1.2.3. ೾਺ۭؒʹ͓͚Δ‫ي‬ಓͱখ‫܈‬࣮ۭؒͷͱ͖ͱಉ༷ɺ⃗k ∈ BZ ΛબΜͰ‫ݻ‬ఆͨ͠ͱ͖ɺू
߹ G⃗k ≡ {g(⃗k) ∈ BZ | g ∈ G} ͸ ⃗k ͷ G ͷ࡞༻ͷ΋ͱͰͷ‫ي‬ಓΛද͢ɻಛʹ೾਺ۭؒͰ͸͜ΕΛ ⃗k ͷ੕ (star) ͱ‫ͼݺ‬ɺS⃗ ͱॻ͘͜ͱʹ͢Δɻ࣮ۭؒͷ৔߹ͱ͸ҟͳΓɺ⃗k ͷ੕ͷkཁૉ਺ |S⃗k | ͸༗‫͋Ͱݶ‬Δɻ·ͨɺ⃗k ∈ BZ Λෆมʹอͭ G ͷ෦෼‫܈‬Λ ⃗k ͷখ‫ ܈‬G⃗k

1.2. ೾਺ۭۭؒؒ‫ ܈‬G ͷද‫ݱ‬͸ɺͦͷෆม෦෼‫͋Ͱ܈‬Δฒਐ‫ ܈‬T ͷද‫ݱ‬Λ‫ߏʹݩ‬੒͞ΕΔɻ͜ͷ T ͷද‫ݱ‬Λಛ௃͚ͮΔͷ͕೾਺⃗k Ͱ͋Δɻ2

ฒਐ‫܈‬ͷ‫ ݩ‬TR
⃗ ∈ T ͷද‫ݱ‬Λ⃗ ⃗U⃗k (TR⃗ ) = e−ik·R(5)⃗aI · ⃗bJ = 2πδI,J Λຬͨ͢Α͏ʹఆٛ͢Δɻ‫ࢠ֨ٯ‬ϕΫτϧ⃗bJ ͸೾਺ۭؒʹ͓͚Δ‫ج‬ຊฒਐϕΫτϧͷ໾ׂΛՌͨ͢ɻ೾਺ۭؒͷ֨ࢠ఺Λ

J

mJ⃗bJ

2

⃗k = ⃗b1 /2 ͸ۭؒ൓సෆมͰ͋Γɺ͜ͷͱ͖ G⃗ ͸ G ࣗ਎ɺkS⃗k = {⃗k} ͱͳΔɻ͜ͷଞͷ ⃗k ʹ͍ͭͯ͸ɺG⃗k ͸ฒਐ‫ ܈‬TͱҰக͠ɺS⃗ = {+⃗k, −⃗k} ͱͳΔɻkͱఆΊΔɻ͜ͷද‫ݱ‬ͷϥϕϧͰ͋ΔϕΫτϧ ⃗k ͷू߹Λ೾਺ۭؒͱ‫Ϳݺ‬ɻ‫ࢠ֨ٯ‬ϕΫτϧ ⃗bJ (J = 1, . . . , d) Λɺ!

k‫ؚ‬Ήɻ͕ͨͬͯ͠ G⃗k ࣗ਎͸఺‫Ͱ܈‬͸ͳ͍ɻ|G⃗k /T | ͱ |S⃗k |ͷੵ͸ɺ⃗k ʹΑΒͣৗʹ |G/T | ʹ౳͍͠ɻྫ ·ͨϩου‫ ܈‬p1̄ ͷྫΛ༻͍ͱɺ⃗b1 = ( 2πa , 0, 0)ɺBZ =
1
1
⃗
⃗
⃗
{k b1 | − < k ≤ } ͱͳΔɻI(k) = −k ͳͷͰ ⃗k = ⃗0 ͱ

1.2.1. ฒਐ‫܈‬ͷද‫ͱݱ‬೾਺ۭؒ⃗ =G⃗ ʹ౳͍͠ɻ
ͱ‫Ϳݺ‬ɻू߹ͱͯ͠͸ {h ∈ G | h(⃗k) = ⃗k (+G)}⃗ฒਐ͸ k Λม͑ͳ͍ͨΊɺG⃗ ͸ඞͣฒਐ‫ ܈‬T Λ෦෼‫ʹ܈‬(6)

1.3. ۭؒ‫܈‬ͷද‫ݱ‬ຊઅͰ͸ಛʹࣹӨද‫ʹݱ‬ॏ఺Λஔ͖ͳ͕Βɺۭؒ‫܈‬ͷද‫ͱ·͍ͯͭʹݱ‬ΊΔɻ

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͢Δ͜ͱʹ͢Δɻಛʹ pg = ± ͷͱ͖͸ θg = 0 ͱ໿ଋ͢1.3.1. ࣹӨද‫ݱ‬Ұൠʹɺ‫ ܈‬G ͷߦྻද‫ͱݱ‬͸ g ∈ G ͔Β u(g) ΁ͷࣸ૾Ͱ′′u(g)u(g ) = u(gg )(10)Δɻ(4) ࣜʹΑΓɺpg ͸ G ͷઢ‫ܕ‬ද‫͋Ͱݱ‬Δɻ⃗x ΁࡞༻͢Δߦྻ pg ∈ O(3) ʹରԠ͠ɺిࢠͷεϐϯʹ

࡞༻͢Δߦྻ psp
g ∈ SU(2) Λ⃗Λຬͨ͢΋ͷͷ͜ͱͰ͋Δɻʢ͜ͷখઅͷ G ͱͯ͠͸ۭؒ−iθg ⃗
ng ·S
psp
g =e

‫ ܈‬G ΍఺‫ ܈‬P Λ૝ఆ͍ͯ͠Δɻʣ͔͠͠όϯυߏ଄ͷද‫ݱ‬⃗ = ⃗σ /2 ͸εϐϯ 1/2 ΛදͱఆΊΔ͜ͱʹ͠Α͏ɻ͜͜ʹ S

Λٞ࿦͢Δࡍʹ͸

(14)

͠ɺ⃗σ = (σ1 , σ2 , σ3 )T ͸ύ΢ϦߦྻͰ͋Δɻ͜ͷߦྻ psp
gu(g)u(g ′ ) = ω(g, g ′ )u(gg ′ )(11)͸ G ͷࣹӨද‫ͳͱݱ‬Γɺ
sp
sp
′ sp
psp
g pg ′ = ω (g, g )pgg ′Λຬࣹͨ͢Өද‫( ݱ‬projective representation) Λߟ͑Δඞ
′(15)

ཁ͕ੜ͡Δɻ͜ͷҐ૬ ω(g, g ) ∈ U(1) ΛࣹӨද‫ݱ‬ͷ৐਺‫ܥ‬Λຬͨ͢ɻӨද‫۠ͱݱ‬ผͯ͠ઢ‫ܕ‬ද‫( ݱ‬linear representation) ͱ‫ݺ‬͹(15) ࣜ͸ిࢠͷεϐϯ 1/2 ʹ‫ى‬Ҽ͢Δ৐਺‫ ܥ‬ω sp (g, g ′ ) =±1 ͷఆٛࣜͱΈΔ͜ͱ͕Ͱ͖Δɻࠓ‫ݱޙ‬ΕΔۭؒ‫ ܈‬G ΍(factor system) ͱ‫Ϳݺ‬ɻ(10) ࣜΛຬͨ͢௨ৗͷද‫ݱ‬͸ɺࣹΕΔɻ৐਺‫ܥ‬͸೚ҙʹબͿ͜ͱ͸Ͱ͖ͣɺߦྻͷੵͷ݁߹ଇ[u(g)u(g ′ )]u(g ′′ ) = u(g)[u(g ′ )u(g ′′ )] ʹରԠͯ͠ɺίαΠΫϧ৚݅ ω(g, g ′ )ω(gg ′ , g ′′ ) = ω(g, g ′ g ′′ )ω(g ′ , g ′′ ) (g, g ′ , g ′′ ∈G) Λຬͨ͞ͳ͚Ε͹ͳΒͳ͍ɻ·֤ͨ g ʹରͯ͠ u(g)′ =u(g)eiϕg ͱऔΓ௚͢ͱ′′ ′u(g) u(g ) = eখ‫ ܈‬G⃗x ɺG⃗k ͷද‫ݱ‬͸͢΂ͯ͜ͷ ω sp (g, g ′ ) Λ৐਺‫͢ͱܥ‬ΔࣹӨද‫͋Ͱݱ‬Δɻ(13) ࣜΛຬͨ֯͢౓ (⃗ng , θg ) ͷબͼํΛม͑Δͱ ω sp (g, g ′ ) ͷ஋΋มΘΔ͕ɺ͜ͷҧ͍͸ 1.3.1 અͰड़΂ͨҐ૬ม‫׵‬ͷࣗ༝౓ʹରԠ͢ΔɻθʔϚϯ࣓৔΍εϐϯ‫ي‬ಓ૬‫͘ͳ͕Ͳͳ༻࡞ޓ‬ɺεϐϯͷ SU(2) ճసରশੑΛԾఆͰ͖Δ৔߹ʹ͸ɺ‫ي‬ಓͱεϐϯi(ϕg +ϕg′ −ϕgg′ )′ ′ͱͳΓɺ৐਺‫ܥ‬͸ ω(g, g ) = e′′ ′ω(g, g )u(gg )i(ϕg +ϕg′ −ϕgg′ )(12)ͷม‫׵‬Λผʑʹߦ͏͜ͱ͕ՄೳͱͳΔɻͦͷ৔߹ɺ‫ي‬ಓͷม‫׵‬Λٞ࿦͢Δࡍʹ͸ҎԼͷٞ࿦Ͱ ω sp (g, g ′ ) = 1 ͱ͢Ε′ω(g, g ) ΁ͱ͹͍͍ɻม‫͞׵‬ΕΔ͜ͱ͕෼͔Δɻ͜ͷΑ͏ʹ u(g) ͷҐ૬ͷऔସϨΠϠʔ‫܈‬΍ϩου‫܈‬͸ͦΕͧΕ 2 ࣍‫ݩ‬ɺ1 ࣍‫ݩ‬ͷฒਐ͑Ͱ݁ͼͭ͘ҧ͍͸ຊ࣭తͰͳ͘ɺ͜ͷҐ૬ม‫ʹ׵‬ΑΔ৐

‫͕͍ͳ͔ͨ࣋͠܈‬ɺ఺‫܈‬෦෼ pg ͱͯ͠͸͍ͣΕʹରͯ͠਺‫ܥ‬ͷಉ஋ྨΛߟ͑Δͷ͕ࣗવͰ͋Δɻ͜ΕʹΑͬͯಘΒ΋ O(3) ߦྻΛߟ͍͑ͯΔ͜ͱΛࢥ͍ग़ͦ͏ɻ͜Ε͸ psp
g ΍

ΕΔຊ࣭తʹҟͳΔ৐਺‫ܥ‬ͷ෼ྨ͸‫܈‬ͷୈ 2 ίϗϞϩδʔ
2

H (G, U(1)) Ͱ༩͑ΒΕΔ͜ͱ͕஌ΒΕ͍ͯΔɻ·ͨɺ͜ͷҐ૬ม‫׵‬ͷࣗ༝౓Λར༻ͯ͠ɺ୯Ґ‫ ݩ‬e ͕͢΂ͯͷ g ∈ Gʹରͯ͠ ω(e, g) = ω(g, e) = 1 Λຬͨ͢Α͏ʹબͿ͜ͱʹ͢Δɻω sp (g, g ′ ) Λಉఆ͢ΔͨΊʹඞཁ͔ͩΒͰ͋Δɻྫ͑͹ۭؒ൓స I : (x, y, z) *→ (−x, −y, −z) ͱ z ࣠·ΘΓͷ π ճస

C2z : (x, y, z) *→ (−x, −y, +z) ͸ x, y ͚ͩΛ‫͍ͯͯݟ‬΋۠ผͰ͖ͳ͍ɻ͔͠͠ిࢠεϐϯʹͱͬͯ͸શ͘ҧ͏ૢ࡞Λspද͓ͯ͠Γɺpsp
I = ɺpC2z = −iσz ͱͳΔɻ఺‫ ܈‬222 ͷྫ

1.3.2. ిࢠεϐϯʹ‫ى‬Ҽ͢Δ৐਺‫ܥ‬εϐϯʹ‫ى‬Ҽ͢Δ৐਺‫ܥ‬ͷͱͯ΋େ੾ͳྫͱͯ͠ɺ఺‫ ܈‬222 Λߟ͑Α͏ɻ͜ͷ‫܈‬ͷҐ਺͸ 4 Ͱɺ୯όϯυߏ଄ʹ‫ݱ‬ΕΔද‫ݱ‬ͷ৐਺‫͕ܥ‬ඇࣗ໌ʹͳΔ‫ʹݯى‬͸େ͖͘෼͚ͯ 2 छྨ͋ΔɻҰͭ໨͕ɺిࢠ͕εϐϯ 1/2 Λ࣋Ґ‫ ݩ‬e ͷଞʹ x, y, z ͦΕͧΕͷ࣠पΓͷ π ճసૢ࡞ C2i(i = x, y, z) Λ‫ͭ࣋ͯ͠ͱݩ‬ɻੵ͸ C2x C2y = C2y C2x = C2zͭ͜ͱʹΑΔɻҰൠʹۭؒ‫ ܈‬G ͷ‫ ݩ‬g ͷ఺‫܈‬෦෼ pg ∈ O(3)

΍ (C2i )2 = e Λຬͨͨ͢Ίɺ͜ͷ‫܈‬͸Մ‫͋Ͱ܈׵‬Δɻ

͸

(13)

g = C2i (i = x, y, z) ͷͦΕͧΕʹର͠ɺ⃗ng Λ (1, 0, 0),(0, 1, 0), (0, 0, 1) ͱ͠ɺθg ͸͢΂ͯ +π ͱબΜͰΈΔɻ͢ͱॻ͘͜ͱ͕Ͱ͖ΔɻॳΊͷූ߸͸ pg ͷߦྻࣜ det(pg ) ʹ−iπσi /2
= −iσi ͱͳΔͨΊɺిࢠͷεϐϯʹ
Δͱ pspC2i = e⃗pg = ±e−iθg ⃗ng ·L

ରԠ͢Δɻࢦ਺ؔ਺ͷ෦෼͸࣠ ⃗ng पΓͷ֯౓ θg ͷճసߦ⃗ = (L1 , L2 , L3 )T ͸֯ӡಈྔԋࢉࢠͷ 3 ࣍‫ݩ‬දྻΛද͠ɺL‫͋Ͱྻߦݱ‬Δʢ۩ମతʹ͸‫׬‬શ൓ରশςϯιϧ ϵ Λ༻͍ͯ(Li )j,k = −iϵijk (i, j, k = 1, 2, 3) ͱ͢Ε͹Α͍ʣɻ(⃗ng , θg )ͷબͼํʹ͸ෆఆੑ͕͋Γɺ(⃗ng , θg ) ͱ (⃗ng , θg + 2πn) ΍‫ى‬Ҽ͢Δ৐਺‫ ܥ‬ω sp ͸ω sp (C2x , C2y ) = ω sp (C2y , C2z ) = ω sp (C2z , C2x ) = +1
(16)
ω sp (C2y , C2x ) = ω sp (C2z , C2y ) = ω sp (C2x , C2z ) = −1(17)(−⃗ng , −θg ) ͸ಉ͡ճసΛද͢͜ͱʹ஫ҙ͍ͨ͠ɻྫ͑͹0 ≤ θg ≤ π ͷΑ͏ʹ੍‫ͯ͜͠ݶ‬ͷෆఆੑΛऔΓআ͜͏ͱ͠

ͯ΋ɺθg = π ͷͱ͖ͷॲཧ͕໰୊ʹͳΔɻԿΕʹͤΑɺ֤g ∈ G ʹରͯ͠ (13) ࣜΛຬͨ͢ (⃗ng , θg ) ΛҰͭબΜͰ‫ݻ‬ఆͱ‫ٻ‬ΊΔ͜ͱ͕Ͱ͖Δɻ

《講義ノート》

物性研究・電子版 Vol. 8, No. 1, 081202（2020年2月号）

1.3.3. ϊϯγϯϞʔϑΟοΫͳૢ࡞ʹ‫ى‬Ҽ͢Δ৐਺‫ܥ‬

1.3.4. ఺‫܈‬ͷ‫ط‬໿ද‫ͦͱݱ‬ͷੑ࣭খ‫ ܈‬G⃗k ͷද‫ͱݱ‬͸͜Ε·Ͱɺ࣮ۭؒʹ͓͚Δ G⃗xʢ1.1.6 અʣͱ೾਺ۭؒʹ͓U⃗k (h)U⃗k (h′ ) = ω sp (h, h′ )U⃗k (hh′ )(18)͚Δ G⃗k /T ʢ1.2.3 અʣͷ 2 ͭͷ఺‫ݱ͕܈‬Εͨɻ෺ཧతԠ༻
ͷࡍʹ͸ɺͦΕͧΕ ω sp (h, h′ ) ͱ ω⃗sp (h, h′ )ω⃗ns (h, h′ ) Λ৐kΛຬࣹͨ͢Өද‫ݱ‬ͷ͏ͪɺฒਐ෦෼‫ ܈‬T ͷද‫( ͕ݱ‬5) ࣜΛຬͨ͢΋ͷͷ͜ͱͰ͋ΔɻG⃗k ͷ‫ط‬໿ද‫ݱ‬Λ‫ٻ‬ΊΔͨΊʹɺ·ͣ͸༗‫ ܈ݶ‬G⃗k /T ͷ‫ط‬໿ද‫ݱ‬Λ‫ٻ‬Ί͓͖ͯɺͦΕʹ‫ͮج‬͍ͯ G⃗k ͷ‫ط‬໿ද‫ݱ‬Λߏ੒͢Δͱ͍͏͜ͱ͕ߦΘΕΔɻͦ
͜Ͱ G⃗k /T ͷද‫ ݱ‬u⃗k Λઅʣɻ͜͜Ͱ͸ΑΓҰൠʹɺ৐਺‫ܥ‬Λ ω(p, p′ ) ͱ͢Δ఺‫ ܈‬PͷࣹӨද‫ݱ‬ͷੑ࣭ʹ͍ͭͯ·ͱΊΑ͏6 ɻP ͷ‫ط‬໿ද‫ݱ‬Λ uα (p)ʢα = 1, . . . , N ʣͱॻ͖ɺͦͷࢦඪ tr[uα (p)] Λ χα (g)ɺ࣍‫ݩ‬Λ Dα = χα (e) ͱ‫͢ه‬ɻࣹӨද‫ݱ‬ͷ৔߹ʹ΋ઢ‫ܕ‬ද‫ݱ‬ͷͱ͖ͱಉ༷ʹද‫ݱ‬ͷ௚ަੑU⃗k (h) = u⃗k (h)e

−i⃗
k·⃗
th

(19)

!

ʹΑͬͯಋೖ͠Α͏ɻฒਐ‫܈‬ͷද‫ݱ‬Λ (5) ࣜͰఆΊ͍ͯΔ
ͨΊʹɺ͜͏͢Δͱ͔֬ʹ h ͱ TR
⃗ h ͸ͱ΋ʹಉ͡ u⃗k Λ༩

[uα (p)]∗i,j [uβ (p)]k,l =p∈P!

(19) Λ (18) ࣜʹ୅ೖ͢Δͱɺu⃗k ͸

p∈P
⃗

⃗

⃗

u⃗k (h)u⃗k (h′ ) = ω sp (h, h′ )e−ik·(ph th′ −th′ ) u⃗k (hh′ )(20)Λຬͨ͞ͳ͚Ε͹ͳΒͳ͍͜ͱ͕Θ͔Δɻ͢ͳΘͪ‫ ܈‬G⃗k /T
⃗

⃗

⃗

k

ͷࣹӨද‫ ݱ‬u⃗k (h) ͕ಘΒΕΕ͹ɺ(19) ࣜΛ༻͍ͯ ω sp (h, h′ )

nα ≡

1 ! α
[χ (p)]∗ χ(p),|P |χ(p) = tr[u(p)](28)͔Β‫ٻ‬ΊΔ͜ͱ͕Ͱ͖Δɻ·ͨɺද‫ݱ‬ͷ࣍‫ ݩ‬Dα ʹରͯ͠΋ઢ‫ܕ‬ද‫ݱ‬ͷͱ͖ͱಉ༷ʹ!
α(Dα )2 = |P |(29)

͕੒ཱ͢Δɻ·ͨɺҟͳΔࣹӨ‫ط‬໿ද‫ ݱ‬uα (p)ʢα = 1, . . . , N ʣͷ਺͸ۭؒ‫ ܈‬P 21 21 21 (No. 19) ͷྫ

(27)

p∈P

Λ৐਺‫͢ͱܥ‬Δ G⃗k ͷࣹӨද‫ߏ͕ݱ‬੒Ͱ͖Δɻω⃗ns (h, h′ ) ͸kۭؒ‫͕܈‬ϊϯγϯϞʔϑΟοΫͰ͔ͭ ⃗k ͕ BZ ද໘ͷ৔߹ͷΈॏཁʹͳΔɻ[χα (p)]∗ χβ (p) = |P |δα,β‫ط‬໿ද‫ ݱ‬uα (p) ͷ਺͸(21)ͷੵʹͳΔɻ‫ ʹٯ‬ω sp (h, h′ )ω⃗ns (h, h′ ) Λ৐਺‫͢ͱܥ‬Δ G⃗k /T(26)

΋ಋ͔ΕΔɻ͕ͨͬͯ͠ɺҰൠͷࣹӨද‫ ݱ‬u(p) ʹ‫·ؚ‬ΕΔͷࣹӨද‫ ݱ‬u⃗k ͷ৐਺‫ܥ‬͸ɺω sp (h, h′ ) ͱω⃗kns (h, h′ ) ≡ e−ik·(ph th′ −th′ )|P |δi,k δj,l δα,βDα͕੒ཱ͢Δɻ͜ΕΑΓ௚ͪʹࢦඪͷ௚ަੑ

͑ɺG⃗k /T ͷද‫͋Ͱ͏ͦ͑ݴͱݱ‬Δɻ͔͠͠ɺ͜ͷఆۭٛࣜؒ‫ ܈‬P 21 21 21 ͸ 2 ͭͷN=ΒͤΜૢ࡞ Sx , Sy ͓Αͼ͜ΕΒͷੵʢྫ͑͹ Sz = (Sx Sy )−1
΍ Tx = Sx2 ʣ͔ΒͳΔϊϯγϯϞʔϑΟοΫۭؒ‫͋Ͱ܈‬Δɻ
͜ΕΒͷૢ࡞ͷ۩ମ‫ܥ‬͸Sx : (x, y, z) *→ (x + ax /2, −y + ay /2, −z),(22)

Sy : (x, y, z) *→ (−x, y + ay /2, −z + az /2),

(23)

Sz : (x, y, z) *→ (−x + ax /2, −y, z + az /2)(24)Ͱ͋Δɻߴରশ೾਺ ⃗k = (πnx /ax , πny /ay , πnz /az )ʢͨͩ͠ nx , ny , nz ͸ 0 ·ͨ͸ 1ʣʹ͓͍ͯ͸ɺখ‫ ܈‬G⃗k ͸ P 21 21 21ࣗ਎ͱҰக͢Δɻ͜ͷ೾਺ʹ͓͍ͯ۩ମతʹ ω⃗ns (h, h′ ) Λk͍͔ͭ͘‫ͯ͠ࢉܭ‬ΈΔͱɺω⃗kns (Sx , Sy ) = eiπ(ny +nz ) ,k਺‫͢ͱܥ‬ΔࣹӨද‫ݱ‬Λߟ͑Δ͜ͱʹͳΔʢ1.3.2 ͓Αͼ 1.3.3

1 ! ω(p, p′ )δpp′ ,p′ p|P | ′ ω(p′ , p)(30)

p,p ∈PͰ༩͑ΒΕΔ4) ɻྫ࠶ͼ఺‫ ܈‬222 Λߟ͑Α͏ɻ͜ͷ‫܈‬͸Մ‫͋Ͱ܈׵‬ΔͨΊɺઢ‫ܕ‬ද‫ͯ͠ͱݱ‬͸ 1 ࣍‫ݩ‬ද‫ ͕ݱ‬4 ͭಘΒΕΔɻ۩ମతʹ͸ɺuα (C2x ) = ±1 ͱ uα (C2y ) = ±1 ΛબͿͱɺuα (C2z ) ͕ࣗಈతʹ uα (C2z ) = uα (C2x C2y ) = uα (C2x )uα (C2y ) ͔Βܾ
·Δɻ͜ͷͱ͖ 12 + 12 + 12 + 12 = 4 ͷΑ͏ʹ (29) ͕ࣜຬͨ͞Ε͍ͯΔ͜ͱʹ஫ҙ͢Δɻ
Ұํɺిࢠͷεϐϯʹ‫ى‬Ҽ͢Δ৐਺‫ ܥ‬ω sp ΛߟྀʹೖΕΔͱɺuα (C2x )uα (C2y ) = −uα (C2y )uα (C2x ) ͷΑ͏ʹද‫ݱ‬͕ඇՄ‫ͳʹ׵‬ΔɻҰൠʹඇՄ‫׵‬ද‫ݱ‬Λ࣮‫͢ݱ‬Δ 1 ࣍‫ݩ‬ද‫ݱ‬ω⃗kns (Sy , Sx ) = eiπnx(25)ͱͳΔɻಛʹ ω⃗ns (Sx , Sy )/ω⃗ns (Sy , Sz ) = eiπ(nx +ny +nz ) =kk͸ଘࡏ͠ͳ͍ɻ࣮ࡍɺ͜ͷ৐਺‫ܥ‬Λ࣋ͭ఺‫ ܈‬222 ͷද‫ݱ‬͸

2 ࣍‫ݩ‬ද‫ ͕ݱ‬1 ͭͰ͋Γɺuα (C2i ) = psp
C2i = −iσi Ͱ༩͑Β
ΕΔɻ͜ͷ৔߹ɺ(29) ࣜ͸ 22 = 4 ͱͯ͠੒ཱ͍ͯ͠Δɻ

−1 ͱͳΔΑ͏ͳߴରশ೾਺఺ʹରͯ͠͸ɺ͜ͷ৐਺‫ܥ‬Λߟྀ͢Δ͜ͱͰ‫͞ڐ‬ΕΔ‫ط‬໿ද‫ݱ‬ͷ࣍‫ݩ‬΍਺͕มԽ͢Δɻྫ͑͹ɺిࢠεϐϯ͕ͳ͍৔߹ʹ͸ 2 ࣍‫ݩ‬ද‫ ͕ݱ‬1 ͚ͭͩͰ1 ࣍‫ݩ‬ද‫ݱ‬͸‫͞ڐ‬Εͳ͘ͳΔʢ1.3.4 અࢀরʣɻ

6 ຊߘͷ໨తʹԊͬͯ఺‫ݶʹ܈‬ఆ͕ͨ͠ɺຊઅͰ঺հ͢Δ಺༰͸Ұൠͷ༗‫ʹ܈ݶ‬ରͯ͠੒ཱ͢Δɻ

《講義ノート》

ྫ

物性研究・電子版 Vol. 8, No. 1, 081202（2020年2月号）

ઌʹड़΂ͨ఺‫ ܈‬222 ͷ৔߹ʹ͸ɺ|P | = 4 Ͱ͋Γɺ4 ࣍‫ݩ‬ͷΤϧϛʔτߦྻ h Λద౰ʹબͿɻ(16),(17) ࣜͷ৐਺‫ܥ‬Λ‫ʹݩ‬ਖ਼ଇද‫ݱ‬Λߏ੒͠ɺh ΛରশԽ͢Δɻ͜ΕΛର֯Խ͢Ε͹ɺΤωϧΪʔ‫ݻ‬༗஋͸ 2 ॏॖୀͨ͠४Ґ͕ೋͭग़ͯ
͘Δɻ͜Ε͸ઌड़ͷ 2 ࣍‫طݩ‬໿ද‫ ͕ݱ‬2 ͭ‫·ؚ‬Ε͍ͯΔ͜
ͱʹରԠ͍ͯ͠Δɻ

2. όϯυߏ଄ͷද‫ͱݱ‬ద߹ؔ܎ʮόϯυߏ଄ʯʹରͯ͠ɺ݁থͷରশੑ͕ͲͷΑ͏ʹӨ‫ڹ‬Λ‫͢΅ٴ‬ͷ͔ʹ͍ͭͯৄࡉʹٞ࿦͢Δɻಛʹۭؒ‫܈‬ͷରশੑ͕‫ڧ‬ଋറ໛‫͏ͲͰܕ‬ද‫͞ݱ‬ΕΔ͔ʹ͍ͭͯಛʹৄ͘͠ղઆ
͢Δɻ

2.1.1. ໛‫ܕ‬ͷఆٛ‫ڧ‬ଋറ໛‫( ܕ‬tight-binding model) ͸཭ࢄ֨ࢠ఺ Λ ⊂ Rd্ʹఆٛ͞Εͨࣗ༝ཻࢠͷ໛‫͋Ͱܕ‬Γɺ͜͜Ͱ͸ಛʹࣗ༝ిࢠͷ৔߹Λٞ࿦͢Δɻ֤୯Ґ๔ʹεϐϯࣗ༝౓Λ‫ؚ‬Ίͯ߹‫ ܭ‬Norb ‫ݸ‬ͷ‫ي‬ಓ͕͋Δͱ͢ΔɻͦΕͧΕͷ୯Ґ๔Λฒ⃗ Ͱࢦఆ͠ɺಛʹ R⃗ = ⃗0 ʹ͋Δ୯Ґ๔Λ UC
ਐϕΫτϧ Rͱ‫͢ه‬ɻUC ಺ͷ s ൪໨ͷ‫ي‬ಓ (s = 1, . . . , Norb ) ͷҐஔΛ

⃗rs ∈ Λ ͱॻ͘ɻԣϕΫτϧ ĉ†⃗ ͸ɺͦͷୈ s ੒෼ (ĉ†⃗ )s ͕ s ൪໨ͷ‫ي‬ಓ
RR⃗ ʹ࡞Δੜ੒ԋࢉࢠͰ͋ΓɺϑΣϧϛΦϯ
ͷిࢠΛ୯Ґ๔ R†
ͷ൓ަ‫({ ܎ؔ׵‬ĉR⃗ )s , (ĉ ⃗ ′ )s′ } = δR,⃗ R⃗ ′ δs′ ,s Λຬͨ͢ͱ͢Δɻ
R‫ڧ‬ଋറ໛‫ܕ‬͸ɺ͜ΕΛ༻͍ͯ

⃗ R
⃗′
R,

=

k

#
d
⃗ ⃗
⃗
† i⃗
k·R
, ĉ†R⃗ = BZ dvBZk e−ik·R c̃ˆ⃗†
⃗ ĉR
⃗e
R
k

"

(vBZ ͸ୈҰϒϦϧΞϯκʔϯ BZ ͷମੵ) ͱఆٛ͢Δͱɺ
ϋϛϧτχΞϯ͸֤ ⃗k ͝ͱʹϒϩοΫର֯Խ͞Εͯ Ĥ =# dd k †c̃ˆ H̃⃗k c̃ˆ⃗k ɺੜ੒ফ໓ԋࢉࢠͷ‫֨ن‬Խ͸ {c̃ˆ⃗k , c̃ˆ⃗† ′ } =BZ vBZ ⃗kk"⃗ ⃗vBZ δ d (⃗k−⃗k ′ ) ͱͳΔɻ͜ͷఆٛͰ͸ H̃⃗k ≡ R⃗ eik·R HR⃗ ͷ೾⃗ ⃗⃗਺ґଘੑ͸ eik·R Λ௨ͯ͠ͷΈ‫ݱ‬ΕΔͷͰɺH̃⃗ ͸ ⃗k → ⃗k+ Gʹରͯ͠ෆมͰ͋Γɺ೾਺ۭؒͷେҬతੑ࣭ʢd ࣍‫ݩ‬τʔϥεͱಉ૬ʣͱ૬ੑ͕ྑ͍ɻ
ϑʔϦΤม‫ʹ׵‬͸ผͷྲّྀ΋ଘࡏ͢ΔɻͦΕ͸୯Ґ๔ͷ⃗ ͷ୅ΘΓʹ‫ي‬ಓͷ࣮ࡍͷҐஔ R⃗ + ⃗rs ∈ Λ Λ༻͍
Ґஔ R
" †
⃗
⃗
⃗
†
†
ik·(R+⃗rs )= (c̃ˆ⃗ )s eik·⃗rs ͱ͢Δ΋ͷͰͯ (ĉ⃗ )s ≡ R⃗ (ĉR⃗ )s ekk͋Δɻ2 ͭͷϑʔϦΤม‫׵‬ͷఆٛ͸ɺྫ͑͹ ĉ† = c̃ˆ† V † ɺ⃗k2.1. ‫ڧ‬ଋറ໛‫ܕ‬Ĥ =ϑʔϦΤม‫׵‬Λ c̃ˆ⃗† ≡kຊઅͰ͸ɺप‫ظ‬ϙςϯγϟϧதͷిࢠঢ়ଶΛಛ௃෇͚Δ!

2.1.2. ϑʔϦΤม‫׵‬

⃗
k ⃗
k

H⃗k = V⃗k H̃⃗k V⃗† ͱ͍ͬͨ۩߹ʹɺϢχλϦߦྻ (V⃗k )s′ ,s =k⃗⃗δs′ ,s e−ik·⃗rs Ͱ݁͹Ε͍ͯΔɻVG⃗ ̸= Ͱ͋ΔͨΊɺH⃗k ͸k →⃗k + G⃗ ʹରͯ͠ෆมͰ͸ͳ͘ɺ୅ΘΓʹ H⃗ ⃗ = V ⃗ H⃗ V †
G

k+G

⃗
k G

⃗ ͷҧ͍Λ‫׬‬શʹແࢹ͢Δ
Λຬͨ͢ɻ͜ͷͨΊɺ೾਺ ⃗k ͷ G͜ͱ͕Ͱ͖ͳ͘ͳͬͯ͠·͏ͷͰ஫ҙΛཁ͢Δɻ͜ͷ 2 ͭͷϑʔϦΤม‫׵‬ͷఆٛͷҧ͍͸༷ʑͳࠞཚͷ‫ݪ‬ҼͱͳΔ͕ɺͦΕͧΕΛίϯγεςϯτʹ࢖͍ͬͯΔ‫ݶ‬Γ෺ཧతͳࠩҟ͸ੜ͡ಘͳ͍ɻຊߘͰ͸Ұͭ໨ͷఆٛͰٞ࿦
ΛਐΊΔʢνϧμ͕ࢸΔॴʹ‫ݱ‬Εͯ͠·͍൥Θ͍͕͠ɺจ‫ݙ‬5) ͷ‫ه‬๏ʹ߹ΘͤΔͨΊʹ͜ͷΑ͏ʹͨ͠ʣɻ

2.1.3. ‫ڧ‬ଋറ໛‫ܕ‬ͷରশੑ‫ڧ‬ଋറ໛‫͍͓ͯʹܕ‬ɺۭؒ‫܈‬ͷ‫ ݩ‬g ∈ G ͸ c̃ˆ⃗† Λkĉ†R⃗ ′ HR−⃗ R⃗ ′ ĉR⃗!!

⃗ R⃗ ′ s′ sR,(ĉR⃗ ′ )†s′ (HR−⃗ R
⃗ ′ )s′ ,s (ĉR⃗ )sĝ c̃ˆ⃗† ĝ −1 = c̃ˆ† ⃗ Ũ⃗k (g),
k

(31)

(32)

g(k)

ͱม‫͢׵‬Δͱ͢Δɻ͜͜Ͱ g(⃗k) = pg⃗k ͸ g ͷ೾਺ ⃗k ΁ͷ࡞༻Ͱ͋Δʢ1.2.2 અʣɻ·ͨ Ũ⃗k (g) ͸ Norb ࣍‫ݩ‬ϢχλϦߦ⃗⃗ ⃗′ͱఆٛ͞ΕΔ7 ɻH∆R
⃗ (∆R = R− R ) ͸ඈͼҠΓੵ෼΍εϐϯ‫ي‬ಓ૬‫Ͳͳ༻࡞ޓ‬Λද͢ Norb ࣍‫ Ͱྻߦݩ‬H † ⃗ = H−∆R
⃗
∆R
⃗
⃗′
Λຬͨ͢ɻฒਐରশੑ T ΛԾఆ͠ɺH∆R⃗ ͕ R ͱ R ͷࠩʹ͔͠ґଘ͠ͳ͍͜ͱΛ‫͍͍ͯ༻ʹط‬Δɻ͞Βʹϋϛϧτχ⃗ ʹରͯ͠े෼ૣ
Ξϯͷ‫ॴہ‬ੑɺͭ·Γ H ⃗ ͕େ͖ͳ |∆R|∆R

͘‫ݮ‬ਰ͢Δ͜ͱ΋ཁ੥͢Δɻ͜Ε͸όϯυߏ଄ͷ࿈ଓੑ΍ղੳੑͱؔ܎͢ΔॏཁͳԾఆͰ͋ΔɻҎԼʹࣔ͢Α͏ʹ Ĥ Λର֯Խ͢Δ͜ͱͰ Norb ຊͷΤωϧΪʔόϯυ͕ಘΒΕΔɻ͜ͷόϯυߏ଄ʹରۭؒ͠‫܈‬ͷରশੑ͔ΒͲͷΑ͏ͳҰൠత੍‫ݶ‬Λ͔͚Δ͜ͱ͕Ͱ͖Δͷ͔͕ຊߘͷয఺ͷҰͭͰ͋ΔɻྻͰ

Ũg′ (⃗k) (g)Ũ⃗k (g ′ ) = ω sp (g, g ′ )Ũ⃗k (gg ′ ),(33)Ũ⃗k+G⃗ (g) = Ũ⃗
k (g)

(34)
⃗ ⃗

−ik·R
Λຬͨ͢ɻྫ͑͹ g ͕ฒਐͷ৔߹͸ Ũ⃗k (TRͰ͋⃗) = e⃗†−i⃗ˆΓɺ͜Ε͸ঢ়ଶ (c̃ )s |0⟩ ͕ฒਐରশੑ T̂ ⃗ ͷ‫ݻ‬༗஋ e k·R⃗kRΛ࣋ͭ‫ݻ‬༗ঢ়ଶͰ͋Δ͜ͱΛҙຯ͍ͯ͠Δɻω sp (g, g ′ ) ͸

1.3.2 અͰٞ࿦ͨ͠ిࢠεϐϯʹ‫ى‬Ҽ͢Δ৐਺‫͋Ͱܥ‬ΓɺిࢠεϐϯΛߟྀ͠ͳ͍৔߹΍εϐϯͷ SU(2) ճసରশੑΛ
Ծఆ͢Δ৔߹ʹ͸ ω sp (g, g ′ ) = 1 ͱ͢Ε͹Α͍ɻŨ⃗k (g) ͸‫ي‬ಓͷҐஔ ⃗rs ͕ଐ͢ΔϫΠίϑҐஔ΍‫ي‬ಓͷछྨͳͲʹґଘ͢Δɻͦͷ۩ମతͳදࣜ͸ଟগࠐΈೖΔͷͰɺ2.3 અͰվΊ
ͯٞ࿦͢Δ͜ͱʹ͢Δɻ

7 ࿈ଓ্ۭؒͷ໛‫ܕ‬ʢΛ → Rd , N

orb → ∞ ʹରԠʣ΋ద౰ͳ཭ࢄԽ

Λ͢Ε͹͜ͷ‫͖Ͱ͕ͱ͘͜ॻʹܗ‬ΔͷͰɺࣗ༝ిࢠ‫ܥ‬ͷ໛‫ͯ͠ͱܕ‬͸ඇৗʹҰൠతͳΫϥεΛѻ͍ͬͯΔ͜ͱʹͳΔɻ·ͨ Ĥ ͸ిࢠ਺ N̂el ≡" dd k †! †c̃ˆ c̃ˆ⃗ Λอଘ͢Δɻ⃗ ĉ ĉ ⃗ =⃗ R
R R

BZ vBZ ⃗k k

Ĥ ͕ରশੑ ĝ ͷ࡞༻ͷ΋ͱͰෆมͰ͋Δ͜ͱɺ͢ͳΘͪ$dd k ˆ†c̃ ⃗ Ũ⃗k (g)H̃⃗k Ũ⃗k (g)† c̃ˆg(⃗k) = Ĥ (35)ĝ Ĥ ĝ −1 =g(k)BZ vBZ

《講義ノート》

物性研究・電子版 Vol. 8, No. 1, 081202（2020年2月号）

Λཁ੥͢Δͱɺ৚݅ Ũ⃗k (g)H̃⃗k Ũ⃗k (g)† = H̃g(⃗k) ͋Δ͍͸Ũ⃗k (g)H̃⃗k = H̃g(⃗k) Ũ⃗k (g)(36)͕֤‫ ݩ‬g ∈ G ʹରͯ͠ಘΒΕΔɻĤ ͸͢΂ͯͷ ⃗k Λ·ͱΊͯΈ͍ͯΔͷͰ ĝ ͱަ‫͢׵‬Δ͕ɺ֤ ⃗k Λผ‫ʹݸ‬Έ͍ͯΔ

࣮ࡍͷ‫Ͱࢉܭ‬͸ Ũ⃗k (h) Λ (37) ࣜͷΑ͏ʹϒϩοΫର֯Խ
͢ΔߦྻΛఱԼΓతʹ‫͚ͭݟ‬Δ͜ͱ͸༰қͰ͸ͳ͍ɻΉ͠
Ζ H̃⃗k Λ (38) ࣜͷΑ͏ʹ௚઀ର֯Խͯ͠‫ݻ‬༗ϕΫτϧ Ψ̃⃗kΛ‫ٻ‬ΊɺͦΕΛ༻͍ͯߦྻཁૉ

[U⃗αn (h)]i′ ,i = ψ̃⃗†Ũ⃗k (h)ψ̃⃗kni(40)H̃⃗k ͸ඞͣ͠΋ Ũ⃗k (g) ͱ͸ަ‫ͣͤ׵‬ɺҰൠʹ͸೾਺͕ g(⃗k)΁ͱҠΓมΘΔɻΛ‫͢ࢉܭ‬Δ͜ͱʹΑͬͯ U⃗αn (h) Λ‫ٻ‬ΊΔ͜ͱ͕Ͱ͖Δɻ

2.1.4. ֤ ⃗k ͰͷରশੑͱϋϛϧτχΞϯͷର֯Խ

2.1.5. ҟͳΔ೾਺Λ݁Ϳ sewing matrixͦ͜Ͱ ⃗k ͷখ‫ ܈‬G⃗k ʢ1.2.3 અʣʹண໨͠Α͏ɻh ∈ G⃗k ʹରͯ͠͸ Ũ⃗k (h) ͱ H̃⃗k ͕ަ‫͢׵‬ΔͷͰɺH̃⃗k ͷ‫ݻ‬༗஋΍‫ݻ‬༗ϕΫτϧʹର੍ͯ͠‫ݶ‬Λ͔͚Δ͜ͱ͕Ͱ͖Δɻ(33) ࣜʹΑΔͱ Ũ⃗k ͸ G⃗k ͷࣹӨද‫͋Ͱݱ‬Δ͕ɺҰൠʹ͸

kni′

k

k

લઅͰ͸খ‫ ܈‬G⃗k ͷ‫͍ͯͭʹݩ‬ٞ࿦ͨ͠ͷͰɺ͜͜Ͱ͸ΑΓҰൠͷ‫ ݩ‬g ∈ G ʹ͍ͭͯߟ͑Α͏ɻH̃⃗k ͷ‫ݻ‬༗ঢ়ଶψ̃⃗kni (i = 1, . . . , D⃗αn ) ʹ Ũ⃗k (g) Λ࡞༻ͤ͞Δͱɺ(8) ࣜ΍k(36) ͔ࣜΒɺŨ⃗k (g)ψ⃗kni ͸ϋϛϧτχΞϯ H̃g(⃗k) ͱฒਐରՄ໿Ͱ͋Γɺ‫ط‬໿ද‫ ݱ‬U⃗α (α = 1, . . . , N⃗k ) ΁ͱ‫ط‬໿෼ղͰশੑ Ũg(⃗k) (TR⃗ ) ͷಉ࣌‫ݻ‬༗ঢ়ଶͰ͋Γɺ‫ݻ‬༗஋͸ͦΕͧΕ͖Δɻͭ·Γɺ͢΂ͯͷ h ∈ G⃗k ʹରͯ͠‫ڞ‬௨ͷϢχλϦߦεg(⃗k)n = ε⃗kn ͱ e−ig(k)·R Ͱ͋Δ͜ͱ͕෼͔Δɻ͕ͨͬͯ͠kྻ Ψ̃⃗k ΛҰͭબΜͰ
⎛
U α1 (h)⎜ ⃗k⎜ 0⎜Ũ⃗k (h) = Ψ̃⃗k ⎜ .
⎜ ..
⎝
0

0

···

U⃗α2 (h)
k
...

···

0

···

⎞

0
0
..
.

αM⃗

U⃗

k

k

(h)

⎟
⎟
⎟ †
⎟ Ψ̃⃗k⎟⎠(37)ͱϒϩοΫର֯Խ͢Δ͜ͱ͕Ͱ͖Δɻ͜ͷ‫ج‬ఈͰ͸ H̃⃗k ͸
⎛
⎞
0
···
0
ε1⃗k
⎜
⎟
⎜ 0
ε2⃗k
···
0 ⎟
⎜
⎟ †
Ψ̃
(38)
H̃⃗k = Ψ̃⃗k ⎜ .
..
.. ⎟
⎜ ..
⎟ ⃗k
.
.
⎝
⎠
0
0
· · · εM⃗ ⃗kkͷΑ͏ʹࣗಈతʹର֯Խ͞Ε͍ͯΔ͜ͱ͕γϡʔΞͷิ୊1) ͔ΒΘ͔ΔɻĤ ʹ෇Ճతͳରশੑ͕ͳ͚Ε͹ɺҰൠʹ ε⃗kn ͸શͯҟͳΔɻ͕ͨͬͯ͠ΤωϧΪʔ‫ݻ‬༗஋ ε⃗kn(n = 1, . . . , M⃗k ) ʹ͔͔Δ୯Ґߦྻͷ࣍‫ݩ‬͸͜ͷΤωϧΪʔ‫ݻ‬༗஋ͷॖୀ౓Λද͠ɺ͜Ε͸‫ط‬໿ද‫ ݱ‬U⃗αn ͷ࣍‫ݩ‬kD⃗αn ͱҰக͢Δɻ΋ͱ΋ͱ H̃⃗k ͕ Norb ࣍‫ݩ‬ͷߦྻͰ͋ͬk"ͨ͜ͱ͔Β໌Β͔ʹ n D⃗αn = Norb ͕ຬͨ͞Ε͍ͯͳ͚kΕ͹ͳΒͣɺ͜ΕͰ‫ ܭ‬Norb ຊͷΤωϧΪʔόϯυ͕ಘΒΕͨ͜ͱʹͳΔɻ֤ ε⃗ Λ ⃗k ͷؔ਺ͱͯ͠ϓϩοτͨ͠΋ͷkn͕ΤωϧΪʔόϯυਤͰ͋Δɻ(38) ࣜʹΑΕ͹ɺΨ̃⃗k ͷ֤ྻ͸ H̃⃗k ͷ‫ݻ‬༗ϕΫτϧΛද
͍ͯ͠Δɻ‫ݻ‬༗஋ ε⃗kn Λ࣋ͭ‫ݻ‬༗ϕΫτϧͷ͏ͪ i ൪໨⃗Ũ⃗k (g)ψ̃⃗kni ͸Ũ⃗k (g)ψ̃⃗kni =Ũ⃗k (h)ψ̃⃗kni =!
i′ψ̃⃗kni′ [U⃗αn (h)]i′ ,i
k

(39)

k

8 u ͕༻͍ΒΕΔ͜ͱ͕ଟ͍͕ɺද‫ݱ‬ͷ u ͱฆΒΘ͍͠ͷͰຊߘͰ͸ψ ͱͨ͠ɻ

ψ̃g(⃗k)ni′ [W⃗kn (g)]i′ ,i(41)ͱల։Ͱ͖Δ͸ͣͰ͋Δɻ͜͜ʹ‫ݱ‬Εͨ W⃗kn (g) ͸ D⃗αn ࣍k‫ݩ‬ϢχλϦߦྻͰɺจ‫Ͱݙ‬͸ sewing matrix ͱ‫ݺ‬͹Ε͍ͯΔ6) ɻW⃗kn (g) ΋ Ũ⃗k (g) ͱಉ༷Wg′ (⃗k)n (g)W⃗kn (g ′ ) = ω sp (g, g ′ )W⃗kn (gg ′ )(42)Λຬͨ͢ɻಛʹ h ∈ G⃗k ʹରͯ͠͸ W⃗kn (h) = U⃗αn (h) ͱkͳΔɻ

2.1.6. ϒϩοϗঢ়ଶ‫ݻ‬༗ϕΫτϧ ψ̃⃗kni Λ༻͍ͯੜ੒ԋࢉࢠ!
γ̂⃗† ≡ c̃ˆ⃗† ψ̃⃗kni =(c̃ˆ⃗k )†s (ψ̃⃗kni )s
kni

k

(43)

s

Λఆٛ͢ΔͱɺϋϛϧτχΞϯ͸ Ĥ =#dd k
BZ vBZ"n,i ε⃗kn†γ̂⃗ γ̂⃗kni ͷΑ͏ʹର֯Խ͞ΕΔɻରԠ͢ΔҰిࢠঢ়ଶ |⃗kni⟩ ≡kniγ̂⃗† |0⟩ ͸ϒϩοϗঢ়ଶ (Bloch state) ͱ‫ݺ‬͹Εɺͦͷ೾ಈ
kni

ؔ਺͸

|⃗kni⟩ =
=

!!
⃗
R

s

!!
⃗
R

s

⃗ ⃗

eik·R (ψ̃⃗kni )s (c†R⃗ )s |0⟩⃗⃗eik·(R+⃗rs ) (ψ⃗kni )s (c†R⃗ )s |0⟩

(44)
(45)

Ͱ༩͑ΒΕΔɻ͜Ε͸ϋϛϧτχΞϯ Ĥ ͱฒਐରশੑ T̂R⃗

͕ै͍ɺ{ψ̃⃗kni }i ͸‫ط‬໿ද‫ ݱ‬U⃗αn ͷ‫ج‬ఈͱͳΔ͜ͱ͕Θ

͔Δɻ

!
i′

(i = 1, . . . , D⃗αn ) ͷ΋ͷΛ ψ̃⃗kni ͱॻ͘͜ͱʹ͠Α͏8 ɻ͢kΔͱ (37) ࣜΑΓ⃗ͷҰిࢠಉ࣌‫ݻ‬༗ঢ়ଶͰ͋ΓɺΤωϧΪʔ‫ݻ‬༗஋ ε⃗kn ͱฒ⃗ ⃗ਐ‫ݻ‬༗஋ e−ik·R Λ࣋ͭɻ·ͨɺ(39) ࣜ΍ (41) ͔ࣜΒ!
|⃗kni′ ⟩[U⃗αn (h)]i′ ,i(46)ĥ|⃗kni⟩ =i′ĝ|⃗kni⟩ =

!
i′

k

|g(⃗k)ni′ ⟩[W⃗kn (g)]i′ ,i(47)

《講義ノート》

物性研究・電子版 Vol. 8, No. 1, 081202（2020年2月号）

͕ಘΒΕɺϒϩοϗঢ়ଶ {|⃗kni⟩}i ͕খ‫ ܈‬G⃗k ͷ‫ط‬໿ද‫ ݱ‬U⃗αnkͷ‫ج‬ఈΛͳ͢͜ͱ΍ɺΑΓҰൠͷ‫ ʹݩ‬g ∈ G ʹΑͬͯͲͷΑ͏ʹ೾਺఺ؒΛҠΓมΘΔͷ͔͕Θ͔Δɻ

‫ ܈‬G ͷද‫ݱ‬Λߏ੒͢Δ͜ͱ΋Ͱ͖Δɻ͜ͷΑ͏ʹɺ෦෼‫ ܈‬H < G ͷද‫ݱ‬Λ‫ߏʹݩ‬੒͞Εͨ G ͷද‫ݱ‬Λ༠ಋද‫ݱ‬(induced representation) ͱ͍͏ɻ͜ͷߟ͑ํ͸ 2.3 અͰ࣮ۭؒͰ΋༻͍ΒΕΔɻ

2.2. ೾਺఺ؒͷؔ܎
͜͜·Ͱ͸‫ج‬ຊతʹ֤೾਺ ⃗k Λผ‫ʹݸ‬ѻ͖ͬͯͨɻ࣍ʹෳ਺ͷ೾਺఺ʹ͓͚Δϒϩοϗঢ়ଶ΍ͦͷද‫ݱ‬ͷؒͷؔ܎ʹ͍ͭͯٞ࿦͠Α͏ɻ(48) ࣜͷ৒༨ྨ෼ղʹΑΕ͹ɺG ͷͲͷ‫ݩ‬΋ gℓ h (h ∈ G⃗k )ͷ‫͖Ͱ͕ͱ͘͜ॻʹܗ‬ΔͷͰɺಛʹ ggℓ ∈ G ʹରͯ͠΋

ggℓ = gℓ′ h ͱͳΔ ℓ′ ͱ h ∈ G⃗k ͕ଘࡏ͠Ұҙʹఆ·Δɻ‫ݴ‬͍‫͑׵‬Δͱɺg ∈ G ͕ ⃗kℓ Λ ⃗kℓ′ ʹҠ͢ʢͭ·Γ ⃗kℓ′ = g(⃗kℓ )ʣ
⃗ͳΒ͹ɺgℓ−1′ ggℓ ͸ k ͷখ‫ ܈‬G⃗k ʹଐ͢Δɻ͢Δͱલઅͱશ

͘ಉ༷ʹ

2.2.1. S⃗k ͷ֤఺ʹ͓͚Δখ‫܈‬ͷද‫ݱ‬ĝ|⃗kℓ ni⟩ =ۭؒ‫ ܈‬G ͷ‫ݩ‬ͷ͏ͪɺখ‫ ܈‬G⃗k ʹ‫·ؚ‬Ε͍ͯͳ͍‫ݩ‬Λ ⃗k ʹ
࡞༻ͤ͞Δͱҧ͏೾਺΁ͱҠΔɻ͜ͷΑ͏ʹ ⃗k ʹ G ͷ‫ݩ‬Λ࡞

i′ ,ℓ′G=ℓgℓ G⃗k(48)

ͱࠨ৒༨ྨ෼ղ͢Δ͜ͱͱಉ͡Ͱ͋Δɻ͍· ⃗k = ⃗k1 ʹ͓͚Δϒϩοϗঢ়ଶ |⃗kni⟩ (i = 1, . . . , Dαn )⃗kͱɺͦͷখ‫ ܈‬G⃗k ͷ‫ط‬໿ද‫ ݱ‬U⃗αn ͕༩͑ΒΕͨͱ͠Α͏ɻ͢kΔͱ೾਺ ⃗kℓ ∈ S⃗ ͷϒϩοϗঢ়ଶ |⃗kℓ nj⟩ (j = 1, . . . , D αn )⃗kk͕࣋ͭখ‫ ܈‬G⃗kℓ ͷද‫ݱ‬͸ɺରশੑͷཁ੥ʹΑΓࣗಈతʹ

ܾ·ͬͯ͠·͏ɻ͜ΕΛΈΔͨΊʹɺ·ͣ h ∈ G⃗k ͷͱ͖h′ ≡ gℓ hg −1 ∈ G⃗ ͱͳΔ͜ͱʹ஫ҙ͠Α͏ɻ͢ͳΘͪ ⃗k ͱℓkℓ⃗kℓ ͷখ‫܈‬͸‫ڞ‬໾ gℓ G⃗ g −1 = G⃗ ͷؔ܎ʹ͋Δɻ͜Ε͸
k ℓ
kℓ
′ ⃗

⃗
⃗
⃗ ⃗
h (kℓ ) = (pgℓ ph p−1
gℓ )(pgℓ k) = pgℓ ph k = pgℓ k = kℓ

(49)

(53)

sp

ℓ

ℓ

× [W⃗kn (gℓ′ )U⃗αn (h)W⃗kn (gℓ )† ]i′ ,i ,kʹରͯ͠ɺ⃗kℓ = gℓ (⃗k) ͱͳΔ gℓ ∈ G ΛબΜͰ‫ݻ‬ఆ͢Δɻಛʹ ⃗k1 = ⃗k ɺg1 = e ͱ͢Δɻ͜Ε͸ G Λ!

|⃗kℓ′ ni′ ⟩[U αn (g)]ℓ′ i′ ,ℓi ,(g,gℓ )
[U αn (g)]ℓ′ i′ ,ℓi = δ⃗k ′ ,g(⃗kℓ ) ωωsp (g′ ,h)༻ͤͯ͞ಘΒΕΔ೾਺ͷू߹Λ ⃗k ͷ੕ S⃗k ͱ͍͏ͷͰ͋ͬͨʢ1.2.3 અʣɻҎԼͷٞ࿦ͷͨΊʹɺ⃗kℓ ∈ S⃗ (ℓ = 1, . . . , |S⃗ |)k!

k

h = gℓ−1
′ ggℓ

(54)
(55)

Λࣔ͢͜ͱ͕Ͱ͖ΔɻU αn (g) ͸ |S⃗k | × D⃗αn ࣍‫ݩ‬ͷϢχλkϦߦྻͰɺ͜Ε͕ U⃗αn ͔Β༠ಋ͞Εͨ G ͷද‫͋Ͱݱ‬Γɺk{|⃗kℓ ni⟩}ℓ,i ͕ͦͷද‫جݱ‬ఈͰ͋Δɻಛʹ g = h′ ∈ G⃗kℓ ͷ৔߹ʹ͸ɺ⃗kℓ ʹ͍ͭͯͷϒϩοΫ͕ (51) ࣜʹ‫ؼ‬ண͢Δ͜ͱ͕‫ͯݟ‬औΕΔɻ

2.2.3. ‫ॴہ‬తద߹ؔ܎
͜Ε·Ͱ͸ ⃗k ͷ੕ S⃗k ʹଐ͢Δ೾਺఺ؒͷؔ܎ʹண໨͍ͯͨ͠ɻ‫ʹٯ‬ɺରশੑͰ݁ͼ͔ͭͳ͍೾਺఺ͷؒʹ͸Կ͕‫͑ݴ‬ΔͩΖ͏͔ɻ⃗k Λߴରশ఺ͱ͢Δɻ⃗k ͔Β΄ΜͷΘ͔ͣʹ཭Εͨ఺ ⃗k ′Ͱ͸Ұൠʹ͍͔ͭ͘ͷରশੑ͕ഁΕͯରশੑ͕Լ͕Δɻ͔͢Β௚ͪʹΘ͔Δɻ࣍ʹࣜ (47) Λ‫ʹٯ‬ղ͍ͯΈΔͱ!
|⃗kℓ nj⟩ =ĝℓ |⃗kni⟩[W⃗kn (g)† ]i,j(50)
iͱͳΓɺ|⃗kℓ nj⟩ Λ {|⃗kni⟩}i ͰදͤΔ͜ͱʹ஫ҙ͢Δɻۭؒ‫܈‬ͷ‫ͯ͠ͱݩ‬ͷؔ܎ࣜ h′ gℓ = gℓ h ͸ɺࣹӨද‫ݱ‬ͷ৐਺‫ܥ‬΋ߟྀ͢Δͱ ωsp (h1′ ,gℓ ) ĥ′ ĝℓ = ωsp (g1 ℓ ,h) ĝℓ ĥ ͱͳΔɻ͜ΕΒͷࣜͱ (46) ࣜΛ༻͍Δͱɺ{|⃗kℓ nj⟩}j ͸খ‫ ܈‬G⃗ ͷͳΘͪখ‫ ܈‬G⃗k′ ͸ G⃗k ͷ෦෼‫ͳͱ܈‬Δɻখ‫ ܈‬G⃗k ͷ‫ط‬໿ද‫ݱ‬U⃗kα ͸ɺͦͷ෦෼‫ ܈‬G⃗k′ ͷද‫ͯ͠ͱݱ‬͸ҰൠʹՄ໿Ͱ͋Γɺ′G⃗k′ ͷ‫ط‬໿ද‫ ݱ‬U⃗kα′ ΁ͱ‫ط‬໿෼ղͰ͖Δɻ͜ͷ‫ط‬໿෼ղʹc ͸ (28) ࣜΛ༻͍Ε͹Α͍ɻ͜ΕʹΑΓɺ⃗k ఺Ͱ Dα ॏʹ⃗kॖୀ͍ͯͨ͠ϒϩοϗঢ়ଶ͕ ⃗k ′ ʹ͓͍ͯͲͷΑ͏ʹ෼྾͢Δ͔Λௐ΂Δ͜ͱ͕Ͱ͖Δɻ

kℓ

sp

′

,gℓ )
αn
†
U⃗αn (h′ ) ≡ ωωsp(hkn (g)U⃗ (h)W⃗kn (g) ,(gℓ ,h) W⃗(51)

h = gℓ−1 h′ gℓ

(52)

kℓ

k

ͱ͍͏‫ط‬໿ද‫ʹݱ‬ଐ͢Δ͜ͱ͕Θ͔Δ9 ɻ

2.2.4. ೾਺ۭؒͷେҬతߏ଄ʹ‫ى‬Ҽ͢Δద߹ؔ܎εΫϦϡʔૢ࡞΍άϥΠυૢ࡞Λ‫ؚ‬ΉϊϯγϯϞʔϑΟο⃗ ΛಉҰࢹ͢
Ϋͳۭؒ‫ʹ܈‬ରͯ͠͸ɺ೾਺ۭؒͷ ⃗k ͱ ⃗k + GΔͨΊʹੜ͡Δ৚͕݅͋Δ7) ɻ͜Ε͸ྫΛ௨ͯ͠ཧղ͢Δ
ํ͕؆୯ͳͷͰɺҰ൪؆୯ͳྫΛΈͯΈΑ͏ɻ

2.2.2. ༠ಋ͞ΕΔۭؒ‫܈‬ͷද‫ݱ‬ྫલઅͰ͸খ‫ ܈‬G⃗k ͷද‫ݱ‬Λ‫ʹݩ‬ɺ⃗k ͷ੕ S⃗k ʹଐ͢Δ೾਺఺ͷখ‫܈‬ͷද‫ݱ‬Λߏ੒͕ͨ͠ɺ͞ΒʹҰาਐΜͰɺۭؒ!

9 ಋ ग़ ͸ ࣍ ͷ ௨ Γɻĥ′ |⃗′⃗kℓ nj⟩ =(g)† ]i,j =i ĥ ĝℓ |kni⟩[W⃗kn!
! ωsp (h′ ,gℓ )αn′⃗⃗(g)† ]i,j =(h′ )]j ′ ,j . Ұi ω sp (gℓ ,h) ĝℓ ĥ|kni⟩[W⃗j ′ |kℓ nj ⟩[U⃗knkℓ†ൠʹද‫ ݱ‬U ͱ W U W ͸ಉ஋ͳͷͰɺ(51) ࣜͷ W ͸ॏཁͰͳ͍ɻ

ϩου‫ ܈‬p21 ͸ɺSx : (x, y, z) *→ (x + a2 , −y, −z) ͱ͍͏εΫϦϡʔૢ࡞ʹΑͬͯੜ੒͞ΕΔɻεΫϦϡʔૢ࡞Λ 2 ճ‫܁‬Γฦ͢ͱ (x, y, z) *→ (x + a, y, z) ͱͳΔͷͰɺۭؒ‫܈‬ͷ‫ ͯ͠ͱݩ‬Sx2 ͸ฒਐ T⃗a1 (⃗a1 ≡ (a, 0, 0)) ͱ౳͍͠ɻ

BZ − πa < kx ≤ πa ʹ͓͚Δখ‫܈‬͸ G = P 21 ࣗ਎Ͱ͋Γɺ

《講義ノート》

物性研究・電子版 Vol. 8, No. 1, 081202（2020年2月号）

અʣΛߟྀ͢Δ͔΍ɺεϐϯ‫ي‬ಓ૬‫༻࡞ޓ‬ΛऔΓೖΕΔ͔

(a)Ͳ͏͔ʢ1.3.2 અʣʹґଘ͢Δɻ

2.3. ‫ڧ‬ଋറ໛‫ܕ‬ͷରশੑͷද‫ݱ‬(b)

2.3.1. site symmetry ͱ‫ي‬ಓྔࢠྗֶͷߨٛͰֶͿΑ͏ʹɺਫૉ‫ࢠݪ‬ͷిࢠͷγϡϨʔσΟϯΨʔํఔࣜͷղ͸ɺΤωϧΪʔ४ҐΛ۠ผ͢Δओྔ
ࢠ਺ n ͷଞʹ‫ي‬ಓ֯ӡಈྔͷྔࢠ਺ ℓ ͱ࣓‫ ਺ࢠྔؾ‬m Ͱϥϕϧ෇͚͞Εɺℓ = 0, 1, 2, . . . Λ࣋ͭిࢠ‫ي‬ಓ͸ͦΕͧΕ

s, p, d, . . . ‫ي‬ಓͱ‫ݺ‬͹ΕΔɻ࣮ࡍʹ‫ڧ‬ଋറ໛‫ܕ‬Λߏ੒͢Δࡍਤ 2 (a) ϩου‫ ܈‬p21 ͷରশੑΛ࣋ͭ‫ڧ‬ଋറ໛‫ܕ‬ͷྫɻ࠷ۙ઀ͷඈͼҠΓੵ෼Λਤͷ޲͖ʹ teiθ ͱͨ͠ɻ(b) ͜ͷ‫ڧ‬ଋറ໛‫ܕ‬ͷόϯυߏ଄εkx n = ±2t sin( kx2 a − θ)ɻkx a = 2θ Ͱ্Լͷόϯυ͕ަ͍ࠩͯ͠Δɻʹ΋͜ͷ‫يࢠݪ‬ಓͷ‫ݴ‬༿‫͍༻͕͍ݣ‬ΒΕΔ͜ͱ͕ଟ͍ɻ͠
͔͜͠ͷϥϕϧ෇͚͸ຊདྷɺ‫ٿ‬ରশੑΛ࣋ͭத৺ྗϙςϯγϟϧதͰͦ͜‫ͳີݫ‬΋ͷͰ͋Γɺ‫ݻ‬ମதͰ͸ۙࣅతʹ͸ਖ਼ͯ͘͠΋ඞͣ͠΋ਖ਼֬ͱ͸‫͍ͳ͑ݴ‬ɻ࣮ࡍɺྫ͑͹఺‫܈‬

Ukx (Sx )2 = −Ukx (T⃗a1 ) = −e−ikx a Λຬͨ͢ɻ͕ͨͬͯ͠
͜ͷখ‫܈‬ͷ 2 ͭͷ‫ط‬໿ද‫ݱ‬͸Oh ͷରশੑΛ࣋ͭ݁থ৔ʹΑͬͯ d ‫ي‬ಓͷॖୀ͕ղ͚ͯɺ
Oh ͷʢઢ‫ܕ‬ʣ‫ط‬໿ද‫͋Ͱݱ‬Δ T2g ʢ3 ࣍‫ݩ‬ද‫ݱ‬ʣ΍ Eg ʢ2࣍‫ݩ‬ද‫ݱ‬ʣΛ࣋ͭ‫ي‬ಓʹ෼྾͢Δͱ͍ͬͨ͜ͱ͕‫͜ى‬Δɻ

i
Ukαx (Sx ) = αie− 2 kx a(56)

Ͱ͋Δʢα = ±1ʣɻ͜͜Ͱࢦ਺ͷ‫ ʹݞ‬12 ͕‫ݱ‬Ε͍ͯΔ͜ͱ

͕ॏཁͰɺ͜ͷͨΊʹ‫ط‬໿ද‫ݱ‬͸ kx → kx + 2π
a ʹର͢Δप‫ظ‬Λຬ͍ͨͯ͠ͳ͍ɻΉ͠Ζ kx → kx + 2π
a ʹΑͬͯೋͭͷද‫͕ݱ‬ೖΕସΘΔ [ਤ 2(b)]ɻ͜ͷ͜ͱ͔Βɺද‫ ݱ‬Uk+x (Sx )͜ͷΑ͏ͳޮՌΛ‫ܥ‬౷తʹऔΓೖΕΔͨΊʹɺ‫يࢠݪ‬ಓͷ͜ͱ͸Ұ୴શͯ๨Εɺରশੑ͚ͩʹ‫͍ͯͮج‬ʮ‫ي‬ಓʯΛߏ੒͢Δ͜ͱʹ͠Α͏ɻ୯Ґ๔ UC ಺ͷҰ఺ ⃗r ʹண໨͠Α͏ɻ఺ ⃗r Λෆಈ఺ʹ࣋ͭ G ͷ෦෼‫܈‬Λ ⃗r ͷখ‫ ܈‬G⃗r ͱ͍͏ͷͰ͋ͬͨʢ1.1.6 અʣɻΛ࣋ͭϒϩοϗঢ়ଶͱ Uk−x (Sz ) Λ࣋ͭϒϩοϗঢ়ଶ͸ඞͣG⃗r ͸͋ΒΏΔ݁থ৔ͷޮՌ΋औΓೖΕ্ͨͰ࠷ऴతʹ࢒ΔϖΞͰ‫ݱ‬ΕɺͦͷΤωϧΪʔόϯυ͸ − πa < kx ≤ πa ͷͲ‫ݺ‬͹ΕΔɻ͕ͨͬͯ͠఺ ⃗r ʹఆٛͰ͖Δ‫ي‬ಓͷछྨ͸ɺω sp‫ط‬໿ද‫ ݱ‬U⃗α ͷඇप‫ظ‬ੑ͸ (34) ࣜͱ͸ໃ६͠ͳ͍ɻ࣮ࡍ

Λ৐਺‫͢ͱܥ‬Δ G⃗r ͷ‫ط‬໿ද‫ ݱ‬u⃗βr (β = 1, . . . , N⃗r ) ʹΑ͔ͬ͜Ͱඞͣަࠩ͠ͳ͚Ε͹͍͚ͳ͍͜ͱ͕Θ͔Δ [ਤ 2(b)]ɻ
k͜ͷରশੑΛ΋ͭϛχϚϜͳ‫ڧ‬ଋറ໛‫[ ܕ‬ਤ 2(a)] Λߟ͑Δ
ͱɺSx ͷද‫ݱ‬͸

Ũkx (Sx ) =+01−e−ikx a0,఺ ⃗r पΓͷ‫ͳີݫ‬ରশੑΛද͓ͯ͠Γɺsite symmetry ͱͯ෼ྨ͞ΕΔɻ͍·খ‫ ܈‬G⃗r ͷ‫ط‬໿ද‫ ݱ‬u⃗βr ΛҰͭબͼɺͦͷ࣍‫ݩ‬Λ D⃗rβ
⃗ = ⃗0 ͷ୯ͱॻ͘ɻੜ੒ԋࢉࢠ (ĉ⃗ )† (i = 1, . . . , Dβ ) ͸ɺR
0 i

(57)

⃗
r

Ґ๔಺ͷҐஔ ⃗r ʹ i ൪໨ͷ‫ي‬ಓΛ࣋ͭిࢠΛ࡞ΔԋࢉࢠͰɺĥ(ĉ⃗0 )†i ĥ−1 =!

(ĉ⃗0 )†i′ [u⃗βr (h)]i′ ,i

(58)

i′

10

ͱͳΔ ͕ɺ͜ͷ Ũkx (Sx ) ͸ (34) ࣜΛຬ͍ͨͯ͠Δɻ

Λຬͨ͢ͱ͠Α͏ɻখ‫ ܈‬G⃗r ͸ۭؒ‫ ܈‬G ͷ෦෼‫ͳ܈‬ͷͰɺ͜

2.2.5. όϯυઈԑମΛ࣮‫͢ݱ‬ΔϑΟϦϯά
͜͜·Ͱɺ೾਺఺ ⃗k ʹ͓͚Δϒϩοϗঢ়ଶ͕ G⃗k ͷ‫ط‬໿ද‫ݱ‬ͷ‫ج‬ఈͱͳΔ͜ͱ΍ɺBZ ಺ͷෳ਺ͷ೾਺఺ʹ͓͚Δද‫ݱ‬ͷؒʹؔ܎͕͋Δ͜ͱΛΈ͖ͯͨɻ͜ΕΒͷؔ܎ʹΑͬͯɺྫ͑͹ਤ 2(b) ʹࣔͨ͠Α͏ͳόϯυަ͕ࠩ‫ݱ‬ΕΔɻόϯυઈԑମΛ࣮‫͢ݱ‬ΔͨΊʹ͸ɺΤωϧΪʔ͕௿͍ํ͔Β ν ຊͷόϯυΛ‫׬‬શʹ઎༗͠ɺ͔ͭඇ઎༗Ͱ͋Δ ν + 1 ຊ໨ͷͷ G⃗r ͷද‫ ݱ‬u⃗βr ͔Β G ͷද‫͕ݱ‬༠ಋ͞ΕɺͦΕ͕ 2.1.3 અͰ༻͍ͨ‫ڧ‬ଋറ໛‫ܕ‬ͷରশੑͷද‫ͳͱݱ‬Δɻ

2.3.2. ‫ڧ‬ଋറ໛‫ܕ‬ͷରশੑͷද‫ݱ‬ͷ۩ମ‫ܗ‬఺ ⃗r ʹ G ͷ‫ݩ‬Λ࡞༻͍ͤͯ͘͞ͱɺG ෆมͳ֨ࢠ Λ⃗r ͕Ͱ͖ΔͷͰ͋ͬͨʢ1.1.6 અʣɻରশੑ G Λ࣋ͭ‫ڧ‬ଋറ໛‫ܕ‬Λߏ੒͢ΔͨΊʹ͸ɺΛ⃗r ͷ֤఺ʹ఺ ⃗r ͷ΋ͷͱʮ౳Ձʯͳ

όϯυͱͷؒʹόϯυΪϟοϓ͕։͍͍ͯΔΑ͏ʹ͢Δඞ

‫ي‬ಓ͕ఆٛ͞Ε͍ͯΔඞཁ͕͋Δɻ͜ΕΛ۩ମతʹߏ੒͢

ཁ͕͋Δɻ͔ۭؒ͠͠‫܈‬ͷରশੑʹΑͬͯอূ͞ΕΔόϯΔͨΊʹɺΛ⃗r ͷ఺ͷ͏ͪ୯Ґ๔ UC ʹଐ͍ͯ͠Δ΋ͷ Λ⃗UCrυަࠩͷͨΊʹɺόϯυઈԑମΛ࣮‫͖Ͱݱ‬Δ ν ͷ஋ʹ͸ۭΛ·ͣߟ͑Δɻ⃗rℓ ∈ Λ⃗UC(ℓ = 1, . . . , |Λ⃗UCrr |) ͷ֤఺ʹରؒ͠‫܈‬ͷ੍‫͔͔͕ݶ‬Δ͜ͱʹͳΔɻ༩͑ΒΕۭͨؒ‫ ܈‬G ʹରͯ͠ɺόϯυઈԑମΛ࣮‫͠ݱ‬ಘΔ઎༗όϯυ਺ ν Λ‫ٻ‬ΊΔ໰୊͸จ‫ݙ‬8) Ͱௐ΂ΒΕͨɻ݁Ռ͸࣌ؒ൓సରশੑʢ2.4.1
10 ࣍અ (64) ࣜͰ G = {e}, uβ (e) = 1, D β = 1ɺΛUC = {⃗r, ⃗r + 12 ⃗a1 },
⃗
r
⃗
r
⃗
r
g2 = Sx ͱ͢Δɻͯɺ⃗rℓ = gℓ (⃗r) ͱͳΔ gℓ ∈ G ΛબΜͰ‫ݻ‬ఆ͢Δɻ͜ΕʹΑ

ΓG͸

G=

!
⃗
R,ℓ

TR⃗ gℓ G⃗r(59)

《講義ノート》

物性研究・電子版 Vol. 8, No. 1, 081202（2020年2月号）⃗ + ⃗rℓ ʹ i ൪໨ͱࠨ৒༨ྨ෼ղ͞ΕΔɻ͜ΕΛ༻͍ͯҐஔ R†ͷ‫ي‬ಓΛ࡞Δੜ੒ԋࢉࢠ (ĉR⃗ )ℓi Λɺ(ĉR⃗ )†ℓi ≡ T̂R⃗ ĝℓ (ĉ⃗0 )†i ĝℓ−1 T̂R⃗−1

(60)

(⃗
r,β)
΍ϊϯγϯϞʔϑΟοΫͳม‫׵‬͸ ⃗tg ̸= ⃗0 ͳͷͰɺU⃗(g)k⃗͸ k ʹґଘ͢Δɻ

ҰํͰɺ(39) ࣜͷ U⃗αn (g) ΍ (41) ࣜͷ W⃗kn (g) ͸ಛʹมkߋΛड͚ͣɺೋͭͷϑʔϦΤม‫׵‬ͷఆٛʹ‫ڞ‬௨Ͱ͋ΔɻͭʹΑͬͯఆΊΔ͜ͱʹ͢Δɻ͜ΕͰ‫ج‬ఈ͕ߏ੒Ͱ͖ͨͷͰɺ·Γϒϩοϗঢ়ଶ͕࣋ͭ‫ط‬໿ද‫ʹݱ‬͸ɺϑʔϦΤม‫׵‬ͷఆ‫ޙ‬͸ g ∈ G ͷ࡞༻ʹΑͬͯ͜ΕΒ͕‫Ͳʹ͍ޓ‬ͷΑ͏ʹҠΓ⃗ + ⃗r⃗ + ⃗rℓ ͕ R̃มΘΔ͔Λௐ΂Ε͹Α͍ɻ͍· g ʹΑͬͯ R

ٛʹґଘ͢Δᐆດ͞͸ଘࡏ͠ͳ͍ɻℓ̃ʹҠΔͱ͢Δɻ͢Δͱ 2.2.2 અͱಉ༷ͷखॱͰĝ(c†R⃗ )ℓi ĝ −1 =! †sp(g,gℓ ) β′(c ⃗ )ℓ̃i′ ωωsp (gr (h)]i ,i ,,h) [u⃗R̃i′ℓ̃h = gℓ̃−1 T ⃗−1 gTR⃗ gℓR̃࠷‫ʹޙ‬ɺ͜͜·Ͱ৮Εͯ͜ͳ͔ͬͨ࣌ؒ൓సରশੑʹͭ

(61)
(62)

Λࣔ͢͜ͱ͕Ͱ͖Δɻ͜ΕΛϑʔϦΤม‫ͯ͠׵‬গ͠ॻ͖‫׵‬͑Δͱ9)ĝ(c̃⃗† )ℓi ĝ −1 =
k

ℓ′ i′

͍ͯ·ͱΊΑ͏ɻ

2.4.1. ࣌ؒ൓సT Λ࣌ؒ൓సૢ࡞ͱ͢Δɻ͜Ε΋ྔࢠྗֶͷߨٛͰश͏Α͏ʹɺT̂ ͸൓ϢχλϦԋࢉࢠͰ͋Γ T̂ (c1 |ψ1 ⟩ + c2 |ψ2 ⟩) =

! †
(⃗
r,β)
(c̃⃗ )ℓ′ i′ [Ũ⃗(g)]ℓ′ i′ ,ℓi ,k2.4. ࣌ؒ൓సͱ࣓‫܈ۭؒؾ‬

(63)

k

⃗
(⃗
r,β)
[Ũ⃗
(g)]ℓ′ i′ ,ℓi = δ⃗r′ ℓ′ ,g(⃗rℓ ) e−ig(k)·(g(⃗rℓ )−⃗rℓ′ )ksp(g,gℓ )× ωωsp (g[u⃗βr (h)]i′ ,i ,ℓ′ ,h)

h = gℓ−1′ T⃗
rℓ′ −g(⃗rℓ ) ggℓ

(64)
(65)

c∗1 T̂ |ψ1 ⟩ + c∗2 T̂ |ψ2 ⟩ Λຬͨ͢ɻ·ͨ |ψ1 ⟩ ͱ |ψ2 ⟩ ͷ಺ੵΛg(|ψ1 ⟩, |ψ2 ⟩) ≡ ⟨ψ1 |ψ2 ⟩ ͱॻ͘ͱ͖ɺg(T̂ |ψ1 ⟩, T̂ |ψ2 ⟩) =g(|ψ1 ⟩, |ψ2 ⟩)∗ ͱͳΔɻ࣌ؒ൓సԋࢉࢠͷࣗ৐͸ిࢠ਺ N̂el ≡
" †
⃗ ĉR
⃗ ʹґଘͯ͠
⃗ ĉR
R
T̂ 2 = (−1)N̂el(67)ΛಘΔɻͨͩ͠ (64) ࣜʹ͓͍ͯɺδ⃗x′ ,⃗y ͸ɺฒਐϕΫτϧ෼

ͱͳΔͨΊɺϋϛϧτχΞϯ͕࣌ؒ൓సରশੑΛ࣋ͭͳΒ

x = ⃗y ͷͱ͖ 1 ͰɺͦΕҎ֎ͷͱ͖ 0 Ͱ͋ͷҧ͍Λআ͍ͯ ⃗͹‫ࢠి਺ح‬ঢ়ଶʹରͯ͠͸ΫϥϚʔεॖୀ͕‫͜ى‬Δ11 ɻ·Δͱఆ͍ٛͯ͠Δɻ(⃗r, β) ͸‫ط‬໿ද‫ ݱ‬u⃗βr ͔Β༠ಋ͞Εͨ΋
(⃗
r,β)
(g) ͕
ͷͰ͋Δ͜ͱΛ໌ࣔ͢ΔͨΊʹఴ͑ͨɻ͜ͷ Ũ⃗kͨ࣌ؒ൓సԋࢉࢠ͸ۭؒ‫܈‬ͷૢ࡞ͱՄ‫׵‬ɺͭ·ΓT̂ ĝ = ĝ T̂

2.1 અͷ Ũ⃗k (g) Ͱ͋Γɺ(33) ࣜ΍ (34) ࣜΛຬ͍ͨͯ͠Δ͜

ͱ͕֬ೝͰ͖Δɻ‫ي‬ಓͷϥϕϧ s ʹ͸ ℓ = 1, . . . , |Λ⃗UCr |ͱ

i = 1, . . . , D⃗rβ ͷϖΞ͕ରԠ͓ͯ͠Γɺ߹‫ܭ‬ͷόϯυ਺͸

Norb = D⃗rβ |Λ⃗UCrʹ
r | ͱͳΔɻ(64) ࣜ͸ɺ୯Ґ๔ UC ಺ͷ఺ ⃗‫ط‬໿ද‫ ݱ‬u⃗βr ʹଐ͢Δ‫ي‬ಓΛ͓͍ͨ৔߹ͷ‫ڧ‬ଋറ໛‫ܕ‬ͷද‫ݱ‬Λ༩͑ΔॏཁͳެࣜͰ͋Γɺ3 અͰόϯυߏ଄ͷτϙϩδʔΛ୳Δࡍʹ‫׆‬༂͢Δɻ͜͜·Ͱ͸୯Ґ๔ UC ಺ͷҰ఺ʹ‫ط‬໿ද‫ي͏ैʹݱ‬ಓΛ(68)Λཁ‫͢ٻ‬Δɻ͢Δͱ (8) ࣜͱಉ༷ʹͯ͠ T (⃗k) = −⃗k ͕ै͏ɻҰిࢠঢ়ଶʹର͢Δද‫ݱ‬ͷϢχλϦߦྻͷ෦෼Λ U (T )ͱॻ͘ͱɺ(67) ࣜɺ(68) ࣜ͸ U (T )U (T )∗ = − ʢͭ·Γω sp (T , T ) = −1ʣɺU (T )U⃗k (g)∗ = U−⃗k (g)U (T ) ͱද‫͞ݱ‬ΕΔɻ͏·͘‫ج‬ఈΛબͿͱ U (T ) ͸ −iσy ⊗ ͱॻ͘͜ͱ͕
Ͱ͖Δɻ

ఆٛͨ͠৔߹Λߟ͑ͨɻΑΓҰൠͷ‫ڧ‬ଋറ໛‫Ͱܕ‬͸ɺಉ͡Ґஔʹෳ਺ͷ‫ط‬໿ද‫ݱ‬Λஔ͘͜ͱ΋Ͱ͖Δ͠ɺΛ⃗r ʹଐ͞ͳ′͍ผͷҐஔ ⃗r ʹαΠτΛ௥Ճͯ͠΋Α͍ɻͦͷ৔߹ɺ‫ڧ‬ଋ(⃗r,β)(g) (m⃗r,β = 0, 1, 2, . . .)kറ໛‫ܕ‬ͷද‫ݱ‬͸௚࿨ ⊕⃗r,β m⃗r,β Ũ⃗Ͱ༩͑ΒΕΔɻ

2.4.2. ࣓‫܈ۭؒؾ‬ͷλΠϓ࣓‫ͱ܈ۭؒؾ‬͸ɺ͜Ε·Ͱٞ࿦͖ͯͨ͠ϢχλϦͳۭؒ‫܈‬G ʹՃ͑ͯɺ൓ϢχλϦͳ‫͔ݩ‬ΒͳΔू߹ A Λۭ࣋ͭؒ‫܈‬
Ͱ͋Δɻg, g ′ ∈ G ͱ a, a′ ∈ A ʹରͯ͠ɺgg ′ ∈ G, ga ∈ A,

ag ∈ A, aa′ ∈ G Ͱ͋ΔͨΊɺA ͷ‫ݩ‬ͷ਺͸ G ͷ‫ݩ‬ͷ਺ͱ౳͍͔͠ɺ͋Δ͍͸ 0 Ͱͳͯ͘͸ͳΒͳ͍ɻจ‫ݙ‬2) ʹशͬ

2.3.3. ϑʔϦΤม‫׵‬ͷೋͭ໨ͷྲّྀͰͷදࣜ
2.1.2 અͰઆ໌ͨ͠ϑʔϦΤม‫׵‬ͷೋͭ໨ͷྲّྀͰ͸ɺ͜
Ε·Ͱͷ Ũ⃗k (g) ΍ ψ̃⃗kni Λ U⃗k (g) = Vg(⃗k) Ũ⃗k (g)V⃗† ΍ ψ⃗kni =kV⃗k ψ̃⃗kni ͷΑ͏ʹมߋ͢Ε͹Α͍͜ͱ͕෼͔Δɻ͢Δͱ (64)
ࣜ͸
sp
⃗ ⃗
(⃗
r,β)
(g,gℓ )(g)]ℓ′ i′ ,ℓi = δ⃗r′ ℓ′ ,g(⃗rℓ ) e−i(pg k)·tg ωωsp (g[u⃗βr (h)]i′ ,ikℓ′ ,h)

[U⃗

(66)

(⃗
r,β)
ͷΑ͏ʹมߋ͞Εɺ⃗tg = ⃗0 Λຬͨ͢‫ʹݩ‬ରͯ͠͸ U⃗(g)k

͸ ⃗k ʹґଘ͠ͳ͍ͱ͍͏ॏཁͳੑ࣭͕ಋ͔ΕΔɻฒਐରশੑ

࣓ͯ‫܈ۭؒؾ‬Λ M = G + A ͱॻ͘͜ͱʹ͢Δɻ

3 ࣍‫͍͓ͯʹۭؒݩ‬ɺ࣓‫܈ۭؒؾ‬͸ 1,651 ‫͋ݸ‬Δͱ‫ݴ‬ΘΕΔɻ͜Ε͸େ͖͘ 4 ͭͷλΠϓʹ෼͚Δ͜ͱ͕Ͱ͖Δ2) ɻλΠϓ I ͷ࣓‫܈ۭؒؾ‬͸ɺ൓ϢχλϦ෦෼ A ΛҰ੾࣋ͨͳ͍ۭؒ‫ ܈‬G ࣗ਎Ͱ͋Γɺ230 छྨ͋ΔɻλΠϓ II ͷ࣓‫ۭؒؾ‬‫܈‬͸ɺۭؒ‫ ܈‬G ͱ࣌ؒ൓సͷ‫ ܈‬ZT2 = {e, T } ͷ௚ੵ G × ZT2

ͰɺA = GT = {gT |g ∈ G} ͷ৔߹ʹରԠ͢Δɻ͜Ε΋ 230छྨଘࡏ͢Δɻ

11 ిࢠεϐϯΛߟྀ͠ͳ͍৔߹͸ T̂ 2 = 1 ͱ͢Δɻ

《講義ノート》

物性研究・電子版 Vol. 8, No. 1, 081202（2020年2月号）

࢒ΓͷλΠϓ III, IV ͷ࣓‫܈ۭؒؾ‬͸ɺA = Gg0 T ={gg0 T |g ∈ G} ͷΑ͏ʹɺG ʹ‫·ؚ‬Εͳ͍ g0 ͱ͍͏ۭؒૢ
࡞͕ඞཁʹͳΔ΋ͷͰ͋ΔɻλΠϓ IVʢ517 छྨʣͰ͸ g0
͸ฒਐ g0 : ⃗x *→ ⃗x + ⃗tg ʹબͿ͜ͱ͕Ͱ͖Δ͕ɺ͜ͷ ⃗tg ͸Α͏ͳࣹӨද‫ڵʹݱ‬ຯ͕͋Δͱ͢Δʢp, p′ ∈ P , a, a′ ∈ Aʣɻ

u(p)u(p′ ) = ω(p, p′ )u(pp′ ),(75)

u(p)u(a) = ω(p, a)u(pa),(76)u(a)u(p)∗ = ω(a, p)u(ap),‫ج‬ຊฒਐϕΫτϧͷ൒෼ͷ‫ܗ‬Λ͍ͯ͠ΔɻλΠϓ IIIʢ674
छྨʣͰ͸ɺg0 ͸ฒਐҎ֎ͷૢ࡞Ͱ g02 ∈ G Λຬͨ͢΋ͷ

′ ∗

′

′

u(a)u(a ) = ω(a, a )u(aa ).

(77)
(78)

Ͱ͋Δɻ

A ͕ՃΘͬͨ͜ͱͰɺP ͷ‫ط‬໿ද‫ ݱ‬uα ͕ͲͷΑ͏ʹมߋΛड͚Δ͔Λௐ΂Δʹ͸ɺ࣍ͷྔΛ‫͢ࢉܭ‬Ε͹Α͍ɻ

2.4.3. ࣓‫΁܈ۭؒؾ‬ͷ֦ுξα ≡͜Ε·Ͱ੔ཧ͖ۭͯͨؒ͠‫܈‬ʢλΠϓ I ͷ࣓‫܈ۭؒؾ‬ʣͷ৔߹ʹ͍ͭͯͷ‫߲ࣄૅج‬ΛɺλΠϓ II-IV ͷ࣓‫΁܈ۭؒؾ‬x ∈ R3 Λͱ֦ு͠Α͏ɻ͍· A ͷ‫ ݩ‬a ͕ۭؒͷ఺ ⃗a(⃗x) = pa ⃗x + ⃗ta ∈ R3

(69)΁ࣸ͢ͱ͢Δͱɺ೾਺ ⃗k ∈ BZ ͸

(70)ͱม‫͞׵‬ΕΔɻෛ߸͸ a ͕࣌ؒ൓సΛ‫ؚ‬Έ൓ϢχλϦͰ͋Δ͜ͱʹ‫ى‬Ҽ͍ͯ͠Δɻ͜Ε·Ͱͷٞ࿦͸શͯɺ͜ͷ A ͷ⃗x ΍ ⃗k ΁ͷ࡞༻Λ༻͍ͯɺ͜Ε·Ͱͷ G Λ୯ʹ M = G + A΁ͱ‫ࣜܗ‬తʹஔ͖‫͑׵‬Ε͹Α͍ɻྫ͑͹ ⃗k ͷখ‫܈‬͸ M⃗k ={h ∈ M | h(⃗k) = ⃗k} ͱఆٛ͞ΕΔɻԾʹ M ͕λΠϓ II ͷ࣓‫ͯ͠ͱͨͬ͋Ͱ܈ۭؒؾ‬΋ɺM⃗k ͸λΠϓ I, II ͷΈͳΒͣλΠϓ III ͷ࣓‫ͳʹ܈ۭؒؾ‬Δ৔߹΋͋Δ͜ͱʹ஫ҙ͢Δɻ

U⃗k (a)H⃗k∗ = Ha(⃗k) U⃗k (a),(71)

Ua(⃗k) (g)U⃗k (a) = ω sp (g, a)U⃗k (ga),(72)

Ug(⃗k) (a)U⃗k (g)∗ = ω sp (a, g)U⃗k (ag),(73)

Ua′ (⃗k) (a)U⃗k (a′ )∗ = ω sp (a, a′ )U⃗k (aa′ )(74)ͷΑ͏ʹมߋ͞ΕΔɻ
ిࢠεϐϯʹ‫ى‬Ҽ͢Δ৐਺‫ ܥ‬ω sp (a, a′ ) ͸ɺ1.3.2 અͱಉ༷ͷͱ͖ pspa ≡e͜͜Ͱ χα (p) = tr[uα (p)] ͸‫ط‬໿ද‫ ݱ‬uα ͷࢦඪͰ͋Δɻξ α͸ඞͣ 0ɺ+1ɺ·ͨ͸ −1 ͷ͍ͣΕ͔ʹͳΔɻͦͷҙຯ͢

Δͱ͜Ζ͸ɺͦΕͧΕҎԼͷ௨ΓͰ͋Δɻ

⃗−iθa ⃗
na ·S(−iσy ) ͱͯ͠

sp ∗
sp
′ sp
psp
a (pa′ ) = ω (a, a )paa′ ʹΑͬͯఆΊΔɻ·ͨ M⃗
k /T ͷද⃗• ξ α = −1: 2 ͭͷ uα ͕ϖΞΛ૊Ήɻ• ξ α = +1: uα ͷ··มԽ͠ͳ͍ɻ
͜ͷ൑ఆ৚݅Λ༻͍ΔͱɺA ͷଘࡏԼͰɺόϯυߏ଄ͷ֤ ⃗kʹ͓͚Δ G⃗k ͷ‫ط‬໿ද‫Ͳ͕ݱ‬ͷΑ͏ʹϖΞΛ૊Ή͔΍ɺ‫ڧ‬ଋറ໛‫ܕ‬Λߏ੒͢ΔࡍʹҐஔ ⃗r ʹఆٛͰ͖Δ‫ي‬ಓʢG⃗r ͷ‫ط‬໿ද‫ݱ‬ʣ͕Ͳ͏ϖΞΛ૊Ή͔Λ؆୯ʹ൑ఆ͢Δ͜ͱ͕Ͱ͖Δɻξ α = 0 ͷྫ ఺‫ ܈‬2 ͸ 2 ճճస C2z ͔ΒͳΓɺP = {e, C2z }Ͱ͋Δɻuα (C2z ) = αi (α = ±) ͷ 2 ͭͷ 1 ࣍‫ݩ‬ද‫͕͋ݱ‬(33) ࣜ΍ (36) ࣜ͸ɺa ͷ൓ϢχλϦੑͷͨΊʹʹɺpa = ±e

(79)

a∈P

• ξ α = 0: uα ͱผͷද‫ ݱ‬uᾱ ͕ϖΞΛ૊Ή13 ɻa(⃗k) = −pa⃗k ∈ BZ⃗
−iθa ⃗na ·L

1 !
ω(a, a)χα (a2 ).
|P |

⃗

⃗

‫ݱ‬Λߟ͑Δࡍʹ͸ ω sp (a, a′ ) ͱ ω ns (a, a′ ) ≡ e−ik·(pa ta′ +ta′ )Δɻ͜͜ʹ࣌ؒ൓సରশੑ T ΛՃ͑ΔͱɺA ͸ A = PT ={T , C2z T } ͱͳΔɻω sp (T , T ) = −1, ω sp (C2z T , C2z T ) =+1, T 2 = (C2z T )2 = e ͳͷͰɺξ α = 12 [−χα (e) + χα (e)] =0. ͕ͨͬͯ͠ u+ ͱ u− ͸࣌ؒ൓సରশੑͷ΋ͱͰϖΞͱͳΓରͰ‫ݱ‬ΕΔɻξ α = −1 ͷྫ఺‫ ܈‬1̄ ͸ۭؒ൓స I ͔ΒͳΓɺP = {e, I}Ͱ͋Δɻuα (I) = α (α = ±) ͷ 2 ͭͷ 1 ࣍‫ݩ‬ද‫͕͋ݱ‬Δɻ

͜͜ʹ࣌ؒ൓సରশੑ T ΛՃ͑ΔͱɺA = PT = {T , IT }ͱͳΔɻω sp (T , T ) = ω sp (IT , IT ) = −1, T 2 = (IT )2 = eͳͷͰɺξ α = 12 [−χα (e) − χα (e)] = −1. ͕ͨͬͯ࣌ؒ͠൓

సରশੑͷ΋ͱͰɺu+ ͸ u+ ͱɺu− ͸ u− ͱϖΞΛ૊ΉɻͷੵΛ༻͍Δʢ1.3.3 અ12 ʣɻω(a, g) ΍ ω(g, a) ΋ಉ༷ʹఆͦͷͨΊɺ൓సରশੑͷ 2 ͭͷ‫ݻ‬༗஋͸ͦΕͧΕඞͣ‫਺ۮ‬ٛͰ͖Δɻ‫ݱݸ‬ΕΔɻξ α = +1 ͷྫ
2.4.4. ࣓‫܈఺ؾ‬ͷද‫ݱ‬͜͜Ͱ͸఺‫ ܈‬P ʹ൓ϢχλϦ෦෼ A ͕ՃΘͬͨ‫܈‬ɺͭ·Γ࣓‫ ܈఺ؾ‬M = P +A ͷද‫ݱ‬Λߟ͑Α͏ɻM = P +A ͷ࣍ͷ

12 ൓ϢχλϦૢ࡞ͷͨΊʹ ⃗ta′ ͷલͷූ߸͕ 1.3.3 અͱ͸‫͍ͯͬͳʹٯ‬Δ͜ͱʹ஫ҙɻ఺‫ ܈‬222 ͷҐ਺͸ 4 Ͱɺ୯Ґ‫ ݩ‬e ͷଞʹx, y, z ͦΕͧΕͷ࣠पΓͷ π ճసૢ࡞ C2i (i = x, y, z)Λ‫ͭ࣋ͯ͠ͱݩ‬ʢ1.3.2 અʣɻ͜͜ʹ࣌ؒ൓సରশੑ T Λ

Ճ͑ΔͱɺA = PT = {T , C2x T , C2y T , C2z T } ͱͳΔɻ

ω sp (T , T ) = −1, ω sp (C2i T , C2i T ) = +1, T 2 = (C2i T )2 =e ͳͷͰɺ2 ࣍‫ݩ‬ද‫ ݱ‬uα (C2i ) = −iσi ʹରͯ͠͸ ξ α =13 A = P a ͱॻ͘ͱ͖ɺuᾱ (p) =ω(p,a)[uα (a−1 pa)]∗ ͱͳΔɻω(a,a−1 pa)

《講義ノート》

物性研究・電子版 Vol. 8, No. 1, 081202（2020年2月号）

1αα
4 [−χ (e) + 3χ (e)] = +1 ͱͳΓɺ࣌ؒ൓సରশੑΛ‫ؚ‬Ίߟྀ͢΂͖೾਺఺ͱͯ͠͸ɺΓ ఺ͳͲͷʮ఺ʯͷଞʹɺͦΕΒΛ݁Ϳʮઢʯ΍ʮ໘ʯ΋ߴରশ೾਺Ͱ͋Ε͹औΓೖΕͯ΋ಛʹ͞ΒͳΔϖΞ͸ੜ͡ͳ͍͜ͱ͕෼͔Δɻ
൑ఆ৚݅͸ΑΓҰൠͷ࣓‫ʹ܈఺ؾ‬ରͯ͠΋੒ཱ͢Δɻిࢠ

Δඞཁ͕͋Δɻઢ΍໘ʹ͸ແ‫ʹݶ‬ଟ͘ͷ೾਺఺͕‫·ؚ‬ΕΔ͕ɺ⃗k ͱ ⃗k ′ ͕Ұͭͷઢ·ͨ͸໘্ͷ 2 ఺Ͱಉ͡খ‫܈‬Λ࣋ͭεϐϯΛߟྀ͢Δ͔൱͔ͷ৘ใ΋͢΂ͯ ω ʹ‫·ؚ‬Ε͍ͯΔɻ

ͱ͖ɺόϯυΪϟοϓͷԾఆ͔Βৗʹ n⃗α = n⃗α′ ͱͳΔɻͦҎ্ɺλΠϓ II ͷ࣓‫܈఺ؾ‬ͷྫΛ 3 ͭ঺հ͕ͨ͠ɺ͜ͷkkͷͨΊɺ࣮ࡍʹ͸֤ઢ΍໘಺ͷ೚ҙͷҰ఺ʹண໨͢Ε͹े

3. τϙϩδʔͷରশੑࢦඪ

෼Ͱ͋ΔɻҎԼͰ͸͜ͷΑ͏ʹબΜͩ༗‫ݸݶ‬ͷߴରশ೾਺

ຊઅͰ͸͜Ε·Ͱͷ४උΛ΋ͱʹɺόϯυߏ଄ͷۭ࣋ͭؒ‫܈‬ͷද‫͍ͯͮجʹݱ‬τϙϩδʔΛௐ΂Δʮରশੑࢦඪͷํ๏ʯʹ͍ͭͯղઆ͢Δɻୈ 3.1 અͰ‫ࣜܗ‬తͳఆٛΛߦͳͬͷू߹Λ K ͱॻ͘͜ͱʹ͢Δɻ߹‫ ܭ‬D ‫ݸ‬ͷ੔਺ͷ૊ {n⃗α }α,⃗k∈K ͕ಘΒΕͨͱԾఆ͠ɺ͜kΕΒΛ 1 ྻʹฒ΂ͨϕΫτϧͨ‫ޙ‬ɺୈ 3.2 અҎ߱Ͱɺి‫ؾ‬෼‫ۃ‬΍νϟʔϯ਺͕ରশੑͷ⃗b = (n1 , n2 , · · · , n1 , n2 , · · · )
⃗
⃗
⃗
⃗
k
k
k
k

ද‫Ͳͱݱ‬ͷΑ͏ʹ݁ͼ͔ͭ͘ʹ͍ͭͯྫΛަ͑ͳ͕Βৄٞ͘͠࿦͢ΔɻಛʹτϙϩδΧϧઈԑମͷதͰ΋Α͘஌ΒΕ
͍ͯΔ 2 ࣍‫ ݩ‬Z2 τϙϩδΧϧઈԑମͳͲͷ‫ج‬ຊతͳ΋ͷʹয఺ΛߜΓɺߴ࣍τϙϩδΧϧઈԑମͳͲΑΓ࠷ۙͷ࿩

1

1

2

2

(80)

Λߟ͑Α͏ɻ⃗b ͸ ZD ͷ೚ҙͷ఺ΛͱΔ͜ͱ͸Ͱ͖ͣɺྫ
͑͹

n⃗α
≥0
k

୊͸࣍અͰٞ࿦͢Δ͜ͱʹ͢Δɻ

(81)΍ɺόϯυͷຊ਺ʹؔ͢Δؔ܎ࣜ

3.1. όϯυઈԑମͷ‫ط‬໿ද‫ʹݱ‬ΑΔಛ௃෇͚
3.1.1. લઅͷ෮शn⃗αDkα = ν
k ⃗

(82)

α

·ͣୈ 2 અͷ಺༰Λ؆୯ʹ෮श͠Α͏ɻରশੑ G Λ࣋ͭ‫ڧ‬ଋറ໛‫ܕ‬ΛҰͭࢦఆ͢Δͱɺ(33) ࣜ΍ (36) ࣜΛຬͨ͠ɺ⃗k ͷؔ਺ͱͯ͠‫׈‬Β͔ͳ Norb ࣍‫ݩ‬Τϧϛʔτߦྻ H̃⃗ ͱkNorb ࣍‫ݩ‬ϢχλϦߦྻ Ũ⃗k (g) ͕ g ∈ G ʹରͯ͠ఆ·Δɻω sp (g, g ′ ) = ±1 ͸ిࢠεϐϯʹ‫ى‬Ҽ͢ΔࣹӨද‫ݱ‬ͷ৐਺‫ܥ‬Ͱ͋Γɺεϐϯ‫ي‬ಓ૬‫ࢠిͲͳ༻࡞ޓ‬εϐϯͷޮՌͷ༗ແΛ۠ผ͢ΔɻϑʔϦΤม‫׵‬͸ɺH̃⃗ ΍ Ũ⃗ (g) ͸ ⃗k Λ‫ࢠ֨ٯ‬ϕkk⃗ ͚ͩͣΒͯ͠΋ෆมͰ͋ΔΑ͏ʹఆ͍ٛͯ͠ΔɻΫτϧ G
H̃⃗k ͷ‫ݻ‬༗ঢ়ଶΛɺΤωϧΪʔ‫ݻ‬༗஋ ε⃗kn Λ۠ผ͢ΔϥϕϧΛ nɺॖୀΛ۠ผ͢ΔϥϕϧΛ i ͱͯ͠ ψ̃⃗kni ͱॻ͘ɻ͜
Εʹখ‫ ܈‬G⃗k ͷ‫ ݩ‬h Λ࡞༻ͤ͞Δͱɺ(41) ࣜͷΑ͏ʹ G⃗k ͷD⃗αn ࣍‫طݩ‬໿ද‫ ݱ‬U⃗αn ʹ͕ͨͬͯ͠ม‫͢׵‬ΔͷͰ͋ͬͨɻ
k!

k

3.1.2. {BS}G≥0 ͷఆٛΛຬͨ͢ɻ͜ͷଞʹ΋ 2.2 અͰٞ࿦ͨ͠‫ॴہ‬త͋Δ͍͸େ‫ہ‬తͳద߹ؔ܎͕ཁ‫͞ٻ‬ΕΔɻ͞Βʹ࣌ؒ൓సରশੑͳͲ
ͷ൓ϢχλϦରশੑ͕ଘࡏ͢Δ৔߹ʹ͸ɺn⃗α ͕‫ͳʹ਺ۮ‬Δk′͜ͱ͕ཁ‫͞ٻ‬ΕͨΓɺn⃗α = nα ⃗ ͷΑ͏ͳ੍‫ͱ͕ͭ͘͜ݶ‬k−k

͕͋Δʢ2.4.4 અʣɻҎԼͰ͸ɺ{n⃗α }⃗k,α ʹ՝͞ΕΔ͜ΕΒkͷ৚݅Λ͢΂ͯͻͬ͘ΔΊͯʮద߹ؔ܎ʯͱ‫͢ʹͱ͜Ϳݺ‬Δɻ༩͑ΒΕۭͨؒ‫ ܈‬G ͷ΋ͱͰՄೳͳɺద߹ؔ܎Λຬͨ͋͢ΒΏΔ ⃗b ͷू߹Λ {BS}G ͱॻ͘͜ͱʹ͢Δɻఆٛʹ
≥0

G
⃗
⃗
ΑΓ ⃗b1 , ⃗b2 ∈ {BS}G≥0 ͷͱ͖ b1 + b2 ∈ {BS}≥0 Ͱ͋Δɻ͜
ʮઈԑମ 1 ͷ ν1 ຊͷ઎༗όϯυʯͱʮઈԑͷ࿨ ⃗b1 + ⃗b2 ͸ɺମ 2 ͷ ν2 ຊͷ઎༗όϯυʯΛ߹ΘͤͯͰ͖Δʮν1 + ν2 ຊͷ઎༗όϯυΛ࣋ͭઈԑମʯͷ࣋ͭ‫ط‬໿ද‫ݱ‬ͷ‫਺ݸ‬Λද͢ɻ

3.1.3. {AI}G≥0 ͷఆٛόϯυઈԑମΛߟ͑ΔͨΊʹɺ֤೾਺఺ͰΤωϧΪʔ͕Ծఆ͞ΕͨશͯͷରশੑΛอͭΑ͏ʹ‫ʹۭ͕ؒࢠݪ‬ฒΜ௿͍ํ͔Β ν ຊ໨ͷόϯυͱ ν + 1 ຊ໨ͷόϯυͱͷؒʹͰ͍Δঢ়‫گ‬Λ૝૾ͯ͠ΈΔɻͦΕͧΕͷ‫ʹࢠݪ‬ଐ͢Δిࢠ

όϯυΪϟοϓ͕։͍͍ͯΔͱԾఆ͢Δɻ‫ ܭ‬ν ຊͷ઎༗όϯ͸ଞͷ‫ͱ΁ࢠݪ‬ඈͼҠΒͣɺ‫͍ͯ͠ࡏہ‬Δ΋ͷͱ͢ΔɻҎυͷ‫ݻ‬༗ϕΫτϧ ψ̃⃗kni Λฒ΂ͨ Norb × ν ߦྻΛ Φ̃⃗k ɺ࢒ΓԼͰ͸͜ΕΛ‫ݶۃࢠݪ‬ͷઈԑମ (atomic insulator) ͱ‫Ϳݺ‬ͷඇ઎༗όϯυͷ‫ݻ‬༗ϕΫτϧΛฒ΂ͨ Norb × (Norb − ν)ߦྻΛ Φ̃⃗c ͱॻ͘͜ͱʹ͢ΔɻΦ̃⃗k ΋ Φ̃⃗k = Φ̃⃗k+G⃗ Λຬͨ͢ɻk⃗খ‫ ܈‬G⃗ ͕ฒਐ‫ ܈‬T Ҏ֎ͷ‫ݩ‬Λ‫ؚ‬Ήͱ͖ɺk Λߴରশ೾਺k͜ͱʹ͢Δɻ‫ݶۃࢠݪ‬ͷઈԑମ͸࣮ۭؒʹ͓͚Δ௚ੵঢ়ଶͰ͋Δ͔Βඇࣗ໌ͳτϙϩδʔΛ࣋ͪಘͳ͍ɻ͜ͷ͜ͱΛ‫ٯ‬खʹͱͬͯɺຊߘͰ͸ʮ͋Δઈԑମ͕‫ݶۃࢠݪ‬ͷઈԑମʹͱ‫͢ʹͱ͜Ϳݺ‬ΔɻΦ̃⃗k ͱ Φ̃⃗c ͷؒʹόϯυΪϟοϓΛԾఆஅ೤઀ଓͰ͖Δ͜ͱʯΛʮͦͷઈԑମͷτϙϩδʔ͕ࣗ໌

͍ͯ͠Δ͜ͱ͔ΒɺΦ̃⃗k ͷதʹ‫·ؚ‬ΕΔ‫ط‬໿ද‫ ݱ‬U⃗α ͷ‫਺ݸ‬

Ͱ͋Δ͜ͱʯͷఆٛͱ͢Δɻ͜͜Ͱ͍͏அ೤઀ଓͱ͸ɺԾn⃗αʢα ͸‫ط‬໿ද‫ݱ‬Λ۠ผ͢Δϥϕϧʣ͕ྑ͘ఆٛ͞Ε͍ͯkఆ͞ΕͨରশੑΛഁΒͣɺ͔ͭྭ‫ى‬ΪϟοϓΛดͣ͡ʹ‫׈‬kkΔɻn⃗α ΛมԽͤ͞Δʹ͸όϯυΪϟοϓΛด͡Δ͔·ͨ͸kΒ͔ʹ‫ڧ‬ଋറ໛‫ܕ‬Λม‫͢ܗ‬Δ͜ͱΛࢦ͢ɻରশੑΛഁΔඞཁ͕͋ΔͨΊɺn⃗α ͸அ೤ม‫Ͱܗ‬͸มΘΒͳͯ͞‫ݶۃࢠݪ‬ͷઈԑମ΋ઈԑମͰ͋Δ͜ͱʹ͸୅ΘΓͳ͍ʮτϙϩδΧϧෆมྔʯͰ͋Δɻ͜ͷ੔਺ͷ૊ {n⃗α }⃗k,αʹΑͬͯόϯυઈԑମΛಛ௃෇͚ɺͦͷτϙϩδʔͷ৘ใ͍ͷͰɺόϯυߏ଄ͱͦΕ͕֤೾਺఺Ͱ࣋ͭ G⃗k ͷ‫ط‬໿ද‫ݱ‬ΛҾ͖ग़͢ͷ͕ରশੑࢦඪͷํ๏9) Ͱ͋ΔɻωϧΪʔόϯυ͸‫׬‬શʹϑϥοτͰ͋Δ͕ɺόϯυߏ଄ʹkkΛٞ࿦͢Δ͜ͱ͕Ͱ͖ΔɻඈͼҠΓੵ෼͕ͳ͍͜ͱ͔ΒΤ

《講義ノート》

物性研究・電子版 Vol. 8, No. 1, 081202（2020年2月号）‫ݱ‬ΕΔ‫ط‬໿ද‫ݱ‬͸೾਺ ⃗k ʹґଘ͢Δʢ3.2.3 અͰ؆୯ͳྫΛѻ͏ʣɻ୯Ґ๔಺ʹ͓͚Δ‫ࢠݪ‬ͷҐஔΛ ⃗rɺͦ͜ʹ‫͢ࡏہ‬Δ
ిࢠͷ‫ي‬ಓ͕খ‫ ܈‬G⃗r ͷ‫ط‬໿ද‫ ݱ‬u⃗βr Ͱ͋Δͱ͢Δͱɺ͜ͷ‫ݶۃࢠݪ‬ͷઈԑମʹ‫·ؚ‬ΕΔ G⃗k ͷ‫ط‬໿ද‫ ݱ‬U⃗α ͷ‫਺ݸ‬͸ɺk(28) ࣜΛ༻͍ͯ(n⃗α)(⃗r,β) =k

1|G⃗k /T |

!

(⃗
r,β)
(h)]
k

tr[U⃗kα (h)]∗ tr[Ũ⃗(83)h∈G⃗k /Tඞͣඇࣗ໌ͳτϙϩδʔΛ༗͢Δ͜ͱ͕෼͔Δɻ‫ʹٯ‬ɺ΋͠ ⃗b ∈ {AI}G ͳΒ͹ɺ‫ط‬໿ද‫ݱ‬ͷ૊Έ߹Θ͚͔ͤͩΒ͸Կ≥0΋‫ͣ͑ݴ‬ɺ‫ݪ‬ཧతʹ͸͜ͷઈԑମ͸ಉ͡ද‫ݱ‬Λ࣋ͭ‫ۃࢠݪ‬‫ݶ‬ͷઈԑମʹஅ೤઀ଓ͞ΕಘΔɻͭ·Γɺ(85) ࣜͷ৚݅͸͜ͷઈԑମ͕τϙϩδΧϧʹඇࣗ໌Ͱ͋ΔͨΊͷे෼৚݅
Ͱ͋Δͱ‫͑ݴ‬Δɻ͜Ε͸ඞͣ͠΋ඞཁ৚݅Ͱ͸ͳ͍ʢͭ·Γɺ⃗b ∈ {AI}G Ͱ΋τϙϩδΧϧʹඇࣗ໌ͳઈԑମ͸ଘࡏ≥0͢Δʣ͕ɺ্ड़ͷΑ͏ͳ໰୊Λ‫ͣ·ؚ‬༰қʹ൑ఆͰ͖Δ͜ͱ(⃗r,β)(h) ͸ (64) ࣜͰ༩͑ͨ‫ڧ‬kଋറ໛‫͚͓ʹܕ‬Δରশੑͷද‫͋Ͱݱ‬Δɻ͜ͷ {(n⃗α )(⃗r,β) }α,⃗k∈K
k
(⃗
r,β)

͕௕ॴͰ͋Δɻۙ೥զʑ͸ʮϑϥδϟΠϧτϙϩδʔʯͱ‫ݺ‬ͱॻ͘͜ͱʹ͠Α͏ɻ͜ΕΛಉఆͰ͖ͨͷ΋͜ͷ൑ఆ๏ͷ؆ศ͕͞‫ͨͬ͋Ͱނ‬ɻ

͜͜·Ͱ͸୯Ґ๔಺ͷҰՕॴʹ‫ط‬໿ද‫ݱ‬Λ࣋ͭిࢠΛ‫ہ‬Ҏ্ɺ઎༗όϯυ͕࣋ͭ‫ط‬໿ද‫͍ͯͮجʹݱ‬ઈԑମ͕࣋

ࡏͤ͞Δ৔߹Λߟ͕͑ͨɺΑΓҰൠʹ͸ɺಉ͡Ґஔʹෳ਺ͭτϙϩδʔΛ൑ఆ͢Δ͜ͱͷ‫ج‬ຊతͳΞΠσΟΞΛड़΂ͷిࢠΛ‫ͤ͞ࡏہ‬Δ͜ͱ΋Ͱ͖Δ͠ɺผͷҐஔ ⃗r′ ʹ‫ࢠݪ‬Λͨɻ࣍અͷٞ࿦͸‫ز‬෼ந৅౓͕૿ͨ͢Ίɺৄࡉʹ‫ڵ‬ຯ͕ͳ௥Ճͯ͠΋Α͍ɻ͜ΕʹରԠ͢Δ‫ط‬໿ද‫ݱ‬ͷ‫਺ݸ‬͸୯ʹ࿨͍ํ͸ඈ͹ͯ͠ 3.2 અҎ߱ʹਐΜͰ͍͍ͨͩͯߏΘͳ͍ɻͱද͢͜ͱ͕Ͱ͖Δɻͨͩ͠ Ũ⃗Λ (80) ࣜͷΑ͏ʹฒ΂ͨ΋ͷΛ ⃗a

⃗a

(⃗
r,β)

(⃗
r ′ ,β ′ )+⃗a͹ΕΔ৽͍͠ΫϥεͷτϙϩδΧϧઈԑମΛఏҊͨ͠10) ͕ɺ+ · · · Ͱ༩͑ΒΕΔɻ͜ΕΒͷ‫ݶۃࢠݪ‬ͷઈԑମʹରԠ͢Δ‫ط‬໿ද‫ݱ‬ͷ૊Έ߹Θ͔ͤΒͳΔू߹Λ-!
.
{AI}G≥0 =m⃗βr,β⃗a(⃗r,β) |m⃗βr ∈ Z, m⃗βr ≥ 0(84)

͢Δ͜ͱͰɺόϯυߏ଄ͷ࣋ͭτϙϩδʔΛද‫͔ݱ‬Βௐ΂ͱॻ͘͜ͱʹ͢Δɻ⃗r ͸Ұ‫͢ݟ‬Δͱ࿈ଓྔ͕ͩɺ࣮ࡍʹ͸

Δ͜ͱ͕Ͱ͖Δ͜ͱΛٞ࿦ͨ͠ɻ͔͠͠ू߹ͷࠩ⃗rG
3.1.5. XBSͷఆٛGલઅͰ͸ɺೋͭͷू߹ {BS}G≥0 ͱ {AI}≥0 ͷҧ͍ʹண໨ҰͭͷϫΠίϑҐஔ͔Β͸୅දͷҰ఺͚ͩΛબ΂͹ྑ͍ͷ

{BS}G≥0 \ {AI}G≥0Ͱ࿨͸཭ࢄతͰ͋Δʢ⃗r ͱ ⃗r′ ͕ಉ͡ϫΠίϑҐஔʹଐ͢Δ′

࣌ɺ(n⃗α )(⃗r,β) = (n⃗α )(⃗r ,β) ͱͳΔ͔Βʣɻ࣌ؒ൓సରশੑͳkkͲͷ൓ϢχλϦରশੑ͕ଘࡏ͢Δ৔߹ʹ͸ɺm⃗βr ͕‫ʹ਺ۮ‬′੍‫͞ݶ‬ΕͨΓɺm⃗βr = m⃗βr ͱ͍͏੍‫͖͕ͭݶ‬ಘΔɻ

(86)͸៉ྷͳ਺ֶతͳߏ଄Λ͓࣋ͬͯΒͣɺ͜ͷ‫ݩ‬Λ‫ܥ‬౷తʹ෼ྨ͢Δ͜ͱ͸೉͍͠ɻͦ͜Ͱू߹ {BS}G≥0 ΛՃ‫ͳͱ܈‬ΔΑ͏ʹ֦ுͯ͠ѻ͍΍͘͢͠Α͏ɻ͜ͷͨΊʹɺ(81) ࣜͷ৚݅ΛҰ୴๨Εͯෛͷ

3.1.4. όϯυτϙϩδʔͷ؆ศͳ൑ఆ๏
ͯ͞ɺ‫ڧ‬ଋറ໛‫ܕ‬΍ີ౓൚ؔ਺๏ͳͲʹΑΔ‫ʹࢉܭ‬Αͬͯɺۭؒ‫ ܈‬G Λ࣋ͭ͋Δ෺࣭ʹ͍ͭͯͷόϯυߏ଄͕༩͑

ΒΕͨͱ͢Δɻಛʹઈԑମʹ͍ͭͯٞ࿦͢ΔͨΊʹɺϑΣϧϛΤωϧΪʔ෇ۙʹେҬతͳόϯυΪϟοϓ͕։͍͍ͯ
ΔͱԾఆ͢Δɻ͜ͷઈԑମ͕ԿΒ͔ͷඇࣗ໌ͳτϙϩδʔΛ͔࣋ͭͲ͏͔Λ؆ศʹ൑ఆ͢Δ͜ͱ͕ʮରশੑࢦඪʯͷ
໨తͰ͋Δɻ΋ͪΖΜɺϒϩοϗؔ਺͕ৄࡉʹ෼͔͓ͬͯ
Γɺؔ܎ͦ͠͏ͳτϙϩδΧϧ਺΋༧૝͕෇͍͍ͯΔͷͰ͋Ε͹ɺ࣮ࡍʹͦͷτϙϩδΧϧ਺ͷ஋Λ‫͑·ͯ͠͠ࢉܭ‬͹Α͍ɻ͔͠͠ɺ֤೾਺఺͝ͱʹಠཱʹର֯Խ͞Εͨ਺஋੔਺΋‫ؚ‬ΊΔ͜ͱʹ͠ɺ͜ΕΛ {BS}G ͱॻ͘ɻ͜͏͢Δͱͭ·Γ ⃗b1 , ⃗b2 ∈ {BS}G ͷͱ͖ࠩ ⃗b1 − ⃗b2 ΋ {BS}G ʹ‫·ؚ‬ΕΔͷͰ {BS}G ͸ʢࣗ༝ʣՃ‫ ܈‬ZdBS ʢ1 ≤ dBS ≤ D ʣͱͳ
Δɻগ͠೉͘͠‫͡ײ‬ΒΕΔ͔΋͠Εͳ͍͕ɺ͜ͷࣄ৘͸ʮෛͰͳ͍੔਺ͷू߹ Z≥0 = {0, 1, 2, · · · } ͸‫Ͱ܈‬͸ͳ͍͕ɺશ

੔਺ͷू߹ Z = {· · · , −2, −1, 0, 1, 2, · · · } ͸Ճ‫͋Ͱ܈‬Δ͜
ͱʯͱશ͘ಉ͡Ͱ͋Δɻಉ༷ʹɺ(84) ࣜӈลͷ m⃗βr ≥ 0 ͱ͍͏৚݅Λ๨Εɺ͋ΒΏ
G

Δ੔਺Λ‫͢ʹͱ͜͢ڐ‬Ε͹Ճ‫{ ܈‬AI}G = ZdAI ʢ1 ≤ dAI ≤Dʣ͕ಘΒΕΔɻఆٛʹΑΓ {AI}G ͸ {BS}G ͷਖ਼‫ن‬෦෼‫܈‬
Ͱ͋Δ͔Βɺ঎‫܈‬‫݁ࢉܭ‬Ռ͔ΒେҬతʹ‫׈‬Β͔ͳϒϩοϗؔ਺Λߏ੒͢Δ͜

G
XBS= {BS}G /{AI}Gͱ΍ɺ۩ମతʹͲͷτϙϩδΧϧ਺Λௐ΂Ε͹͍͍ͷ͔ఱ

(87)ԼΓʹ༧૝͢Δ͜ͱ͸Ұൠʹ೉͍͠ɻG͕ఆٛͰ͖Δɻ‫ݪ‬ཧతʹ͸ dG
BS > dAI ͱͳΓಘΔ͕ɺ࣮ࡍͦ͜Ͱɺ͜ͷόϯυߏ଄ͷ࣋ͭ G⃗k ͷ‫ط‬໿ද‫ݱ‬ͷ৘ใΛߴରশ೾਺ ⃗k ∈ K ͰͦΕͧΕநग़͢Δ͜ͱʹΑΓɺ⃗b ∈G͸ৗʹΔ9) ɻ͜ͷͨΊɺXBS{BS}G≥0 Λߏ੒͠ɺ͜Ε͕ {AI}G≥0 ʹ΋‫·ؚ‬Ε͍ͯΔ͔Ͳ͏͔ௐ΂Δɻ͍·Ծʹ⃗b ∈/ {AI}G≥0(85)ʹ͸ 230 ‫ݸ‬શͯͷۭؒ‫ʹ܈‬ରͯ͠ dBS = dAI ͕੒ཱ͍ͯ͠G
XBS= Z n 1 × Z n2 × · · · × Z nN

(88)

ʢni ͸ 2 Ҏ্ͷ੔਺ɺN ͸ 1 ≤ N ≤ dG
BS ͷൣғͷ੔਺ʣͱ͍͏‫ͳʹܗ‬Δɻ

ͩͱ͢Δͱɺ‫ط‬໿ද‫ݱ‬ͷ૊Έ ⃗b Λ࣋ͭΑ͏ͳઈԑମͷ‫ࢠݪ‬

230 छྨͷۭؒ‫܈‬΍ɺ1651 छྨͷ࣓‫ʹ܈ۭؒؾ‬ରͯ͠ɺ‫ݶۃ‬͸ͦ΋ͦ΋ଘࡏ͠ͳ͍ͷͰɺ͍·ߟ͍͑ͯΔઈԑମ͸GdGBS = dGAI ͷ஋΍ XBS
ͷ۩ମ‫ܗ‬Λ‫ٻ‬ΊΔ໰୊͸จ‫ݙ‬9) Ͱղ《講義ノート》

物性研究・電子版 Vol. 8, No. 1, 081202（2020年2月号）

ܾ͞Εͨɻྫ͑͹ฒਐରশੑʹՃۭ͑ͯؒ൓సରশੑͷΈϕΫτϧ͸ ψ⃗kni = V⃗k ψ̃⃗kni Ͱ༩͑ΒΕΔɻψ⃗kn Λ༻͍ΔͱΛ࣋ͭ 3 ࣍‫܈ۭؒݩ‬ʢP 1̄ʣͰ͸࣌ؒ൓సରশੑ΍ిࢠεϐ(92) ࣜ͸G= (Z2 )3 × Z4 ͱ͍͏݁Ռ͕ಘΒϯͷ༗ແʹؔΘΒͣ XBS

dd k †
ψ i∇⃗ ψ⃗vBZ ⃗kn k knͱίϯύΫτʹද‫͖Ͱݱ‬Δɻ

G
͕ԿΛද͢ͷ͔Λ۩ମྫΛΕͨɻҎԼͰ͸ɺ۩ମతʹ XBS

௨ͯ͠ΈͯΏ͘ɻ⟨⃗x⟩n =$

(93)͜ͷ݁ՌΛ ν ຊͷ઎༗όϯυ͔ΒͳΔઈԑମ΁ͱҰൠԽ

3.2. ෼‫ͱۃ‬ද‫ݱ‬ͷؔ܎
3.2.1. ϕϦʔҐ૬ͱ෼‫ۃ‬όϯυߏ଄͕࣋ͭ‫ط‬໿ද‫ݱ‬ͷ৘ใ {n⃗α }⃗k,α ͕όϯυߏ଄kͷτϙϩδʔͱਂؔ͘Θ͍ͬͯΔ͜ͱΛ۩ମྫΛ௨ͯ͠ೲ
ಘ͢ΔͨΊʹɺ·ͣ͸֤छͷτϙϩδΧϧ਺Λٞ࿦͢Δ্Ͱ΋ͬͱ΋‫ૅج‬తͳి‫ؾ‬෼‫( ۃ‬electric polarization) ʹ͍ͭ͠Α͏ɻϫχΤத৺ͷ࿨͸$ d /0d k ⃗˜A⃗k + tr[Φ̃⃗†⃗rΦ̃⃗k ]⟨⃗x⟩ =
k
vBZ
$ d
d k⃗
A⃗
=
vBZ k
ͱɺ઎༗όϯυͷϕϦʔ઀ଓ (Berry connection)⃗˜⃗ = tr[Φ̃† i∇⃗ Φ̃⃗ ],
A
⃗
k
k k

ͯٞ࿦͠Α͏ɻి‫ؾ‬෼‫ۃ‬Λ‫͢ࢉܭ‬Δʹ͸ɺຊདྷ͸෺࣭தͷk⃗A⃗k = tr[Φ⃗† i∇⃗k Φ⃗k ]kਖ਼ిՙͱෛిՙͷҐஔͷζϨΛௐ΂Δͳ͚Ε͹ͳΒͳ͍͕ɺ࣮ࡍʹ͸ిࢠͷࣗ༝౓ͷΈʹண໨͠ɺҎԼͰઆ໌͢ΔϫχΤத৺ (Wannier center) Λ༻͍ͨٞ࿦͕ͳ͞ΕΔ͜ͱ͕ଟ

(94)

(95)
(96)

Λ༻͍ͯද‫͢ݱ‬Δ͜ͱ͕Ͱ͖ΔʢΦk ≡ Vk Φ̃k ʣɻͭ·Γೋͭ5)໨ͷϑʔϦΤม‫׵‬ͷఆٛΛ༻͍ΔͱʮϕϦʔ઀ଓͷ BZ ۭ·ͣ؆୯ͷͨΊʹΤωϧΪʔ ε⃗kn Λ࣋ͭ 1 όϯυ͕ଞͷx⟩ Λද͢ʯ5) ͜ͱʹͳΔɻ΋ͪΖؒͰͷฏ‫͕ۉ‬ϫχΤத৺ ⟨͍⃗ ɻόϯυ͔ΒόϯυΪϟοϓͰִͯΒΕ͍ͯΔ৔߹Λߟ͑ΔɻΜϑʔϦΤม‫׵‬ͷҰͭ໨ͷఆٛʹ‫ͮ͘ج‬ϕϦʔ઀ଓΛ༻͍͜ͷόϯυͷϒϩοϗঢ়ଶΛΔ͜ͱ΋Ͱ͖Δ͕ɺͦͷ৔߹͸୯Ґ๔಺ͷҐஔΛද͢ ⃗r ͷ|⃗kn⟩ =!
⃗R,s‫د‬༩ΛՃ͑ΔͷΛ๨Εͳ͍Α͏ʹ͠ͳ͚Ε͹ͳΒͳ͍ɻ·⃗ ⃗⃗eik·R (ψ̃⃗kn )s |Rs⟩(89)

⃗ ≡ (ĉ ⃗ )† |0⟩ ͸୯Ґ๔ R⃗ = " nI ⃗aI ʹଐ͠࠲
ͱ͢Δɻ|Rs⟩I
R s⃗ + ⃗rs ʹҐஔ͢Δ‫ي‬ಓ s (s = 1, 2, . . . , Norb ) ͷҰి
x=Rඪ⃗⃗ 0 ʹଐ͢ΔϫχΤঢ়ଶͱϫχΤؔࢠঢ়ଶͰ͋Δɻ୯Ґ๔ R

਺ (Wannier function) ͸

$

(91)

෺ཧతҙຯ͸ͳ͍͜ͱʹ஫ҙ͕ඞཁͰ͋Δɻ͜ͷϕϦʔҐ૬ͱ෼‫ۃ‬ͷؔ܎͸ɺ೾਺ ⃗k ͷ୅ΘΓʹͻͶ

3.2.2. ରশੑʹΑΔ෼‫ۃ‬ͷྔࢠԽۭؒ‫܈‬ͷରশੑͷ΋ͱͰ෼‫Ͳ͕ۃ‬ͷΑ͏ʹม‫͞׵‬ΕΔ͔

ඪʢ= ϫχΤঢ়ଶͰͷҐஔͷ‫ظ‬଴஋ʣ͸Λ‫ݟ‬ΔͨΊʹɺ2.1.5 અͰٞ࿦ͨ͠ sewing matrix Λࢥ͍ग़ͦ͏ɻ͜Ε͸ G ͷ֤‫ ݩ‬g ʹରͯ͠ߦྻཁૉ

[W⃗k (g)]n′ i′ ,ni = ψ † ⃗g(k)n′ i′!
⃗ R⃗ + ⃗rs )⟨Rs|W⃗⟨⃗x⟩n =
⟨W⃗0n |Rs⟩(⃗0n ⟩U⃗k (g)ψ⃗kni(97)ʹΑͬͯఆٛ͞ΕΔɻ͜ͷ ν ࣍‫ݩ‬ϢχλϦߦྻ W⃗k (g) Λ༻

⃗
R,s

=

x⟩ ͷฒਐϕΫτϧͷҧ͍ʹ͚มԽ͠ಘΔ14 ɻ͕ͨͬͯ͠ ⟨⃗(90)Ͱఆٛ͞Εɺ‫֨ن‬௚ަ৚݅ ⟨WR⃗ 0 δn′ ,n Λ
⃗ ′ n′ |WR⃗ 0 n ⟩ = δR
⃗ ′ ,R00⃗ຬͨ͢ɻ͕ͨͬͯ͠ɺಛʹ R0 = ⃗0 ͷϫχΤؔ਺ͷத৺࠲$ͱมԽ͠ɺ݁Ռͱͯ͠ ⟨⃗x⟩ ͸‫ج‬ຊฒਐϕΫτϧͷ੔਺ഒͩΓ‫ڥ‬ք৚݅Λ༻͍Δ͜ͱͰ૬‫ͱ΁ܥ༻࡞ޓ‬ҰൠԽͰ͖Δ11) ɻdd k −i⃗k·R⃗ 0 ⃗
|kn⟩
e
vBZ
$ d
d k −i⃗k·(R⃗ 0 −R)
⃗
⃗
⟩
=
e
(ψ̃⃗kn )s
⟨Rs|W
⃗ 0n
R
vBZ
|WR⃗ 0 n ⟩ =ͨɺν ࣍‫ݩ‬ϢχλϦߦྻ v⃗k ʹΑΔม‫ ׵‬Φ⃗kn → Φ⃗kn v⃗kʢͨͩ⃗⃗ + i∇⃗ log det v⃗⃗ ͸A⃗⃗ → A͠ v⃗k+G⃗ = v⃗k ʣͷ΋ͱͰɺA⃗
k
k
k
k
k

dd k †ψ̃ (⃗r + i∇⃗k )ψ̃⃗kn
vBZ ⃗kn(92)ͱද͢͜ͱ͕Ͱ͖Δɻ͜͜ʹ ⃗r ͸୯Ґ๔಺ͷҐஔΛද͢ର֯ߦྻͰ (⃗r)s′ ,s ≡ δs′ ,s⃗rs Ͱఆٛ͞ΕΔɻ͜ͷࣜͷಋग़ʹ͸" i⃗k·R⃗" ⃗ i⃗k·R⃗= vBZ δ d (⃗k) ΍ R⃗ Re= −vBZ i∇⃗k δ d (⃗k) Λ༻⃗ eR͍Ε͹Α͍ɻ

2.1.2 અͰઆ໌ͨ͠Α͏ʹɺϑʔϦΤม‫ʹ׵‬͸୯Ґ๔ͷҐ⃗ Λ༻͍ͯ c̃ˆ† ≡ " ⃗ ĉ† ei⃗k·R⃗ ͱ͢Δ΋ͷͱɺ࣮ࡍͷ‫ي‬ಓ
ஔR
⃗
⃗
R R
k
⃗ rs )⃗ + ⃗rs Λ༻͍ͯ (ĉ† )s ≡ " ⃗ (ĉ† )s ei⃗k·(R+⃗
ͷҐஔ R
ͱ͢Δ
⃗
⃗
R R
k

΋ͷͷೋ͕ͭ͋ΔɻೋͭͷϑʔϦΤม‫׵‬ͷදࣜ͸ɺϢχλ⃗Ϧߦྻ V⃗k = e−ik·⃗r Ͱ݁͹Ε͓ͯΓɺH⃗k = V⃗k H̃⃗k V⃗† ͷ‫ݻ‬༗k͍ΔͱɺΦ⃗k ͷ g ∈ G ͷ΋ͱͰͷม‫׵‬ଇ͸U⃗k (g)Φ⃗k = Φg(⃗k) W⃗k (g)(98)ͱද‫͖Ͱݱ‬Δɻಛʹ h ∈ G⃗k ʹରͯ͠͸ (41) ࣜʹ‫ؼ‬ண͠

[W⃗k (g)]n′ i′ ,ni = δn′ ,n [U⃗αn (g)]i′ ,i ͱͳΔɻ2.3.3 અͰΈͨΑk͏ʹ ⃗tg = ⃗0 ͱͳΔ‫͍ͯͭʹݩ‬͸ U⃗k (g) = Vg(⃗k) Ũ⃗k (g)V⃗† ͸k⃗⃗ ͷ೾਺ ⃗k ʹґଘ͠ͳ͍ɻ͜ͷ৔߹͸ (117) ࣜΛ༻͍ͯ A
k

ม‫׵‬ଇ

⃗ ⃗ = pg A⃗⃗ − pg i∇⃗ log det W⃗ (g)
A
g(k)
k
k
k
!

(99)

⃗
= eiθ⃗k ΍ ⃗k =J qJ bJ (−0.5 ≤ qJ ≤ 0.5) ͱ" dd k" dॻ͘ͱ ⃗bJ ·∇⃗k θ⃗k =d q∂qJ θ⃗k = 2πn ͱͳΔ͜ͱ͔ΒɺvBZ" dd k⃗ ͱҰக͢Δ͜ͱ͕෼͔Δɻ
i∇⃗k log det v⃗k ͕͋ΔฒਐϕΫτϧ Rv
14 det v⃗
k

BZ

《講義ノート》

物性研究・電子版 Vol. 8, No. 1, 081202（2020年2月号）

͕ಘΒΕΔɻ͜ͷಋग़ʹ͸ϢχλϦߦྻ v⃗k ʹରͯ͠ tr[v⃗† ∇⃗k v⃗k ] =ඇࣗ໌ͳ෼‫ۃ‬Λੜ͡Δྫ ΑΓඇࣗ໌ͳྫͱͯ͠ɺ࣍ͷk
Norb = 2 όϯυ‫ڧ‬ଋറ໛‫ܕ‬Λߟ͑Α͏ɻt > 0 ΛΤωϧtr[∇⃗k log v⃗k ] = ∇⃗k log det v⃗k ͱͳΔ͜ͱΛ༻͍ͨɻ͢Δͱ$ dΪʔͷ࣍‫ݩ‬Λ࣋ͭఆ਺ɺm Λແ࣍‫ݩ‬ύϥϝʔλͱͯ͠
d k
⃗ ⃗ )
⃗⃗ − Apg ⟨⃗x⟩ − ⟨⃗x⟩ =
(pg A
g(k)
k
vBZ
Hk = τx t sin k + τz t(m − cos k)
(103)
$ d
d k
i∇⃗ log det W⃗k (g)
(100)
= pg
vBZ k
ͱ͢ΔɻҎԼɺτ0 ΍ σ0 ͸ 2 ࣍‫ݩ‬୯Ґߦྻɺ⃗τ = (τx , τy , τz )ͱม‫͖Ͱܗ‬Δ͕ɺ࠷‫ޙ‬ͷࣜͷӈล͸‫٭‬஫ 1 Ͱड़΂ͨΑ͏ʹ⃗ ͱ౳͍͠ɻͭ·Γʮ⃗tg = ⃗0 ͱͳΔ‫ ݩ‬g
͋ΔฒਐϕΫτϧ R⃗ Λຬͨ͢Α͏ʹྔࢠͷଘࡏԼͰ͸ɺ⟨⃗x⟩ ͸ pg ⟨⃗x⟩ − ⟨⃗x⟩ = RԽ͞Ε͍ͯΔʯ͜ͱ͕෼͔Δɻ΍ ⃗σ = (σx , σy , σz ) ͸ύ΢Ϧߦྻͱ͢Δɻ͜ͷ໛‫Ͱܕ‬͸֤୯Ґ๔ʹ͋Δ 2 αΠτ͸ͱ΋ʹ rs = 0 ʹ͋ΔͱԾఆͨͨ͠
ΊɺVk = 1 ͱͳΓೋͭͷϑʔϦΤม‫׵‬ͷ۠ผ͸ͳ͍ɻରশ
1
ੑ g ͸ Uk (g) = u2 τz ͱද‫͞ݱ‬ΕΔɻu2 ≡ ω sp (g, g) ͸ g͕ۭؒ൓సͷ৔߹ʹ͸ 1, g ͕ 2 ճճస΍‫ڸ‬өରশੑͷ࣌͸

i ͱ͢Δɻm ̸= ±1 ͷͱ͖ 2 όϯυؒʹόϯυΪϟοϓ͕։

3.2.3. 1 ࣍‫͚͓ʹܥݩ‬Δ෼‫ͱۃ‬ද‫ݱ‬ͷؔ܎x Λ −x ΁ͱม‫͢׵‬Δରশૢ࡞ g Λ࣋ͭΑ͏ͳ 1 ࣍‫ܥݩ‬Λߟ͑Α͏ɻϩου‫͜Ͱ܈‬ΕΛ࣮‫͢ݱ‬Δʹ͸ɺx ࣠ํ޲ͷฒਐରশੑʹՃ͑ͯ‫ڸ‬ө Mx : (x, y, z) *→ (−x, y, z) ͔ 2

ճճస C2z : (x, y, z) *→ (−x, −y, z)ɺ΋͘͠͸ۭؒ൓సI : (x, y, z) *→ (−x, −y, −z)ʢୈ 1 અͰٞ࿦ͨ͠ p1̄ ʹରԠʣΛ࣋ͭ‫܈‬Λߟ͑Ε͹Α͍ɻ͜ΕΒͷରশੑͷ΋ͱͰ͸ɺલઅͷ݁Ռ͔Β 2⟨x⟩ = an (n ∈ Z) ͱͳΓɺ⟨x⟩ ͸ 0 ·ͨ͸

a/2 (mod a) ΁ͱྔࢠԽ͞ΕΔɻࣜΛ؆୯ʹ͢ΔͨΊʹҎԼͰ͸֨ࢠఆ਺ a Λ 1 ͱ͢Δɻ͞Βʹ (99) ࣜΛ༻͍Δͱ
$ π
1
⟨x⟩ =
dk(Ak + A−k )
2π 0
$ π
1
=
dki∂k log det Wk (g)
2π 0
ζ0 (g)
1
log
,
(101)
=
2πi
ζπ (g)
͋Δ͍͸(−1)2⟨x⟩ = ζζπ0 (g)
(g)

(102)

ͱॻ͘͜ͱ͕Ͱ͖Δ12) ɻ͜͜Ͱ g ෆมͳ೾਺ k = 0, π ʹ͓͍ͯ ζk (g) ≡ det Wk (g) ͸ʮ֤όϯυͷ g ͷ‫ݻ‬༗஋Λશͯͷ઎༗όϯυʹ౉ͬͯੵΛͱͬͨ΋ͷʯΛද͢ɻͭ·Γରশੑ g ͷ‫ݻ‬༗஋ͷੵͷൺ͕෼‫ۃ‬ͷྔࢠԽ஋Λܾఆ͢ΔͷͰ͋Δɻ͘ͷͰɺν = 1 ͷઈԑମΛߟ͑Δ͜ͱ͕Ͱ͖Δɻg ෆมͳ೾਺ k = 0, π Ͱ͸ Hk ͷୈ 1 ߲͕ফ͑ͯର֯ͳ
ୈ 2 ߲ͷΈ͕࢒ΔͨΊɺͦͷ܎਺ t(m − cos k) ͷූ߸ʹண໨͢Ε͹Α͍ɻ|m| > 1 ͷͱ͖͸ k = 0 ͱ π Ͱಉූ߸ͳͷͰ઎༗όϯυͷ g ͷ‫ݻ‬༗஋͸౳͘͠ɺ(102) ࣜΑΓ ⟨x⟩ = 0(mod 1) ͱͳΔɻҰํ |m| < 1 ͷͱ͖͸ k = 0 ͱ π ʹ͓͚Δ઎༗όϯυͷ g ͷ‫ݻ‬༗஋͸‫ٯ‬ɺͭ·Γ ζ0 (g) = −ζπ (g) ͱͳΔͨΊɺϫχΤத৺͸ ⟨x⟩ = 1/2 (mod 1) ʹҐஔ͢Δɻ͜ͷ৔߹͸αΠτͷҐஔ rs = 0 ͱϫχΤத৺ͷҐஔ͕ͣΕ͓ͯΓɺඇࣗ໌ͳ෼‫ۃ‬Λ͍࣋ͬͯΔͱ‫͑ݴ‬Δɻ

3.2.4. 1 ࣍‫Ͱݩ‬͸ରশੑࢦඪ͕ͳ͍͜ͱ
લઅͰ঺հͨ͠ 1 ࣍‫ݩ‬ͷʮ‫ݶۃࢠݪ‬ͷઈԑମͷྫʯͱʮඇ
ࣗ໌ͳ෼‫ۃ‬Λੜ͡Δྫʯ͸ɺ࢒೦ͳ͕Βରশੑͷ‫ݻ‬༗஋͚ͩΛ‫͍ͯͯݟ‬΋۠ผͰ͖ͳ͍ɻk = 0 ͱ k = π ʹ͓͚Δ gͷ‫ݻ‬༗஋͕ҟූ߸Ͱ͋ͬͨͱͯ͠΋ɺ‫ݩ‬ʑrs = 1/2 ʹαΠτ͕͚͋ͬͨͩͷࣗ໌ͳঢ়ଶͳͷ͔ɺͦΕͱ΋‫ݩ‬ʑrs = 0ʹαΠτ͕ఆٛ͞Εͨ໛‫Ͱܕ‬ϫχΤத৺ͷҐஔ͕ ⟨x⟩ = 1/2(mod 1) ʹͣΕͨඇࣗ໌ͳঢ়ଶͳͷ͔෼͔Βͳ͍͔ΒͰ͋ΔɻΑΓҰൠʹɺ1 ࣍‫Ͱݩ‬શ 75 छྨ͋Δϩου‫ʹ܈‬ରͯ͠ɺG
G
XBS
͸ৗʹࣗ໌ʢXBS= 1ʣͰ͋Δ9) ɻ3.3. 2 ࣍‫ݩ‬ͷτϙϩδΧϧ਺ͱද‫ݱ‬ͷؔ܎‫ݶۃࢠݪ‬ͷઈԑମͷྫ֤୯Ґ๔಺Ͱ r = ξ/2 (ξ = 0 ·ͨ͸ 1) ʹిࢠ͕‫͍ͯ͠ࡏہ‬Δঢ়‫گ‬Λߟ͑ΔɻͦΕͧΕ 1.1.7

GΛ࣋ͭྫͱͯ͠ɺ2 ࣍‫ܥݩ‬࣍ʹඇࣗ໌ͳରশੑࢦඪ XBS

ͷ֤छτϙϩδΧϧ਺ͱͷؔ܎Λٞ࿦͠Α͏ɻUCઅͷ Λ⃗UCxa = {(0, 0, 0)} ΍ Λ⃗
xb = {(1/2, 0, 0)} ͱ͍͏֨ࢠʹ

ରԠ͢Δɻ͜ͷ‫ي‬ಓͷ࣋ͭ g ͷ‫ݻ‬༗஋Λ u(g) ͱॻ͘ɻ͜ΕΛ ν = Norb = 1 ͷ 1 όϯυ໛‫ܕ‬ʢH̃k = Hk = 0ʣͩͱΈͳͯ͜͠Ε·Ͱͷ݁ՌΛద༻ͯ͠ΈΔͱɺŨk (g) = e−iξk u(g),ψ̃k = 1ɺVk = e−iξk/2 ɺUk (g) = u(g), ψk = e−iξk/2 ͱͳ

3.3.1. νϟʔϯ਺࣌ؒ൓స͕ഁΕͨ‫Ͱܥ‬ॏཁʹͳΔνϟʔϯ਺ (Chern num-ber) ͸ɺϕϦʔ‫( ཰ۂ‬Berry curvature)ΔɻಛʹඈͼҠΓੵ෼ͳͲ΋‫͋Ͱྫͳ໌͍ࣗͳ·ؚ‬Δ͕ɺϕ#πϦʔҐ૬ (1/2π) −π dkAk = ξ/2 (mod 1) ͸ਖ਼͘͠ϫχΤ˜⃗˜⃗ ,F⃗⃗k = ∇⃗k × Akத৺ͷҐஔ ⟨x⟩ Λද͍ͯ͠Δɻg ͷ‫ݻ‬༗஋͸ ζ0 (g) = u(g),⃗⃗F⃗⃗k = ∇⃗k × Akξζπ (g) = (−1) u(g) Ͱ͋Γɺ(102) ࣜͷ੒ཱ΋֬ೝͰ͖Δɻ͜ͷ໛‫Ͱܕ‬͸ɺαΠτͷҐஔʢਖ਼ిՙͷҐஔʣͱϫχΤத৺ͷҐஔʢෛిՙͷҐஔʣ͕Ұக͍ͯ͠ΔͷͰɺඇࣗ໌ͳ
෼‫ۃ‬͸ੜ͡ͳ͍ɻ

(104)
(105)

Λ༻͍ͯ

Cxy = −

12π$

1d2 k(F⃗⃗k )z = −
2π
BZ

$

BZ

˜
d2 k(F⃗⃗k )z(106)

《講義ノート》

物性研究・電子版 Vol. 8, No. 1, 081202（2020年2月号）

ͱఆٛ͞ΕΔɻνϟʔϯ਺ Cxy ∈ Z ͱϗʔϧ఻ಋ౓ σxy ͱͷ

(a)

(b)

(c)

(d)

ؒʹ͸ σxy = (e /2π")Cxy ͷؔ܎͕͋ΓɺCxy ͕ৗʹ੔਺஋

+

+

+

+

ϗʔϧ఻ಋ౓ͷ੔਺ྔࢠԽΛઆ໌͢Δ13) ɻҎԼɺද‫ه‬Λ؆୯

‒

+

+

+

2

ΛͱΔ͜ͱ͕प‫ظ‬తϙςϯγϟϧதͷࣗ༝ిࢠ‫͚͓ʹܥ‬Δ

ʹ͢ΔͨΊ‫ج‬ຊฒਐϕΫτϧΛ ⃗ax = (1, 0, 0), ⃗ay = (0, 1, 0)
ͱ͢Δɻ⃗⃗ + i∇⃗ log det v⃗ ʹ⃗⃗ → Aήʔδม‫ ׵‬Φ⃗kn → Φ⃗kn v⃗k ɺAkkkk⃗ ⃗ )y = (A⃗⃗ ⃗ )y ͔ͭ (ii) BZ ಺ʹ A⃗⃗ ͷಛҟ
Αͬͯɺ(i) (A
k

k+G

k

⃗ ⃗ )x ͱ఺Λ࣋ͨͳ͍Α͏ʹ͢Δɻ͜ͷήʔδͰ͸Ұൠʹ (Ak⃗(A⃗ ⃗ )x ͸Ұகͤͣɺνϟʔϯ਺͸
k+G

$ π

$ π

= (⟨xπ ⟩ − ⟨x−π ⟩)$ π−π⃗ (k ,k ) ]x
dkx [Ax yͷे෼৚݅Ͱ͋Δ͕ඞཁ৚݅Ͱ͸ͳ͍͜ͱͷྫͰ͋Δɻ⃗⃗ ͕ࣔͯ͠ΈΑ͏ɻ(99) ࣜʹΑΓ F
kF⃗g(⃗k) = F⃗⃗k(107)ͷΑ͏ʹɺ֤ ky ʹରͯ͠ఆٛͨ͠෼‫⟨ ۃ‬xky ⟩1⟨xky ⟩ ≡2π

͠·͏ɻ͜Ε͕ (85) ͕ࣜτϙϩδΧϧʹඇࣗ໌Ͱ͋ΔͨΊ(109) ࣜͷ؆୯ͳಋग़͸จ‫ݟʹݙ‬ΒΕͳ͍ͷͰɺ͜͜Ͱ

1dkx
dky (F⃗⃗k )z
2π −π
−π
$ π
1
⃗ (k ,π) − A⃗ (k ,−π) ]x=dkx [Axx2π −π

Cxy = −

ਤ 3 (a): C2 ճస΍ۭؒ൓సͰෆมͳ೾਺ɻ (b) 0 < m′ < 2ʢCxy = 1ʣͷͱ͖ͷ઎༗όϯυͷۭ࣋ͭؒ൓సରশੑͷ‫ݻ‬༗஋ɻ(c) 2 < m′ʢCxy = 0ʣͷͱ͖ͷ઎༗όϯυͷۭ࣋ͭؒ൓సରশੑͷ‫ݻ‬༗஋ɻ

(108)ͷࠩͰද‫͢ݱ‬Δ͜ͱ͕Ͱ͖Δɻ

3.3.2. νϟʔϯ਺ͱճసରশੑͷ‫ݻ‬༗஋z ࣠पΓͷ n ճճసରশੑ Cnz Λ࣋ͭ‫Ͱܥ‬͸ɺߴରশ೾
਺ʹ͓͚Δճసରশੑͷ‫ݻ‬༗஋ʹΑͬͯ xy ໘ͷνϟʔϯ਺ Cxy Λ mod n Ͱ‫͢ࢉܭ‬Δ͜ͱ͕Ͱ͖Δ14) ɻྫ͑͹ n = 2

ͷ৔߹ʹ͸

(111)

Λຬͨ͢͜ͱʹ஫ҙ͢Δɻಛʹ g = C2z ճసͷ৔߹Λߟ͑
Δͱɺ࣍ͷΑ͏ʹม‫͖Ͱܗ‬Δɻ
$ π
$ π
1
Cxy = −dkx
dky (F⃗⃗k + F⃗−⃗k )z
2π −π
0
$ π
$
1 π
=−
dkx
dky (F⃗⃗k )z
π −π
0
$
1 π
⃗ (k ,π) − A⃗ (k ,0) ]x=
dkx [Axxπ −π= 2(⟨xπ ⟩ − ⟨x0 ⟩).

(112)͜Εʹ (102) ࣜΛద༻ͯ͠গ͠ॻ͖‫͑׵‬Δͱ (109) ࣜͱͳ
Δ 15 ɻ2 ճճసͰ͸ͳ͘ n ճճసͷ৔߹ͷࣜ΋ಉ༷ʹͯ͠ಋ͘͜ͱ͕Ͱ͖Δɻ͜ͷճసରশੑͷ‫ݻ‬༗஋ͱνϟʔϯ਺ͷؔ܎΋ɺͻͶΓ

(−1)

Cxy

ζ
(C2z ) ζ(π,0) (C2z )= ζ(0,π)(0,0) (C2z ) ζ(π,π) (C2z )(109)ʹΑͬͯ Cxy ͷ‫حۮ‬ΛܾఆͰ͖Δɻζ⃗k (C2z ) ≡ det W⃗k (C2z )͸ɺ೾਺

K2D ≡ {(nx π, ny π, 0)|nx , ny = 0, 1}(110)ʢਤ 3(a)ʣʹ͓͚Δ C2z ରশੑͷ‫ݻ‬༗஋Λશͯͷ઎༗όϯυʹ౉ͬͯੵΛͱͬͨ΋ͷͰ͋Δɻ

n ճճసରশੑ͕͋Δ৔߹ͱ‫ͯͬݴ‬΋ɺxy ໘ʹରͯ͠ճసରশੑͷΑ͏ʹৼΔ෣͑͹े෼Ͱɺ‫ ʹີݫ‬3 ࣍‫ۭؒݩ‬ͷճసରশੑͰ͋Δඞཁ͸ͳ͍ɻྫ͑͹ C2z ͷ୅ΘΓʹۭؒ൓స I ɺC4z ͷ୅ΘΓʹ 4 ճճ൓ S4z = IC4z ɺC6z ͷ୅ΘΓʹ 3 ճճ൓ S3z = IC3z Λ༻͍ͯ΋ྑ͍ɻ͜ΕΒ޿͍ҙຯͰͷ n ճసରশੑͷ‫ݻ‬༗஋ͱ n Λ๏ͱͨ͠νϟʔϯ਺ͷ‫ڥ‬ք৚݅Λ༻͍Δ͜ͱͰ૬‫ͱ΁ܥ༻࡞ޓ‬ҰൠԽͰ͖Δ15) ɻνϟʔϯઈԑମͷྫg Λ 2 ճճసରশੑ·ͨ͸ۭؒ൓సରশੑͱ͢Δɻ͜ͷରশੑΛ࣋ͭ 2 ࣍‫ݩ‬ͷ 2 όϯυ‫ڧ‬ଋറ໛‫Ͱܕ‬ɺrs = 0 ʹαΠτ͕͋Δ΋ͷΛߟ͑Α͏ɻH⃗k = τx t sin kx + τy t sin ky + τz t(m′ − cos kx − cos ky ).
(113)′m ͸ແ࣍‫ݩ‬ύϥϝʔλͰ ⃗τ = (τx , τy , τz ) ͸ύ΢ϦߦྻͰ
͋Δɻ͜͜Ͱ΋ରশੑ g ͸ U⃗k (g) = u2 τz ͱද‫͞ݱ‬ΕΔɻ

m′ ̸= 0, ±2 ͷͱ͖͸શ೾਺఺ͰόϯυΪϟοϓ͕։ͨ͘Ίɺν = 1 ͷઈԑମΛߟ͑Δ͜ͱ͕Ͱ͖Δɻ͜ͷ 2 ࣍‫ݩ‬໛‫ܕ‬͸ky = 0, π Ͱ͸ (103) ࣜͷ 1 ࣍‫ݩ‬໛‫ؼʹܕ‬ண͠ɺky = 0 Ͱ͸m = m′ −1ɺky = π Ͱ͸ m = m′ +1 ͱରԠ͢Δɻ͕ͨͬ͠

͕ؔ܎͍ͯ͠Δ͜ͱ͔ΒɺฒਐରশੑʹՃ͑ͯ͜ΕΒͷର

ͯɺ|m′ | < 2 Ͱ͸ʢόϯυΪϟοϓ͕ด͡ΔҰ఺ m′ = 0 ΛG= Zn ͱͳΔɻ͜ͷ८ճশੑ͚ͩΛ࣋ͭ 2 ࣍‫Ͱܥݩ‬͸ XBS

਺͕‫͋Ͱ਺ح‬Δ͜ͱ͕෼͔Δɻಉ͜͡ͱ͕ͩɺ(109) ࣜͷެ‫܈‬͸ Cxy = 1 mod n ͷνϟʔϯઈԑମͰੜ੒͞ΕΔɻ

ճసରশੑͷ‫ݻ‬༗஋Λௐ΂Δ͚ͩͰνϟʔϯ਺ͷ৘ใΛಘΔ͜ͱ͕Ͱ͖Δͷ͸ศར͕ͩɺCxy ͷ n ͷҧ͍Λແࢹ͠

ͯ͠·͏ͷ͕ܽ఺Ͱ͋Γɺྫ͑͹ Cxy = +n > 0 ͩͱͯ͠΋όϯυߏ଄ͷ࣋ͭද‫ݱ‬͸ Cxy = 0 ͷ৔߹ͱಉ͡ʹͳͬͯআ͍ͯʣ⟨x(π)⟩ − ⟨x(0)⟩ = 1/2 (mod 1) ͱͳΓɺνϟʔϯ͔ࣜΒ΋௚ͪʹ (−1)Cxy = −1 ͕ಘΒΕΔɻʢg ͕ۭؒ൓స

ରশੑͷ৔߹ͷ઎༗όϯυͷ‫ݻ‬༗஋Λਤ 3(b,c) ʹࣔͨ͠ɻʣ
࣮ࡍʹ Cxy Λ‫ͯ͠ࢉܭ‬ΈΔͱɺ0 < m′ < 2 ͳΒ Cxy = 1ɺ
15 (112) ࣜͷ 3,4 ߦ໨ʹ͸ɺC⃗ ʹੜ͡Δಛҟੑͷͨ
xy ̸= 0 ͷͱ͖ʹ A⃗kΊʹ‫਺ۮ‬෼ͷෆఆੑ͕͋Δ͕ɺ͜Ε͸ Cxy ͷ‫ʹحۮ‬͸Ө‫͍ͳ͠ڹ‬ɻ

《講義ノート》

物性研究・電子版 Vol. 8, No. 1, 081202（2020年2月号）−2 < m′ < 0 ͳΒ Cxy = −1ɺ|m′ | > 2 ͳΒ Cxy = 0 ͰٛͰ͖Δ17 ɻνxy ͸͜ͷύϑΟΞϯΛ༻͍ͯ͋Δɻ(−1)νxy =

2

⃗
k∈K2D

3.3.3. νϟʔϯઈԑମͷΧΠϥϧ୺ঢ়ଶPf[W⃗k (T )]1
det W⃗k (T )ͱఆٛ͞ΕΔɻ͜ͷ͕ࣜҙຯΛ࣋ͭͨΊʹ͸

(118)
1

det W⃗k (T )(113) ࣜͷ໛‫ܕ‬ͷɺCxy = 1 ͷ૬ʢ0 < m′ < 2ʣͱ Cxy =͕࿈ଓͱͳΔΑ͏ʹେҬతʹήʔδΛ‫ݻ‬ఆ͢Δඞཁ͕͋Γɺ3(d) ͷΑ͏ʹɺn̂ = (cos φ, sin φ, 0) Λ๏ઢϕΫτϧɺℓ̂ =Ұํɺ࣌ؒ൓సରশੑʹՃۭ͑ͯؒ൓సରশੑ I ͕͋Δ

0 ͷ૬ʢ2 < m′ ʣͷք໘ʹ‫ݱ‬ΕΔ୺ঢ়ଶΛௐ΂Α͏ɻਤ(− sin φ, cos φ, 0) Λ‫ڥ‬քʹԊ͏ํ޲ͱ͢Δ௚ઢతͳ‫ڥ‬քΛߟ
x = (x, y, 0) Λ ⃗x = xn n̂+x∥ ℓ̂ ͱ෼ղ͢ΔɻCxy = 1͑Α͏ɻ⃗͜ͷఆٛʹैͬͯ νxy Λ‫͢ࢉܭ‬Δͷ͸༰қͰ͸ͳ͍ɻ৔߹͸ɺFu-Kane ެࣜ(−1)νxy =ͱ Cxy = 0 ͷೋͭͷ૬ͷ‫ڥ‬ք͕ xn = L ͱͳΔΑ͏ʹ͢Δ

2

ζ⃗k′ (I)

(119)

⃗
k∈K2D

ͨΊʹɺ(113) ࣜͷ H⃗k ʹ‫·ؚ‬ΕΔύϥϝʔλ m′ ʹ xn ґଘੑΛ࣋ͨͤΔɻδmxn ≡ m′xn − 2 ͕ xn < L ͷͱ͖ҰఆΛ༻͍Δ͜ͱͰ νxy Λ؆୯ʹ‫͢ࢉܭ‬Δ͜ͱ͕Ͱ͖Δ16) ɻͨΑ͏ʹɺྫ͑͹ δmxn = m0 tanh xna−L ͱ͢Ε͹Α͍ɻH⃗ ͸ m′ = 2 ͷͱ͖ʹ ⃗k = 0 ͰΪϟοϓΛด͡ΔͷͰɺબΜͰ I ͷ‫ݻ‬༗஋ͷੵΛऔΔ͜ͱΛҙຯ͢Δɻ͜ͷۙ๣ʹண໨ͨ͠௿ΤωϧΪʔ༗ޮ৔ͷཧ࿦Λߟ͑Δɻ
⃗k = kn n̂ + k∥ ℓ̂ ͷҰ࣍·Ͱల։ͨ͠ϋϛϧτχΞϯΛ n̂ ํͷͰɺ͜͜Ͱ͸ུ֓ͷΈ঺հ͢Δɻରশੑ IT ͷ sewing

஋ −m0 ʢ0 < m0 < 2ʣʹɺxn > L ͷͱ͖ +m0 ʹۙͮ͘kͩ͠ ζ⃗′ (I) ͸ɺ઎༗‫ي‬ಓͷ֤ΫϥϚʔεର͔Βยํ͚ͩΛk

Fu-Kane ެࣜͷূ໌͸‫ݪ‬࿦จ16) ΍ͦͷղઆ17) ʹৄ͍͠
matrix ͸W⃗k (IT ) = W−⃗k (I)W⃗k (T )޲ʹ࣮ۭؔͯؒ͠΁໭͢ͱ

Hn̂,xn ,k∥ = −ita⃗τ · n̂∂xn + ta⃗τ · ℓ̂k∥ + τz tδmxn(114)

ͱදͤΔɻW⃗k (IT ) ͸൓ରশߦྻͰ͋Γɺશ೾਺఺ͰύϑΟΞϯ͕ఆٛͰ͖Δɻ(120) ࣜʹద౰ͳ‫ج‬ఈΛͱͬͯ‫ࢉܭ‬

ͱͳΔɻ͜ͷϋϛϧτχΞϯͷ‫ݻ‬༗ؔ਺ͷ͏ͪ xn = L ෇ۙʹ‫͢ࡏہ‬Δ΋ͷ͸ɺ‫֨ن‬Խఆ਺Λআ͍ͯ
+
,
! xn
′
1
1
−a
δm
dx
′
x
0
ψn̂,xn ,k∥ = e
ieiφ

(120)

͢Δͱɺ࣌ؒ൓సରশ೾਺ K2D Ͱ͸ Pf[W⃗k (T )] = ζ⃗′ (I)kPf[W⃗k (IT )] ͕੒ཱ͢Δ͜ͱ͕෼͔Δɻଞํ W−⃗k (IT ) =(115)

W⃗k (T ) W⃗k (IT )∗ W⃗k (T )T ʹΑΓશ೾਺఺Ͱ Pf[W−⃗k (IT )] =Ͱ͋Δɻ͜Ε͸ΤωϧΪʔ‫ݻ‬༗஋ εk∥ = tak∥ Λ࣋ͪɺℓ̂ ͷΔΑ͏ʹ͢ΔͱɺPf[W⃗k (T )] = ζ⃗′ (I) ͔ͭ det[W⃗k (T )] = +1ํ޲΁ͷ଎౓ vk∥ = !1 ∂k∥ εk∥ = ta
! > 0 Λ࣋ͭΧΠϥϧ୺

ঢ়ଶͰ͋Δɻ(114) ࣜΛ ψn̂,xn ,k∥ ͷۭؒʹࣹӨ͢Δ͜ͱͰɺ͜ͷΧΠϥϧ୺ঢ়ଶΛ‫ه‬ड़͢Δ༗ޮ 1 ࣍‫ݩ‬໛‫ܕ‬eﬀ
Hn̂,k= tak∥∥

det[W⃗k (T )]Pf[W⃗k (IT )]∗ ͕੒ཱ͢Δ18 ɻ͍·ద౰ͳήʔδม‫ ׵‬Φ⃗ → Φ⃗ v⃗ ʹΑ֤ͬͯ఺ ⃗k Ͱ Pf[W⃗ (IT )] = +1 ͱͳ
k

k k

k

k

ͱͳΔͨΊɺ(118) ͔ࣜΒ (125) ͕ࣜ௚ͪʹै͏ɻ͜ͷ Fu-Kane ެ͔ࣜΒ΋ࣔࠦ͞ΕΔΑ͏ʹɺۭؒ൓సରG͸ৗʹ Z2শੑͱ࣌ؒ൓సରশੑΛ࣋ͭ 2 ࣍‫Ͱܥݩ‬͸ XBS(116)ͱ͍͏ϑΝΫλʔΛ‫ؚ‬Ήɻ͕ಘΒΕΔɻ

3.3.5. ϛϥʔνϟʔϯ਺
3.3.4. ྔࢠεϐϯϗʔϧઈԑମ࣌ؒ൓సରশੑ T Λ༗͢Δ 2 ࣍‫ʹܥࢠిݩ‬ରͯ͠͸ Z2τϙϩδΧϧ਺ νxy (= 0 ·ͨ͸ 1) Λఆٛ͢Δ͜ͱ͕Ͱ͖

࣌ؒ൓సରশੑͷ୅ΘΓʹ‫ڸ‬өରশੑ Mz : (x, y, z) *→(x, y, −z) ΛԾఆͯ͠ΈΑ͏ɻMz ͸೾਺Λม͑ͳ͍ͨΊɺU⃗k (Mz ) ͸ BZ ͷ֤఺Ͱ H⃗k ͷରশੑͱͳΔɻ͕ͨͬͯ͠ҰൠੑΛࣦ͏͜ͱͳ͘

Δɻ͜Ε͸࣍ࣜͰఆ·Δ࣌ؒ൓సରশੑͷ sewing matrixW⃗k (T ) Λ༻͍ͯఆٛ͞ΕΔɻ
(117)

࣌ؒ൓సରশੑͷԼͰ͸ɺ(99) ࣜ΍ (111) ࣜʹରԠͯ⃗⃗͠ + i∇⃗ log det W⃗ (T ), F⃗ ⃗ = −F⃗⃗ ͕੒ཱ͢Δͨ⃗ ⃗ =A
A
k

k

k

−k

k

Ίɺνϟʔϯ਺ Cxy ͸ৗʹθϩʹͳΔɻ͔͠͠ W⃗k (T ) ͸࣌ؒ൓సରশ೾਺ K2D [(110) ࣜ] ʹ͓͍ͯ͸൓ରশߦྻͱͳ
ΔͨΊ16 ɺ୅ΘΓʹύϑΟΞϯ (Pfaﬃan) Pf[W⃗k (T )] ͕ఆ

16 ূ ໌ ͸ ɺ[W (T )] ′ ′=ψ † ⃗ ′ ′ [U (T )]ψ⃗∗=⃗n i ,ni
k
kni
−kn i
†
∗
T
ψ⃗ [−U (T )]ψ ⃗ ′ ′ = −[W−⃗k (T )]ni,n′ i′ ɻͨͩ͠ U (T ) = −U (T )
−kn i
kni

ʢ2.4.1 અʣΛ༻͍ͨɻ

H⃗

(+)
k

0

0

U⃗k (Mz ) = i

+
0

H⃗
k
,
0
−

H⃗k =

U (T )Φ⃗∗k = Φ−⃗k W⃗k (T ).

−k

+

+

(−)

,

,

(121)
(122)

(ξ)
(ξ = ±) ͷͦΕͧkͱϒϩοΫର֯ԽͰ͖ΔɻͦͷͨΊ H⃗

Εʹରͯ͠νϟʔϯ਺ C (ξ) Λఆٛ͢Δ͜ͱ͕ՄೳͰɺ͜Ε
17 2n ࣍‫ݩ‬ͷ൓ରশߦྻ a (a= −aij ; 1 ≤ i, j ≤ 2n) ʹରͯ͠ɺύ! jiϑΟΞϯ͸ Pf[a] ≡ 2n1n!
σ∈S2n sgn(σ)aσ1 σ2 aσ3 σ4 · · · aσ2n−1 σ2n ͱఆٛ͞ΕΔɻͨͩ͠ S2n ͸ 2n ࣍ͷஔ‫͔׵‬ΒͳΔू߹Ͱ͋ΔɻύϑΟΞϯͱߦྻࣜͷؒʹ͸ det[a] = Pf[a]2 ͱ͍͏ؔ܎͕͋Δɻ
18 Ұൠʹ a ͕൓ରশߦྻͷͱ͖ waw T ΋൓ରশߦྻͰɺPf[waw T ] =det[w]Pf[a].

《講義ノート》

物性研究・電子版 Vol. 8, No. 1, 081202（2020年2月号）

Β͸Ұൠʹϛϥʔνϟʔϯ਺ (mirror Chern number) ͱ‫ݺ‬
(+)

(−)

+C
͸0
(+)
(−)
Ͱͳ͚Ε͹ͳΒͳ͍͕ɺC= −C͸೚ҙͷ੔਺஋Λ͹ΕΔɻ࣌ؒ൓సରশੑͷ΋ͱͰ͸ C = CऔΓಘΔɻ͜ͷ৔߹ɺC(ξ)ͷ‫حۮ‬͸ νxy ͱҰக͢Δɻ

(k =0)

Εͨऑ͍τϙϩδΧϧ૬Ͱ͸ νxyz࠷‫ޙ‬ͷ Z4 ͱ͍͏ϑΝΫλʔ͸ਅʹ 3 ࣍‫ݩ‬తͳτϙϩδʔ
ͱؔ܎͢Δɻ͜ΕΛཧղ͢ΔͨΊʹ

1 !
(n⃗+ − n⃗− ) mod 4
k
k
4

κ1 ≡
(123)

= +1 mod2 ͱͳΔɻಉ͜͡ͱ͕ x ࣠ɺy ࣠ʹ΋੒ཱ͢Δɻ‫ڸ‬өରশੑ Mz ʹՃ͑ͯ n ճճసରশੑ Cnz ΋Ծఆ͢Δ
ͱɺU⃗k (Cnz ) ͸ U⃗k (Mz ) ͱަ‫͠׵‬
,
+ (+)
0
U⃗ (Cnz )kU⃗k (Cnz ) =(−)
0U⃗ (Cnz )(k =π)= νxyz

(126)

⃗
k∈K3D

ͱ͍͏ྔΛಋೖ͠Α͏18) ɻ͜͜Ͱ K3D ͸ 8 ͭͷ࣌ؒ൓సର
শͳ೾਺ͷू߹ʢਤ 4(a) ͷཱํମͷ௖఺ʣΛද͢ɻ͜Ε͸kͱϒϩοΫର֯Խ͞ΕΔɻ͢Δͱߴରশ೾਺ʹ͓͚Δճస

઎༗όϯυͷ࣋ͭ൓సରশੑͷ‫ݻ‬༗஋ͷʢੵͰ͸ͳ͘ʣ࿨(ξ)

G
Λ 1/4 Ͱׂͬͨ΋ͷͰ͋Δɻ͜ΕΒͷྔΛ༻͍ͯɺXBS=ରশੑͷ‫ݻ‬༗஋Λ༻͍ͯ C

Λ mod n Ͱܾఆ͢Δ͜ͱ͕(k =0)

G= (Zn )2 ͱͳΔɻͰ͖ΔͨΊɺXBSಛʹ‫ڸ‬өରশੑ Mz ͱ࣌ؒ൓సରশੑ T ɺ͞Βʹ 2 ճճసରশੑ C2z ͕͋Δ৔߹͸ɺI = Mz C2z ʹΑΓ
2 (ξ)
2 (ξ)
(−1)νxy =ζ⃗ (C2z ) =ζ⃗ (I)
k

k

⃗
k∈K2D

(124)

(Z2 )3 × Z4 ͷ‫ݩ‬͸Ұൠʹ (νyzx(k =0), νzxy(k =0), νxyz, κ1 ) ͱॻ͘͜ͱ͕Ͱ͖Δɻ

κ1 ͷҙຯΛ୳ΔͨΊʹɺ·ͣ࣍ͷؔ܎ࣜʹ஫ҙ͢Δɻ
2(kz =0)(kz =π)(−1)κ1 =ζ⃗k′ (I) = (−1)νxy(−1)νxy
(127)
⃗
k∈K3D

⃗
k∈K2D

ͱͳΓɺξ = ± ͷͲͪΒ͔ҰͭͷηΫλʔͰ‫ͨ͠ࢉܭ‬൓సԾʹ κ1 ͕‫͋Ͱ਺ح‬Δͱ͢Δͱɺkz = 0 ͱ kz = π ͷೋͭͷରশੑͷ‫ݻ‬༗஋ͷੵ͕ νxy Λ༩͑Δɻ͜Ε͸‫ڸ‬өରশੑ Mz

໘Ͱͷ Z2 ਺ νxy ͕‫ͳʹٯ‬ΔͨΊɺ3 ࣍‫ݩ‬తʹඇࣗ໌ͳߏ଄

ΛԾఆͨ͠৔߹ͷ (125) ࣜʹଞͳΒͳ͍ɻΛ࣋ͭτϙϩδΧϧ૬͕࣮‫͍͖ͯͰݱ‬Δ͜ͱ͕෼͔Δʢਤ

Ҏ্ɺ2 ࣍‫ݩ‬ͷ֤छτϙϩδΧϧ૬ͱ઎༗όϯυ͕࣋ͭ4(b)ʣɻ͜Ε͸࣌ؒ൓సରশੑʹΑͬͯकΒΕͨ‫͍ڧ‬τϙϩখ‫܈‬ͷද‫ͱݱ‬ͷؔ܎ʹ͍ͭͯɺFu-Kane ެࣜͳͲ͜Ε·ͰδΧϧઈԑମ (strong topological insulator) Ͱ͋Γɺ(127)஌ΒΕ͍ͯͨ݁Ռ͕ରশੑࢦඪͷํ๏ʹ‫·ؚ‬Ε͍ͯΔ͜ͱࣜ͸·͘͞͠ 3 ࣍‫ݩ‬൛ͷ Fu-Kane ެࣜͰ͋Δɻ͔͠͠ (126)Λ‫͖ͨͯݟ‬ɻ͜Ε͚ͩͰ͋Ε͹‫ط‬ଘͷ݁Ռͷম͖௚͠Ͱ͋

ࣜͷ κ1 ͸ mod 4 Ͱఆٛ͞ΕΔͨΊɺ࿩͸͜ΕͰऴΘΓͰ

Δ͕ɺରশੑࢦඪͷํ๏ʹΑΔҰൠతͳऔΓѻ͍Λ௨ͯ͠͸ͳ͍ɻҎԼʹ‫ݟ‬ΔΑ͏ʹɺ࣮͸ (0, 0, 0, 2) ʹଐ͢Δ૬͸ 1ॳΊͯ໌Β͔ʹͳͬͨ৽͍͠τϙϩδΧϧ૬΋͋Δɻ͜Ε࣍‫ݩ‬తͳΪϟοϓϨεද໘ঢ়ଶΛ༗͢Δߴ࣍τϙϩδΧϧ

ʹ͍ͭͯ͸࣍અͰղઆ͢Δ͜ͱʹ͢Δɻઈԑମ (higher-order topological insulator) Ͱ͋Δ18) ɻ

4. ৽͍͠τϙϩδΧϧ૬΁ͷԠ༻τϙϩδΧϧઈԑମͷྫຊઅͰ͸͜Ε·Ͱٞ࿦͖ͯͨ͠ରশੑࢦඪͷํ๏Λʮߴ࣍τϙϩδΧϧઈԑମʯ΍ʮ੬ऑͳτϙϩδΧϧઈԑମʯτϙϩδΧϧઈԑମͷྫͱͯ͠ɺ࣍ͷ 4 όϯυ໛‫ܕ‬Λٞ࿦͠Α͏ɻͱ͍ͬͨ৽͍͠τϙϩδΧϧ૬΁ͱԠ༻͢Δɻ

4.1. ۭؒ൓సରশੑΛ࣋ͭ 3 ࣍‫ݩ‬τϙϩδΧϧઈԑମ
4.1.1. ۭؒ൓సରশੑͷ‫ݻ‬༗஋ʹΑΔ෼ྨ

ۭؒ൓సରশੑΛ࣋ͭ 3 ࣍‫ ݩ‬Z2

H⃗k =

!

I=x,y,z/0!
τx σI t sin kI a + τz σ0 t m −cos kI a .
I=x,y,z(128)

͜ͷ໛‫ۭؒͰܕ‬൓సରশੑ͸ U⃗k (I) = τz σ0 ɺ࣌ؒ൓సରশ

ۭؒ൓సରশੑͱ࣌ؒ൓సରশੑΛ࣋ͭࡾ࣍‫ݩ‬ͷిࢠ‫ܥ‬Λߟ͑Δɻ͜ͷରশੑʹରͯ͠͸ 3.1.5 અͰٞ࿦ͨ͠ରশ
G
͸ (Z2 )3 × Z4 ͱͳΔɻ͜ͷ݁ՌͷҙຯΛཧղ
ੑࢦඪ XBS͢Δͷ͕ຊઅͷ໨తͰ͋Δɻੑ͸ U (T ) = −iτ0 σy ͱද‫͞ݱ‬ΕΔɻ࣌ؒ൓సෆมͳ೾਺

K3D ʹ͓͚ΔϑΣϧϛ໘ҎԼ 2 όϯυͷۭؒ൓సରশੑͷ‫ݻ‬༗஋Λௐ΂Δʹ͸ɺ࠷‫ޙ‬ͷ߲ͷ܎਺ʹண໨͢Ε͹Α͍ɻ"
1 < m < 3 ͷͱ͖ m −cos kI a ͸ ⃗k = (0, 0, 0) ͷ
I=x,y,z࢝Ίͷ 3 ͭͷ Z2 ͱ͍͏ϑΝΫλʔ͸ 2 ࣍‫ݩ‬ͷ Z2 τϙϩ

1 ఺ͰͷΈෛͱͳΓɺ࢒Γͷ 7 ఺Ͱਖ਼ͱͳΔʢਤ 4(b)ʣɻҰδΧϧઈԑମΛ 3 ࣍‫ݩ‬తʹੵ૚͢Δ͜ͱͰಘΒΕΔऑ͍τ

ํ 3 < m Ͱ͸ 8 ͭશͯͷ೾਺Ͱਖ਼Ͱ͋Δʢਤ 4(c)ʣɻͨ͠

ϙϩδΧϧઈԑମ (weak topological insulator) Λද͢ɻ۩͕ͬͯ (νyzxମతʹٞ࿦͢ΔͨΊɺ͜͜Ͱ΋ ⃗ax = (a, 0, 0), ⃗ay = (0, a, 0),(1, 1, 1, 3)ɺ3 < m ͷͱ͖ (0, 0, 0, 0) ͱͳΔɻ

(k =0)(k =0), νzxy(k =0), νxyz, κ1 ) ͸ 1 < m < 3 ͷͱ͖⃗az = (0, 0, a) ͱ͢Δɻkz = 0 ͱ kz = π ͷ 2 ࣍‫ݩ‬໘ʹͦΕͧΕ (125) ࣜͷ Fu-Kane ެࣜ(−1)νxy =2

4.1.2. τϙϩδΧϧઈԑମͷද໘ঢ়ଶζ⃗k′ (I)

(125)

⃗
k∈K2D

ʢζ⃗′ (I) ͸֤ΫϥϚʔεର͔Βยํ͚ͩΛબΜͰͦͷ I ͷ‫ݻ‬k༗஋ͷੵΛͱͬͨ΋ͷʣΛద༻͢Δͱɺz ࣠ํ޲ʹੵ૚͞લઅͷྫͷ 1 < m < 3 ͷ૬ͱ 3 < m ͷ૬ͷؒͷ 2 ࣍‫ݩ‬ք໘ʹ‫ݱ‬ΕΔද໘ঢ়ଶΛɺ3.3.3 અͱಉ͡ํ๏Ͱղੳ͠Α͏ɻ
͍·๏ઢϕΫτϧΛ n̂ ͱ͢Δฏ໘Λߟ͑ɺδmxn = m − 3
͕ xn ≡ n̂ · ⃗x = L ͷલ‫ Ͱޙ‬−m0 ͔Β +m0 (0 < m0 < 2)

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物性研究・電子版 Vol. 8, No. 1, 081202（2020年2月号）

(a)

(b)

++

(c)

++

++

‒‒

++

++
++

(d)

++

++

++

++
++

(e)

++++
++++
++
++++
++++

++
++

++

‒‒‒‒
++++
++++
++++

࣌ؒ൓సରশੑʹकΒΕΔ‫͍ڧ‬τϙϩδΧϧઈԑମ͸ Z2
Ͱ෼ྨ͞ΕΔɻ͜ͷࣄ࣮͸ɺH eﬀ⃗ ୯ମʹ͸࣭ྔ߲Λ‫ؚ‬Ίn̂,k∥Δ͜ͱ͸Ͱ͖ͳ͍͕ɺ͜ΕΛೋͭॏͶͨ (H eﬀ⃗ )′ = H⃗eﬀ ⊕

n̂,k∥
k
eﬀ
H⃗k ʹ͸࣭ྔ߲ Mn̂ n̂ · ⃗σ λy ΛೖΕΒΕΔ͜ͱ͔ΒཧղͰeﬀ′

͖Δɻ࣮ࡍ͜ͷ߲Λ‫ؚ‬Ίͯ΋࣌ؒ൓సରশੑ U(f)(T ) =−iσy λ0 ͸ഁΕͳ͍͕ɺද໘ঢ়ଶͷ෼ࢄؔ܎ʹ͸ εn̂,⃗k∥ =3± (ta|⃗k∥ |)2 + Mn̂2 ͷΑ͏ʹΪϟοϓ͕։͘ɻ͔ۭؒ͠͠൓సରশੑΛཁ‫͢ٻ‬Δ৔߹͸‫׬‬શʹΪϟοϓ

Λ։͚Δ͜ͱ͸Ͱ͖ͳ͍ɻͳͥͳΒ (132) ࣜͷؔ܎Λอͭ
ਤ 4 (a)–(d): ࣌ؒ൓సෆมͳ೾਺ͱ઎༗όϯυͷ࣋ͭ൓సରশੑͷ‫ݻ‬༗஋ɻ(e): 1 < m < 3 ͷ૬ͱ 3 < m ͷ૬ͷք໘ɻ(f): ߴ࣍τϙϩδΧ
ϧઈԑମͷྫɻ੺৭ͷԁ͕ϔϦΧϧ୺ঢ়ଶΛද͢ɻͨΊʹ͸ɺ࣭ྔ߲ͷ܎਺͕M−n̂ = −Mn̂(133)Λຬͨ͞ͳ͚Ε͹ͳΒͣɺMn̂ ͷ஋͕ਖ਼͔Βෛ΁ͱมԽͱ΁ͱมԽ͢Δͱ͢Δʢਤ 4(e)ʣɻ⃗k = kn n̂ + ⃗k∥ ʹ͍ͭͯઢ‫ܕ‬Խ͞ΕͨϋϛϧτχΞϯΛ n̂ ํ޲ʹͷΈ࣮ۭؒʹ໭͢ͱ

Hn̂,⃗k∥ ,xn = −itaτx⃗σ · n̂∂xn + taτx⃗σ · ⃗k∥ + τz σ0 tδmxn (129)ͱͳΔɻ͜Εʹνϟʔϯઈԑମͷͱ͖ͱಉ༷ͷղੳΛߦ͏ͱZ2 τϙϩδΧϧઈԑମͷΑ͏ʹ 1 ࣍‫ݩ‬ͷϔϦΧϧ୺ঢ়ଶ͕‫ݱ‬ΕΔʢਤ 4(f) ͷ੺ԁʣɻ3 ࣍‫ݩ‬όϧΫͷද໘ʹ 1 ࣍‫ݩ‬తͳΪϟοϓϨεྭ‫ى‬ঢ়ଶΛ࣋ͭͨΊɺ͜Ε͸ߴ࣍τϙϩδΧ

(130)

∥

ͱ͍͏ද໘ঢ়ଶΛ‫ه‬ड़͢Δ༗ޮ໛‫͕ܕ‬ಘΒΕΔɻ͜ͷ༗ޮ໛‫ܕ‬΋࣌ؒ൓సରশੑΛอ͓ͬͯΓɺU eﬀ (T ) = −iσy ʹର

ͯ͠

eﬀ
eﬀ
U eﬀ (T )(Hn̂,)∗ = Hn̂,−U eﬀ (T )
⃗
⃗
k
k
∥

(131)

͕੒ཱ͢Δɻ͜ͷରশੑͷͨΊʹ࣭ྔ߲ Mn̂ n̂·⃗σ Λ‫ؚ‬ΊΔ͜ͱ͸‫͞ࢭې‬Ε͓ͯΓɺH eﬀ ͸ ε ⃗ = ±ta|⃗k∥ | ͱ͍͏σΟn̂,⃗k∥ੜͣΔ͔ΒͰ͋Δ18, 19) ɻ͜ͷυϝΠϯ΢Υʔϧʹ͸ 2 ࣍‫ݩ‬ϧઈԑମͷྫͰ͋Δɻeﬀ
Hn̂,= ta⃗k∥ × n̂ · ⃗σ⃗k∥͜ΖͰඞͣ Mn̂ = 0 ͱͳΔ 1 ࣍‫ݩ‬తͳυϝΠϯ΢Υʔϧ͕n̂,k∥ϥοΫ‫ܕ‬෼ࢄؔ܎Λ࣋ͭΪϟοϓϨεද໘ঢ়ଶΛ‫ه‬ड़͢Δɻ
͜ͷද໘༗ޮ໛‫ܕ‬͸࣌ؒ൓సରশੑ͚ͩͰͳ͘
eﬀ
eﬀ
Hn̂,
= H−n̂,−⃗⃗k∥k∥

4.1.4. Ұ༷࣓৔ԼͷτϙϩδΧϧઈԑମ
3 ࣍‫ݩ‬ͷߴ࣍τϙϩδΧϧઈԑମͷྫ͸ɺ3 ࣍‫ݩ‬ͷ‫͍ڧ‬τϙϩδΧϧઈԑମʹҰ༷࣓৔Λ͔͚ͯ΋࡞Δ͜ͱ͕Ͱ͖
Δɻ࣌ؒ൓సରশੑ͕ഁΕΕ͹ (130) ࣜͷ H eﬀ⃗ ʹ࣭ྔ߲n̂,k∥Mn̂ n̂ · ⃗σ ΛՃ͑Δ͜ͱ͕Ͱ͖Δ͕ɺۭؒ൓సରশੑͱ੔߹

͢ΔͨΊʹ͸΍͸Γ (133) ͕ࣜຬͨ͞Εͳ͚Ε͹ͳΒͳ͍ɻ͜ͷ৔߹ɺMn̂ = 0 ͱͳΔ 1 ࣍‫ݩ‬తͳυϝΠϯ΢Υʔϧʹ
͸ 2 ࣍‫ݩ‬νϟʔϯઈԑମͷΑ͏ʹΧΠϥϧ୺ঢ়ଶ͕‫ݱ‬ΕΔɻ

(132)

࣮͸͜ͷ 3 ࣍‫࣍ߴݩ‬τϙϩδΧϧઈԑମ͸ 2 ࣍‫ݩ‬νϟʔϯઈԑମ΁ͱஅ೤઀ଓͰ͖Δ20) ɻΛຬ͍ͨͯ͠Δ͜ͱʹ஫ҙ͢Δɻ͜Ε͸ +n̂ Λ๏ઢͱ͢Δ
ද໘ͱ −n̂ Λ๏ઢͱ͢Δද໘͕ۭؒ൓సରশੑͰ݁ͼͭ͘͜ͱΛ͍ࣔͯ͠Δɻ֤ද໘Λ‫ݸ‬ผʹΈΔͱۭؒ൓సରশੑ

4.2. 2 ࣍‫ݩ‬ͷ੬ऑͳτϙϩδΧϧઈԑମG= {BS}G /{AI}G ͷඇࣗ໌ͳ஋ʹରԠ
͜͜·Ͱ͸ XBS͸ഁΕ͍ͯΔ͕ɺਤ 4(f) ͷΑ͏ʹۭؒ൓సରশੑΛอͭΑ͢ΔઈԑମΛߟ͖͑ͯͨɻ͜ΕΒͷτϙϩδΧϧઈԑମ͸͏ʹ༗‫ܥݶ‬Λ੾Γग़͢ͱɺද໘શମͱͯ͠͸ۭؒ൓సରশ

νϟʔϯ਺΍ Z2 ෆมྔͱ͍͏Ճ๏తͳτϙϩδΧϧ਺Λੑ͕อͨΕΔͷͰ͋Δɻ

༗͍ͯͨ͠ɻՃ๏తͳτϙϩδΧϧ਺͸ɺఆٛʹΑΓɺ‫ݪ‬ࢠ‫ݶۃ‬ͷઈԑମͱॏͶ߹Θͤͯ΋஋Λม͑ͳ͍ɻ
G
Ͱ‫ݟ‬Δͱࣗ໌͕ͩ
͜͜Ͱ͸‫ʹٯ‬ɺXBS

4.1.3. (0, 0, 0, 2) ૬ͷද໘ঢ়ଶ͜ͷ (1, 1, 1, 3) ૬ΛʮೋͭॏͶͨʯઈԑମΛߟ͑Α͏ɻΑΓਖ਼֬ʹ͸ɺ(128) ࣜͷ H⃗k Λ 2 ͭର֯ʹฒ΂ͯ H⃗′ =kH⃗k ⊕ H⃗k = H⃗k ⊗ λ0ʢλ ΋ σ ͱಉ༷ʹύ΢ϦߦྻΛද͢ͱ͢Δʣͱ͠ɺ઎༗όϯυͱඇ઎༗όϯυͷ਺ΛͦΕͧΕഒʹ͢Δɻ͢Δͱ઎༗όϯυͷ൓సରশੑͷ‫ݻ‬༗஋ ±1 ͷ਺͸୯ʹ(k =0)(k =0)(k =0), κ1 ) = (1, 1, 1, 3)+
ഒͱͳΔͨΊɺ(νyzx , νzxy , νxyz

(1, 1, 1, 3) = (0, 0, 0, 2) ͱͳΔʢਤ 4(d)ʣɻ͜ͷঢ়ଶͷද໘Λ‫ه‬ड़͢Δ༗ޮ໛‫ܕ‬͸ (H eﬀ )′ = H eﬀ ⊕ H eﬀ = ta⃗k∥ × n̂ ·⃗σ λ0Ͱ༩͑ΒΕΔɻ

n̂,⃗
k∥

⃗
k

⃗
k

⃗b ∈
/ {AI}G≥0(134)

Ͱ͋Δ‫ط‬໿ද‫ݱ‬ͷ૊Έ߹Θͤ ⃗b ∈ {BS}G≥0 Λ࣋ͭઈԑମΛߟ͑Α͏ɻ͜ͷ৔߹ɺ͋Δ ⃗a ∈ {AI}G≥0 ͕ଘࡏͯ͠

⃗b + ⃗a ∈ {AI}G
≥0

(135)

ͱͳΒͳ͚Ε͹ͳΒͳ͍ɻ(134)ɺ(135) ࣜʹΑΕ͹ɺ͜ͷઈԑମͦΕࣗମ͸ରԠ͢Δ‫͕ݶۃࢠݪ‬ଘࡏ͠ͳ͍ͨΊඞͣඇࣗ໌ͳτϙϩδʔΛ༗͢Δ͕ɺద౰ͳ‫ݶۃࢠݪ‬ͷઈԑମ

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ΛॏͶ߹ΘͤΔ͚ͩͰτϙϩδΧϧʹࣗ໌ʹͳΔՄೳੑ͕͋ΔɻҰൠʹɺ‫ݶۃࢠݪ‬Λ૊Έ߹ΘͤΔ͚ͩͰࣗ໌ʹͳͬ
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͋Δͱ͠Α͏ɻ͢ͳΘͪ

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⃗b = (n+ , n− , n+ , n− , n+ , n− , n+ , n− )
Γ
Γ
X
X
Y
Y
M
M
= (2, 0, 2, 0, 2, 0, 0, 2) ∈ {BS}G≥0(136)Ծʹ࣌ؒ൓సରশੑ΋͋Δ৔߹ʹ͸ɺFu-Kane ެࣜ (125)ʹΑΔͱ͜ͷ ⃗b ͸ 2 ࣍‫ ݩ‬Z2 τϙϩδΧϧઈԑମʹରԠ͠
͍ͯΔ͜ͱ͕෼͔Δɻ͔͠͠ࠓ͸࣌ؒ൓సରশੑΛԾఆ͍ͯ͠ͳ͍ͷͰɺZ2 ෆมྔ͸ఆٛͰ͖ͳ͍ɻ͔࣮͠͠͸͜ͷ⃗b ʹ͍ͭͯ͸ (134) ͕ࣜ੒ཱ͓ͯ͠Γɺۭؒ൓సରশੑͷԾఆͷ‫Ͱݩ‬͸೗ԿͳΔ‫ʹݶۃࢠݪ‬΋݁ͼ͔ͭͳ͍ɻ͜ͷ ⃗b ͱҰக͢Δۭؒ൓సରশੑͷද‫ݱ‬Λ࣋ͭઈԑମͷྫ͸ɺΧΰϝ֨ࢠʹ७‫਺ڏ‬ͷඈͼҠΓੵ෼Λಋೖ͢Δ͜ͱͰ࣮‫͢ݱ‬Δ͜ͱ͕Ͱ͖Δ10, 21) ɻ͜͜ʹ⃗a = (1, 0, 0, 1, 0, 1, 1, 0) ∈ {AI}G≥0(137)ΛՃ͑ͨ ⃗b + ⃗a = (3, 0, 2, 1, 2, 1, 1, 2) ͸ {AI}G≥0 ʹଐ͍ͯ͠

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