《講義ノート》

物性研究・電子版 Vol. 10, No. 1, 101215(2022年3月号)

૬ؔిࢠ‫ܥ‬ͷඇฏߧ෺ੑ‫ڀݚ‬ͷ࠷લઢଜ্ɹ༤ଠ౦‫ۀ޻ژ‬େֶ ཧֶӃ෺ཧֶ‫ܥ‬֓ཁฏߧঢ়ଶͰ͸ɺిࢠ૬ؔʹΑΓ΋ͨΒ͞ΕΔ෺ੑ΍ྔࢠ૬ʹΑΓ‫ݻ‬ମ෺ཧ͕࠼ΒΕ͖ͯͨɻҰํɺ૬ؔిࢠ‫ܥ‬Λ֎৔Ͱ‫ۦ‬ಈ͢Δ͜ͱͰΑΓଟ࠼ͳ෺ੑͷൃ‫ظ͕ݱ‬଴͞Εɺۙ೥ɺ࣮‫ݧ‬ɾཧ࿦ͷ྆ଆ໘͔Β‫ٸ‬଎ʹ‫͕ڀݚ‬ਐΊΒΕ͍ͯΔɻ
ඇฏߧ෺ੑͷ‫Ͱڀݚ‬͸ɺ‫ޫ͍ڧ‬Λ༻͍ͯ෺࣭Λʮ‫ݟ‬Δʯͦͯ͠ʮૢΔʯͱ͍͏͜ͱ͕େ͖ͳ໨తͱͳ͍ͬͯΔɻຊߘͰ͸ɺ͜ͷ̎఺ΛΩʔϫʔυʹͯ͠ɺ͍͔ͭ͘ͷ‫ڀݚ‬ΛৼΓฦΓͭͭඇฏߧ෺ੑ‫ڀݚ‬ͷಋೖΛߦ͏ͱͱ΋ʹɺզʑͷ࠷ۙͷ‫঺͍ͯͭʹڀݚ‬հ͢Δɻʮ‫ݟ‬Δʯͱ͍͏‫Ͱ఺؍‬͸ɺMott ઈԑମ͔Βͷߴ࣍ߴௐ೾ൃੜͷ‫ڀݚ‬Λ঺հ͠ɺHHG
ʹΑΓଟମ‫ܥ‬ͷྭ‫ߏى‬଄͕ଊ͑ΒΕΔ͜ͱΛղઆ͢ΔɻʮૢΔʯͱ͍͏‫͔఺؍‬Β͸ɺPrethermal ঢ়ଶʹ͓͚Δྭ‫ࢠى‬ઈԑ૬ͷ༠‫ى‬ɺͦͯ͠ɺ༠‫ʹى‬ΑΓੜ੒͞ΕΔޫΩϟϦΞ͔Βੜ͡Δඇฏߧடং૬ʹؔ͢Δ‫ڀݚ‬Λ঺հ͢Δɻ·ͨɺඇฏߧঢ়ଶʹ͓͚Δిࢠ૬ؔͷӨ‫ڹ‬Λௐ΂Δ্Ͱཧ࿦ख๏ͷ։ൃ͸ॏཁͰ͋Δɻͦ͜Ͱɺ༗ྗͳཧ࿦ख๏Ͱ͋ΔඇฏߧάϦʔϯؔ਺๏ɺಛʹඇฏߧಈతฏ‫ۉ‬৔ཧ࿦ɺͷ‫ۙ࠷ͱૅج‬ͷਐలͷղઆ΋ߦ͏ɻ

1 ͸͡Ίʹ
ిࢠ૬ؔͷ‫Ͱܥ͍ڧ‬͸ɺిՙɺεϐϯɺ‫ي‬ಓɺϑΥϊϯͳͲ͕ෳࡶʹབྷΈ߹͏ɻͦΕʹΑΓ‫ܥ‬͸༷ʑͳ෺࿦ (DMFT)ɺͷൃల͕஫໨ʹ஋͢Δɻඇฏߧ DMFT
ͷൃలʹΑΓɺ‫ڧ‬૬ؔిࢠ‫ܥ‬ͷඇฏߧμΠφϛΫεΛ
ৄࡉʹٞ࿦Ͱ͖ΔΑ͏ʹͳΓɺ࣮‫ͱݧ‬ཧ࿦ͱͷൺֱ΋ͳ͞Εͭͭ͋Δɻ

ੑͱடং૬Λࣔ͠ɺฏߧ‫͚͓ʹܥ‬Δ‫ݻ‬ମ෺ཧΛ๛͔ʹຊߘͷ໨త͸ɺ૬ؔిࢠ‫ܥ‬ͷඇฏߧ෺ੑͷ‫ڀݚ‬ͷָ͖ͯͨ͠ɻ͜ͷΑ͏ͳ‫ܥ‬͸Թ౓΍ѹྗͳͲͷ֎෦ύϥ͠͞Λ఻͑Δ͜ͱͰ͋ΔɻͦͷͨΊɺຊߘͰ͸·ͣඇϝʔλʹରͯ͠େ͖ͳԠ౴Λࣔ͠ɺ෺ੑ͕େ͖͘มԽ

ฏߧ෺ੑΛ୳Δ্Ͱͷಈ‫ػ‬΍ߟ͑ํͷ֓‫͔؍‬Β࢝ΊΔɻͨ͠Γɺ૬సҠ͕‫͖ى‬ΔɻͰ͸ɺ࣌ؒʹґଘ͢Δ‫͍ڧ‬·ͨɺ૬ؔిࢠ‫ܥ‬ͷඇฏߧΛѻ͏্Ͱద੾ͳཧ࿦ख๏

֎৔Ͱ‫ܥ‬Λ‫ۦ‬ಈ͢ΔͱԿ͕‫͖ى‬Δͷ͔ʁɹۙ೥ɺ૬ؔ

Λ༻͍Δ͜ͱ͸ॏཁͰ͋Δɻͦ͜ͰɺඇฏߧάϦʔϯిࢠ‫͚͓ʹܥ‬Δඇฏߧ‫ݱ‬৅͕஫໨ΛूΊɺ༷ʑͳ‫ݱ‬৅ؔ਺๏Λத৺ʹཧ࿦ख๏Λ঺հ͠ɺ࠷৽ͷ։ൃঢ়‫گ‬Λ͕༧‫ݴ‬ɾ‫؍‬ଌ͞Ε͍ͯΔɻඇฏߧঢ়ଶͰ͸ɺฏߧঢ়ଶ

঺հ͢Δɻͦͯ͠ɺ࠷ۙͷ‫͔ڀݚ‬Βɺ‫ڧ‬૬ؔిࢠ‫ʹܥ‬Ͱ͸ಘ೉͍‫ܥ‬ͷ৘ใͷ֫ಘ͕ՄೳʹͳͬͨΓɺ‫ܥ‬ͷੑ͓͚Δߴ࣍ߴௐ೾ൃੜɺͦͯ͠ɺඇ೤తͳடং૬ͷ༠࣭Λେ͖͘ม͑Δࢥ͍΋‫د‬Βͳ͍‫ݱ‬৅͕‫͖ͨى‬Γ͢Δ‫ؔ͢ʹى‬Δ‫ڀݚ‬Λ঺հ͢Δɻ͜ͱ͕Θ͔͖ͬͯͨ [1–5]ɻ
͜ΕΒͷਐలͷഎ‫ͯ͠ͱܠ‬ɺϨʔβʔٕज़ͷൃలʹ

2 ෺࣭ + ‫? = ޫ͍ڧ‬ΑΓɺ༷ʑͳ৚݅Ͱྭ‫؍ͱى‬ଌ͕Մೳʹͳͬͨ͜ͱ͕

ඇฏߧ෺ੑͷ‫ڀݚ‬ͷେ͖ͳಈ‫ʹػ‬͸ʮ‫ݟ‬Δʯͱʮૢ‫͛ڍ‬ΒΕΔɻ͞Βʹɺ෺࣭͚ͩͰ͸ͳ͘ɺૢ࡞ੑͷߴ

Δʯͱ͍͏ଆ໘͕͋Δɻʮ‫ݟ‬Δʯͱ͸ɺ෺࣭ʹ‫ޫ͍ڧ‬Λ͍ྫྷ٫‫ؾࢠݪ‬ମͰ΋ଟମ‫ܥ‬ͷඇฏߧ‫ݱ‬৅͕γϛϡϨʔ

͋ͯΔ͜ͱͰɺ(ऑ͍ޫͰ͸) ͦΕ·ͰಘΒΕͳ͔ͬͨτՄೳͰ͋Δ͜ͱ΋ҰཌྷΛ୲͍ͬͯΔɻ‫ܥ‬ͷ৘ใɺྫ͑͹‫ݻ‬༗ͷྭ‫ى‬ঢ়ଶͷಛੑɺ͕‫؍‬ଌͰ͖Δ·ͨɺཧ࿦తʹ͸͜ͷे਺೥ͷؒʹɺඇฏߧ‫ݱ‬৅ΛΑ͏ʹͳΔ͜ͱͰ͋ΔɻʮૢΔʯͱ͸ɺ‫ޫ͍ڧ‬Λ༻͍ͯ

ଊ͑Δ্Ͱͷ༷ʑͳߟ͑ํ΍ཧ࿦ख๏ͷ։ൃ͕ਐల

‫ܥ‬Λฏߧঢ়ଶ͔Βେ͖͘‫ۦ‬ಈ͢Δ͜ͱͰɺฏߧͱ͸ҟ͖ͯͨ͠ɻલऀͷҰྫ͸ɺFloquet ΤϯδχΞϦϯά

ͳΔ෺ੑΛࣔ͢ঢ়ଶΛ࣮‫͢ݱ‬Δͱ͍͏͜ͱͰ͋Δɻ͜ͱ‫ݺ‬͹ΕΔߟ͑ํͰ͋Γɺप‫֎ظ‬৔Λ‫ʹܥ‬রࣹͨ࣌͠

͜Ͱ͸ɺඇฏߧ෺ੑ‫΁ڀݚ‬ͷಋೖͱͯ͠ɺ
ʮ‫ݟ‬Δʯͱʮૢ

ʹɺ༗ޮతʹ‫ܥ‬ͷϋϛϧτχΞϯΛมߋͰ͖Δͱ͍͏

Δʯͱ͍͏‫͔఺؍‬Βɺ͍͔ͭ͘ͷ‫ྫڀݚ‬Λ‫ͭͭ͛ڍ‬ɺ༗΋ͷͰ͋Δɻ͜ͷߟ͑ํʹ‫༷͍ͯͮج‬ʑͳ෺࣭ͷૢ࡞༻ͳ֓೦Λ঺հ͍ͨ͠ɻඇฏߧ‫ڀݚ‬͸‫ڧ‬૬ؔిࢠ‫ʹܥ‬๏͕ཧ࿦తʹ༧‫͞ݴ‬Ε͓ͯΓɺྫྷ٫‫ؾࢠݪ‬ମͷ࣮‫Ͱݧ‬‫ͨͬݶ‬΋ͷͰ͸ͳ͍͕ɺ༷ʑͳछྨͷ૬‫͕༻࡞ޓ‬བྷΈͦͷ࣮‫͕ݱ‬ใࠂ͞Ε͍ͯΔɻཧ࿦ख๏։ൃͱ͍͏఺Ͱ

߹͍ɺෳ਺ͷடং͕‫͢߹ڝ‬Δ‫ڧ‬૬ؔిࢠ‫ܥ‬͸྆ଆ໘͔͸ɺඇฏߧάϦʔϯؔ਺๏ɺಛʹඇฏߧಈతฏ‫ۉ‬৔ཧΒ֨޷ͷ୊ࡐʹͳ͍ͬͯΔɻ

《講義ノート》

物性研究・電子版 Vol. 10, No. 1, 101215(2022年3月号)

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൒ಋମʹ͓͚Δߴ࣍ߴௐ೾ൃੜͷϝΧχζϜͷ

֓೦ਤɻ

ͱͳΔɻ͜͜Ͱɺĉ ͸ϑΣϧϛཻࢠͷফ໓ԋࢉࢠɺi, j͸αΠτΛද͠ɺij ͸࠷ۙ઀αΠτͷϖΞɺq ͸ిՙɺrij ͸ j αΠτ͔Β i αΠτ΁ͷූ߸෇‫͋Ͱ཭ڑ‬Δɻ( = 1 ͱ͍ͯ͠Δ) A(t) ͸ϕΫτϧϙςϯγϟϧ
Λද͠ɺి৔ E(t) ͱ͸ E(t) = −∂t A(t) Ͱ݁ͼͭ͘ɻ

2.1 ‫Ͱޫ͍ڧ‬෺Λ‫ݟ‬Δ෺࣭ͷੑ࣭Λௐ΂Δ্Ͱޫ͸ॏཁͳπʔϧͰ͋Δɻ
ޫֶԠ౴ɺϥϚϯࢄཚɺ֯౓෼ղޫిࢠ෼ޫ͸͢΂ͯޫͱ෺࣭ͷ૬‫༻࡞ޓ‬Λհͯ͠෺࣭ͷྭ‫ى‬ঢ়ଶͷ৘ใΛಘΔख๏Ͱ͋Δɻ͜ΕΒͷଌఆͰ༻͍Δޫ͸͋·Γ‫ڧ‬

eiqA(t)rij ͸ Peierls Ґ૬ͱ‫ݺ‬͹Εɺtight-binding ໛‫ܕ‬ʹ͓͚Δޫͱ෺࣭ͷ݁߹Λද߲͢ͷҰͭʹͳΔɻ؆୯ͷͨΊ q = 1ɺ֨ࢠִؒ΋ ̍ ͱͯ͠ɺϋϛϧτχΞϯ
Λӡಈྔදࣔ͢Δͱɺ(k) = −2v cos(k) Λ༻͍ͯ͘ͳ͍ɻ͔͠͠ɺΑΓ‫ޫ͍ڧ‬Λ༻͍Δ͜ͱͰɺͦΕ·Ĥ(t) =

Ͱ‫͔ͨͬͳ͑ݟ‬ɺ΋͘͠͸‫ܥ͔ͨͬ͘ʹ͑ݟ‬ͷ৘ใΛಘΔ͜ͱ͕ՄೳʹͳΔɻ(k − A(t))ĉ†k ĉk(2)kͱද͞ΕΔɻ࣌ࠁ t = 0 Ͱ͸֎৔͕ͳ͘ɺॳ‫ظ‬ঢ়ଶͦͷҰͭͷྫ͕ɺ௒఻ಋதͷूஂྭ‫ى‬ϞʔυͰ͋Δ

ͱͯ͠ӡಈྔ k0 ͷཻࢠ͕Ұͭ͋Δঢ়‫گ‬Λߟ͑Δɻ͜

Higgs Ϟʔυͷ‫؍‬ଌͰ͋Δ [6]ɻHiggs Ϟʔυ͸ిՙΛͷঢ়ଶ͸ ĉk0 |vac ͱද͞Εɺ࣌ؒґଘ͠ͳ͍͜ͱ͕Θ࣋ͨͳ͍ͨΊɺ௨ৗͷ௒఻ಋͰ͸ޫֶԠ౴΍ϥϚϯࢄ

͔Δɻ͔͠͠ɺ֎৔ʹΑΓରԠ͢Δӡಈྔ͸࣌ؒมԽཚͰ͸ଊ͑ΒΕͳ͔ͬͨɻ(ͨͩ͠ɺిՙடংͱ௒఻

͠ k(t) ≡ k0 − A(t) ͱͳΓɺͦΕʹ൐͍ΤωϧΪʔಋͷ‫ڞ‬ଘ૬Ͱ͸ϥϚϯ‫׆‬ੑʹͳΓ‫؍‬ଌ͞Ε͍ͯͨɻ)͔͠͠ɺ‫ ͍ڧ‬THz ޫΛ༻͍ͨ௒఻ಋମͷϙϯϓɾϓ(k(t)) ΍ిࢠ଎౓ v(k(t)) = ∂k (k(t)) ΋มԽ͢Δɻ†ిྲྀ͸ ĵ(t) = q k v(k − A(t))ĉk ĉk (= −∂A Ĥ) Ͱ͋Δϩʔϒ࣮‫ Ͱݧ‬Higgs Ϟʔυʹ༝དྷ͢ΔৼಈΛ‫؍‬ଌͨ͠ͷͰɺ্ͷঢ়ଶͰ͸ j(t) = v(k(t)) ͕‫؍‬ଌ͞ΕΔɻ†Γ [7]ɺޫ༠‫͞ى‬ΕΔ Higgs Ϟʔυͷৼಈʹ༝དྷ͢Δ̏

͜͜Ͱɺ୯৭ޫͷ֎৔ A(t) = A0 sin(Ωt) ͕͔͔ͬͯ

࣍ͷߴௐ೾ൃੜΛ‫؍‬ଌ͢Δ͜ͱ [8] ͰɺHiggs ϞʔυΛ͍Δͱ͢Δͱɺ(k) ͷඇௐ࿨ੑ͔Βɺv(k(t)) ʹ͸ nΩ࣮‫ݧ‬తʹଊ͑Δ͜ͱ͕Ͱ͖Δɻ͜ͷํ๏ʹΑΓɺैདྷͷ੒෼͕‫·ؚ‬ΕΔ͜ͱ͕Θ͔ΔɻA0 ͷେ͖͞ʹΑͬ‫ܕ‬௒఻ಋ͚ͩͰ͸ͳ͘ɺ࠷ۙͰ͸ಔࢎԽ෺ߴԹ௒఻ಋ

ͯɺBrillouin κʔϯͷͲͷҐஔ·Ͱిࢠ͕Ҡಈ͢Δ͔

ͷ Higgs Ϟʔυ‫؍‬ଌ΋ਐΜͰ͍Δ [9]ɻ͕ܾ·Δɻখ͚͞Ε͹ (k) ͸ k 2 ͰۙࣅͰ͖ඇௐ࿨ੑ

·ͨผͷྫͱͯ͠ɺߴ࣍ߴௐ೾ൃੜ (HHG) ͱ‫ݺ‬͹Ε͸ޮ͔ͳ͍ͷͰ HHG ͸ൃੜ͠ͳ͍ɻA0 ͕े෼େ͖͘

Δ‫ݱ‬৅͕‫͛ڍ‬ΒΕΔɻHHG ͱ͸ɺप೾਺ Ω ͷޫΛ෺ඇௐ࿨ੑ͕ޮ͍ͯ͘Δӡಈྔ·ͰిࢠΛྭ‫͖Ͱى‬Ε͹࣭ʹೖΕͨ࣌ɺ੔਺ഒͷप೾਺ nΩ Λ࣋ͭޫ͕෺࣭͔

ͦΕʹԠͯ͡ HHG ͕ൃੜ͢Δɻ͜ͷ HHG ͷൃੜϝΒ์ࣹ͞ΕΔ‫ݱ‬৅Ͱ͋Δ [10, 11]ɻ͜ͷ‫ݱ‬৅͸ɺ‫ࢠݪ‬ɾΧχζϜ͸ɺిࢠͷόϯυ಺ͷӡಈΛ‫͢ͱݯى‬Δόϯ෼ࢠΨεͰ‫͞ڀݚ‬Ε͖ͯͯɺϨʔβʔͷप೾਺ม‫ʹ׵‬υ಺‫ݺͱߏػ‬͹ΕΔ΋ͷͰ͋Δ (ਤ̍ࠨ)ɻ͜ΕҎ֎ʹ࣮ࡍʹ࢖ΘΕ͍ͯΔɻۙ೥ɺ‫ݻ‬ମʹ͓͍ͯ΋‫؍‬ଌ͞Εɺిࢠͱϗʔϧͷ࠶݁߹ʹ༝དྷ͢Δ์ࣹΛද͢όϯυؒ‫ݻ‬ମதͷ HHG ͕େ͖ͳ஫໨ΛूΊ͍ͯΔ [12, 13]ɻ෺

‫͕͋ߏػ‬Δ (ਤ̍ӈ) [12, 14, 15]ɻԿΕʹͤΑɺిࢠ͕࣭͔Β์ࣹ͞ΕΔޫ͸෼‫ۃ‬ͷมԽ͢ͳΘͪిྲྀ (j(t))όϯυ෼ࢄΛ൓ө͠ͳ͕Βಈ͘͜ͱͰ HHG ͕ൃੜ͢

ʹ༝དྷ͢ΔɻΑΓਖ਼֬ʹ͸ɺిࢠͷՃ଎౓ͱؔ܎͠ɺ

ΔɻͦͷͨΊɺ‫ݻ‬ମதͷ HHG ͸प೾਺ม‫׵‬ͷૉࡐͩ์ࣹޫͷ‫౓ڧ‬෼෍͸ I(ω) = |ωj(ω)| ͱද͞ΕΔɻ‫ݻ‬͚Ͱ͸ͳ͘‫ݻ‬ମͷ෺ੑΛଌఆ͢Δπʔϧͱͯ͠΋஫໨ମதͰ͸ɺిࢠ͸प‫ظ‬తͳ‫ࢠݪ‬഑ྻʹ༝དྷ͢Δϙςϯ͞Ε͍ͯΔɻ

2γϟϧΛ‫͕ͳ͡ײ‬Βಈ͘ɻप‫ظ‬ϙςϯγϟϧͷޮՌ͸Ҏ্ͷྫͷଞʹ΋ɺޫྭ‫ʹى‬ΑΔடং૬ͷ༥ղաఔόϯυ෼ࢄʹ൓ө͞ΕΔͷͰɺ֎৔ʹΑΓ༠‫͞ى‬ΕΔ

Λ‫ݟ‬Δ͜ͱͰடং૬Λࢧ഑͢Δ૬‫༻࡞ޓ‬Λಛఆͨ͠ిྲྀͦͯ͠ HHG ͸όϯυ෼ࢄΛ൓ө͢Δ΋ͷͱͳΔɻΓ [16]ɺྭ‫ޙى‬ͷॳ‫ظ‬ঢ়ଶ΁ͷճ෮աఔΛଌఆ͢Δ͜͜ͷ͜ͱ͸ɺ‫঺Ͱޙ‬հ͢Δ‫ͱڀݚ‬΋ؔΘͬͯ͘ΔͷͰɺͱͰ‫ܥ‬ͷ૬‫༻࡞ޓ‬ͷେ͖͞Λ‫ੵݟ‬΋Δ͜ͱ΋ՄೳͰ࣮͋ࡍͷ໛‫ܕ‬Λ‫͕ͳݟ‬Βগ͠ߟ͑ͯΈΔɻ

Δ [2]ɻ͜ͷΑ͏ʹ༷ʑͳ৘ใ͕֫ಘͰ͖Δ͜ͱ͕ɺ‫ڧ‬؆୯ͷͨΊɺ·ͣ̍࣍‫ݩ‬ɺ̍όϯυͷ૬‫༻࡞ޓ‬ͷͳ͍ޫͰ෺ੑଌఆΛߦ͏͜ͱͷ‫ڧ‬Έͱͳ͍ͬͯΔɻ

《講義ノート》

物性研究・電子版 Vol. 10, No. 1, 101215(2022年3月号)

্‫ه‬ͷྫ͸ɺిࢠͷϗοϐϯάͷ࣌ؒεέʔϧ (਺

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fs) ʹൺ΂ͯɺे෼௕͍࣌ؒ (਺ ps) ग़‫͢ݱ‬Δ͕ɺ֨ࢠͷߏ଄มԽΛ൐͏ͱඇฏߧঢ়ଶ͕ඇৗʹ҆ఆʹͳΔ͜
ͱ΋͋Δɻྫ͑͹ɺ1T −TaS2 ͷྫ͕͋Γɺͦ͜Ͱ͸ɺฏߧঢ়ଶͰ͸ग़‫͍ͳ͠ݱ‬ύλʔϯΛࣔ͢ిՙடং૬͕ɺ
ޫʹΑΓඇৗʹ௕࣌ؒ҆ఆͯ͠ग़‫͢ݱ‬Δ͜ͱ͕ใࠂ͞


Ε͍ͯΔ [25]ɻ͜ͷঢ়ଶ͸ޫΛ͋ͯΔ͜ͱͰฏߧঢ়ଶ

La1.675 Eu0.2 Sr0.125 CuO4 ʢԹ౓ 10Kʣʹ͓͚Δɺ(a) ޫྭ‫ޙى‬ͷ limω→0 ωσ2 (ω)ɺ͓Αͼ (b) ൓ࣹ཰ɻ͜͜Ͱ σ2 (ω) ͸ޫֶ఻ಋ౓ͷ‫ڏ‬෦Ͱ͋Γɺ༗‫ݶ‬ͷ limω→0 ωσ2 (ω) ͸ ޫֶ఻ಋ౓ͷ‫ڏ‬෦͕ 1/ω తʹൃࢄ͢Δ͜ͱΛද͢ɻ൓ࣹ཰ʹ͓͍ͯ͸ 60 cm−1ͷடং૬ʹ໭͢͜ͱ΋Ͱ͖ɺͦΕΛ༻͍ͨ‫ه‬ԱσόΠ෇ۙͰϓϥζϞϯΤοδͱ͞ΕΔ‫ݟ͕ݞ‬ΒΕΔɻจ

తͳΞϓϩʔνͱͯ͠͸ɺHubbard ໛‫Ͳͳܕ‬ͷϛΫϩ‫[ ݙ‬17] ΑΓɻείϐοΫͳ໛‫ه͍ͨͮجʹܕ‬ड़΍ Ginzburg-Landauਤ2εͳͲ΁ͷԠ༻΋‫ظ‬଴͞Ε͍ͯΔɻޫ༠‫ى‬૬సҠ͸༷ʑͳ෺࣭Ͱใࠂ͞Εɺଟ‫ʹذ‬Θ͓ͨͬͯΓɺҟͳΔϝΧχζϜʹΑͬͯൃ‫͏͠ݱ‬Δɻཧ࿦ཧ࿦ͷΑ͏ͳ‫ݱ‬৅࿦Λ༻͍ͨ‫ه‬ड़͕͋Γɺ‫ݸ‬ʑͷ‫ݱ‬৅

2.2 ‫Ͱޫ͍ڧ‬෺ੑΛૢ࡞͢Δ

ͷཧ࿦తͳઆ໌ʹ࢖ΘΕ͖ͯͨɻҰํɺޫ༠‫ى‬૬సҠฏߧঢ়ଶͰ͸ɺԹ౓ɺѹྗɺ෺࣭ͷ૊੒Λม͑Δ͜ͱͷ౷Ұతͳཧղ͸ɺ(චऀͷ஌Δ‫ݶ‬Γ) ಘΒΕ͍ͯͳ͍ɻ

Ͱɺ༷ʑͳঢ়ଶ͕࣮‫͞ݱ‬ΕΔɻ͜ͷΑ͏ͳ͜ͱ͕ɺ‫ڧ‬ࠓ‫ޙ‬΋‫ݸ‬ʑͷ‫ݱ‬৅ʹؔ͢Δ஌‫ݟ‬Λ஝ੵ͠ͳ͕Βɺ౷Ұ͍ޫΛ༻͍ͯߴ଎ʹߦ͑ͳ͍͔ɺͱ͍͏‫͔఺؍‬Βɺ࣮తͳཧղΛ໨ࢦ͢ඞཁ͕͋Δɻ‫ݧ‬ɺཧ࿦ͱ΋ʹ‫͕ڀݚ‬ਐΊΒΕ͍ͯΔɻ

2.2.2 Floquet ΤϯδχΞϦϯάͱ͍͏ύϥμΠϜ

2.2.1 ޫ༠‫ى‬૬సҠ্‫ه‬ͷΑ͏ͳޫ༠‫ى‬૬సҠ͸ɺ‫ج‬ຊతʹ͸֎৔ʹΑͬޫ༠‫ى‬૬సҠͱ͸ɺޫΛরࣹ͢Δ͜ͱͰฏߧঢ়ଶͷͯྭ‫ʹޙͨ͠ى‬ग़‫͢ݱ‬Δ‫ݱ‬৅Ͱ͋ͬͨɻҰํͰɺ֎৔

͋Δ૬͔Βผͷ૬΁‫ܥ‬ΛసҠͤ͞Δ͜ͱͰ͋ΔɻͦͷΛ౰ͯଓ͚ͨͱ͖ɺ΋͘͠͸౰͍ͯͯΔؒʹ໘ന͍‫ݱ‬৅தʹ͸ɺޫ༠‫ى‬ͷઈԑମ-ۚଐసҠ [18–20]ɺޫ༠‫ى‬௒͕ग़‫͔͍ͳ͠ݱ‬ɺͱ͍͏ٙ໰͕༙͍ͯ͘Δɻ͜ͷٙ໰

఻ಋ (?) [17, 21–24] ΍ޫ༠‫ిى‬ՙடং [25] ͳͲ͕‫͛ڍ‬ʹ౴͑Λ༩͑Δͷ͕ɺFloquet ཧ࿦ʹཱ‫͢٭‬Δ FloquetΒΕΔɻྫ͑͹ɺઈԑମ-ۚଐసҠͰ͸ɺޫྭ‫ʹى‬ΑΓΤϯδχΞϦϯάͱ͍͏ߟ͑ํͰ͋Δ [4]ɻిՙΩϟϦΞ͕ੜ੒͞ΕͨΓ [18, 19]ɺ‫ܥ‬ͷύϥϝʔλ(Hubbard ໛‫ܕ‬ͷ͜ͱͰ͍͏ͱϗοϐϯά΍૬‫)༻࡞ޓ‬͕ޫʹΑΓมԽ͢Δ͜ͱͰ [20]ɺΪϟοϓΛࣔ͠ઈԑମతͰ͋ͬͨޫֶԠ౴͕มԽ͠ɺΪϟοϓ͕ຒ·Γۚ

Floquet ཧ࿦ͱ͸ɺ࣍ͷΑ͏ͳ΋ͷͰ͋Δɻ΋͠ϋϛϧτχΞϯ Ĥ(t) ͕ T ͷप‫ظ‬ੑΛ͍࣋ͬͯͨ࣌ɺ

|Φα (t) = e−iα t |uα (t) ୠ͠ |uα (t) = |uα (t + T )(3)

ଐతͳৼΔ෣͍Λࣔ͢Α͏ʹͳΔɻ
·ͨɺޫʹΑΓ௒఻ಋ͕༠‫͞ى‬ΕΔՄೳੑ΋ࢦఠ͞ΕͷΑ͏ʹɺҐ૬෦෼ e−iα t ͱप‫ظ‬෦෼ |uα (t) ʹ෼͚͍ͯΔɻྫ͑͹ಔࢎԽ෺ La1.675 Eu0.2 Sr0.125 CuO4 ͷΒΕΔղ͕ଘࡏ͠ɺ೚ҙͷ೾ಈؔ਺͸ͦΕΒͷ࿨Ͱද࣮‫Ͱݧ‬͸ɺ‫ܥ‬ͷϑΥϊϯΛ‫ڞ‬໐తʹྭ‫͢ى‬Δ͜ͱͰɺ௒

͞ΕΔɻα ͸ quasi-energy ͱ‫ݺ‬͹ΕΔɻ͜ͷཧ࿦͔఻ಋ Tc Ҏ্Ͱ΋ɺޫֶ఻ಋ౓ σ(ω) ͷ‫ڏ‬෦ σ2 (ω) ͕௿Βɺ࣌ࠁ T ͓͖ʹ‫ݟ‬Ε͹ɺঢ়ଶ͕͔͋ͨ΋ α ΛΤωप೾ʹ޲͔ͬͯ 1/ω ͷΑ͏ʹ૿େ͢ΔৼΔ෣͍͕༠‫ى‬ϧΪʔʹ࣋ͭΑ͏ͳ࣌ؒมԽ͠ͳ͍ϋϛϧτχΞϯʹ

͞ΕΔ͜ͱɺ͞Βʹ͸൓ࣹ཰ଌఆʹ͓͍ͯ௒఻ಋঢ়ଶಛ

ैͬͯಈ͍͍ͯΔΑ͏ʹΈͳͤΔɻผͷ‫ํ͍ݴ‬Λ͢Δ

༗ͷ Josephson ϓϥζϞϯΤοδ͕ൃ‫͢ݱ‬Δ͜ͱ͕ใͱɺ࣍ͷΑ͏ʹͳΔɻ͋Δ࣌ࠁ t0 Λ‫ج‬४ʹͯ͠ɺ࣌ࠂ͞Ε͍ͯΔ [17]ɻ͜ͷঢ়ଶ͸ൺֱత௕ण໋ (10ps ҎؒൃలԋࢉࢠΛ Û (t, t0 ) ≡ T exp(−i t0 dt̄Ĥ(t̄)) ͱ্͠) ͱ‫ݴ‬ΘΕ͍ͯΔɻࣅͨΑ͏ͳৼΔ෣͍͸ɺಔࢎԽ෺

ͨ࣌ɺ

YBa2 Cu3 O6+δ [21, 22] ΍ϑϥϨϯ௒఻ಋ K3 C60 [23]ͳͲͰ΋͕ใࠂ͞Ε͍ͯΔɻͨͩ͠ɺ͜ΕΒͷঢ়ଶ͕te−iT Ĥeff (t0 ) ≡ Û(t0 + T , t0 )(4)ຊ౰ʹ௒఻ಋͳͷ͔൱͔ɺ͸ࠓͰ΋ٞ࿦͕ଓ͍͍ͯΔɻʹΑͬͯ Ĥeﬀ (t0 ) Λಋೖ͢Δɻ͜ͷͱ͖ɺt0 + nT ʹྫ͑͹ɺ௒఻ಋͰ͸ͳ͘௕ण໋ͷΩϟϦΞ͕ੜ੒͞Ε

͓͚Δঢ়ଶΛ‫ݟ‬Δ‫ݶ‬Γɺ‫ ͕ܥ‬Ĥeﬀ (t0 ) ʹैͬͯ࣌ؒ

͍ͯΔՄೳੑͳͲ΋ࢦఠ͞Ε͍ͯΔɻ͍ͣΕʹͤΑԿൃల͢Δ͜ͱͱಉ౳ʹͳΔɻ͜ͷ Ĥeﬀ (t0 ) ͸֎৔ʹ͔͠Βͷඇࣗ໌ͳඇฏߧঢ়ଶ͕࣮‫͍ͯ͠ݱ‬Δ͜ͱ͸ؒґଘ͢ΔͨΊɺ֎৔ʹΑͬͯ‫ܥ‬Λࢧ഑͢Δϋϛϧτχ

ҧ͍ͳ͘ɺͦͷ‫ݯى‬ͷղ໌͕‫ٻ‬ΊΒ͍ͯΔɻ

ΞϯΛ༗ޮతʹมߋͰ͖Δͱ͍͏͜ͱʹͳΔɻ͜Ε͕《講義ノート》

物性研究・電子版 Vol. 10, No. 1, 101215(2022年3月号)

ΊΔ [26,27]ɻ͜ΕΒͷํ๏͸ɺ֎৔ͷप೾਺ (Ω = 2π
T )͕‫ܥ‬ͷయ‫ܕ‬తͳΤωϧΪʔεέʔϧΑΓେ͖͍৔߹ʹ

Ĥeﬀ (t0 ) ͸ɺMagnus ల։ͳͲͷߴप೾ల։Λ༻͍ͯ‫ٻ‬



!
  Floquet ΤϯδχΞϦάͷߟ͑ํͰ͋Δɻଟ͘ͷ৔߹ɺ   %$
%

# 
  "༗ޮͰ͋Δɻ

࣍ͷޮՌʹͳΓɺ༗ޮϋϛϧτχΞϯΛ࣌ؒฏ‫ۙͰۉ‬ࣅɺଈͪ Ĥeﬀ ∼1
T t0 +Tt0

dtĤ(t) ͱ͢Δ͜ͱͰ‫ه‬ड़͞ΕΔɻ࣮ࡍʹࣜ (2) Λ A(t) = A0 sin(Ωt) ͱͯ࣌ؒ͠ฏ‫ͯ͠ۉ‬ΈΔͱɺĤeﬀ = J0 (A0 )†k (k)ĉk ĉk ͱ͍







ࡏͰ͋Δ [28]ɻ͜ͷޮՌ͸ɺߴप೾ల։Ͱ͍͏ͱ࠷௿



Ω
W (t)

着目する系

I(t)



प‫֎ظ‬৔ͷ΋ͨΒ͢‫ݱ‬৅ͷҰ൪؆୯ͳྫ͸ɺಈత‫ہ‬



熱浴
時間



ਤ3

(a) ࣌ؒൃల (ա౉աఔ) ͷ֓೦ਤɻ(b)(ࠨ) ඇฏߧఆৗঢ়ଶɻ(ӈ) ண໨͢Δ‫ͱܥ‬೤ཋɺW (t) ͸֎৔͏݁ՌʹͳΔɻ͜͜ͰɺJ0 ͸̌࣍ͷϕοηϧؔ਺Ͱ͔͋ΒͷΤωϧΪʔྲྀೖɺI(t) ͸೤ཋ΁ͷࢄҳΛද͢ɻΔɻ͜ͷࣜ͸ɺ֎৔Λ‫͘͢ڧ‬Δͱ J0 (A0 ) ෼͚ͩόϯ੨ઢ͸ඇฏߧঢ়ଶͷฏߧঢ়ଶ͔ΒͷͣΕɺΦϨϯδઢυ෯͕খ͘͞ͳΔɺͭ·Γɺཻࢠ͕‫͢ࡏہ‬Δͱ‫ͱ͜͏ݴ‬͸Ϩʔβʔʢ֎৔ʣΛද͢ɻΛද͍ͯ͠Δɻ͜ͷޮՌ͸ɺ࣮ࡍʹྫྷ٫‫ؾࢠݪ‬ମ [29]΍༗‫ػ‬ಋମ [30] ͳͲͷ࣮‫Ͱݧ‬ใࠂ͞Ε͓ͯΓɺDMFT͞Βʹ͸ిࢠ֨ࢠؒͷࢄཚ͕ຊ֨తʹ࢝·Δલʹ‫͕ܥ‬ͱྫྷ٫‫ؾࢠݪ‬ମͰ௚઀ൺֱ΋ߦΘΕ͍ͯΔ [29]ɻ͞Βۙͮ͘ɺॳ‫ظ‬ঢ়ଶΛ‫͘ڧ‬൓өͨ͠ঢ়ଶͰ͋Δɻྫ͑͹ɺʹ֎৔ͷߴ࣍ͷޮՌʹΑͬͯɺΑΓ‫ڵ‬ຯਂ͍‫ݱ‬৅͕ग़͜ͷΑ͏ͳঢ়ଶ͸ɺ૬‫͕༻࡞ޓ‬ൺֱతऑ͍৔߹ [39, 40]‫͢ݱ‬Δɻྫ͑͹ɺԁภ޲ͨ͠ޫΛάϥϑΣϯʹরࣹͦͯ͢͠൓‫࣓ڧ‬ੑঢ়ଶ΍௒఻ಋঢ়ଶͷμΠφϛΫεͰใΔͱτϙϩδΧϧͳੑ࣭Λࣔ͢ Haldane ໛‫͕ܕ‬༗ޮతࠂ͞Ε͍ͯΔ [41]ɻ·ͨɺୈ 4 ষͰ঺հ͢Δඇฏߧடʹ࣮‫͖Ͱݱ‬Δ (Floquet τϙϩδΧϧঢ়ଶ) ͜ͱ͕ཧ࿦

ংͷ༠‫ى‬ͷٞ࿦΋͜ͷྖҬΛର৅ͱͨ͠΋ͷͱͳΔɻతʹ༧‫͞ݴ‬Εͨ [31, 32]ɻ͞Βʹɺ‫ڧ‬૬ؔ‫ܥ‬Λ͋ΒΘͦ͢ͷ‫ޙ‬ɺిࢠؒ΍ిࢠ֨ࢠؒͷࢄཚʹΑΔ؇࿨͕ਐੱྗ Hubbard ໛‫Ͱܕ‬͸ɺप‫֎ظ‬৔ʹΑΓεϐϯͷަΉɻࢄཚʹΑΓɺॳ‫ظ‬ঢ়ଶͷ৘ใ͸ࣦΘΕ͍ͯ͘ɻయ

‫׵‬૬‫༻࡞ޓ‬ͷૢ࡞͕ՄೳʹͳΓɺεϐϯ૬͕ؔ൓‫࣓ڧ‬‫ܕ‬తʹ͸͜ͷաఔͷ࣌ؒεέʔϧ͸਺े fsʙ਺ඦ fs Ͱੑత͔Β‫࣓ڧ‬ੑతʹͳΔ͜ͱ͕ཧ࿦తʹ༧‫͞ݴ‬Ε͍ͯ͋Δɻͦ͜Ͱ͸ɺྭ‫͞ى‬Εͨిࢠ͕ӡಈΤωϧΪʔΛΔ [33]ɻ͜ΕΒͷ༧‫ݴ‬͸ྫྷ٫‫ؾࢠݪ‬ମΛ༻͍࣮ͨ‫Ͱݧ‬ు͖ग़ͯ͠ӡಈΤωϧΪʔͷ௿͍ঢ়ଶʹ͏ͭͬͨΓɺઈ‫͞ূݕ‬Ε͍ͯΔ [34, 35]ɻ·ͨɺલऀͷํͰ͸࣮ࡍʹάԑମ͔Βͷྭ‫͋Ͱى‬Ε͹ిࢠͱϗʔϧ͕ରফ໓ͳͲΛϥϑΣϯΛ༻͍࣮ͨ‫ݧ‬΋ใࠂ͞Εͯɺޫ༠‫ى‬ͷτϙϩ‫͢͜ى‬ɻ௨ৗͷۚଐͰ͸ɺϘϧπϚϯํఔࣜʹΑͬͯδΧϧঢ়ଶΛࣔࠦ͢Δ݁Ռ͕ಘΒΕ͍ͯΔ͕ [36]ɺ࣮

‫ه‬ड़͞ΕΔ‫ݱ‬৅Ͱ͋Δɻ‫ڧ‬૬ؔిࢠ‫Ͱܥ‬͸ɺྭ‫ى‬Ωϟࡍʹ Floquet τϙϩδΧϧঢ়ଶ͕࣮‫͍ͯ͠ݱ‬Δ͔͸ٞϦΞͷ෼෍ʹΑͬͯΤωϧΪʔεϖΫτϧ͕େ͖͘ม

࿦ͷ༨஍Λ࢒͍ͯ͠Δ [37]ɻ

Խ͠͏ΔͨΊɺͦͷ‫ه‬ड़͸ΑΓෳࡶʹͳΔɻ

2.2.3 ࣌ؒεέʔϧͷ໰୊ిࢠ෺ੑ͸ిࢠঢ়ଶΛ൓өͯ࣌͠ʑࠁʑมԽ͓ͯ͠

෺࣭Λ֎৔Ͱ‫ۦ‬ಈ͢Δͱ࣌ʑࠁʑঢ়ଶ͕มԽ͍ͯ͠

Γɺ্ͷΑ͏ͳ prethermal ͳঢ়ଶ͞Βʹ͸ɺ؇࿨աఔ

͘ [38]ɻ͜ΕΒͷయ‫ܕ‬తͳ෺ཧաఔΛਤࣜతʹ‫ͯݟ‬Έͷ్தʹɺॳ‫ظ‬ঢ়ଶͱ͸ҟͳΔඇࣗ໌ͳ෺ੑΛࣔ͢ඇΑ͏ (ਤ 3(a))ɻ͜ΕʹΑΓɺ͜Ε·Ͱ‫͖֓ͨͯݟ‬೦ͷฏߧঢ়ଶ΋ੜ͡͏Δɻ·ͨ͜Εͱ͸ผʹɺిࢠࣗ༝౓

ͭͳ͕Γ͕‫ͯ͑͘ݟ‬Δɻ·ͣɺ௨ৗͷ࣮‫Ͱݧ‬͸ϙϯϓ͕ɺ֨ࢠ࿪Έ΍ిՙடংͳͲͷฏ‫ۉ‬৔తͳடংͱ͍ͬ

ޫʹΑͬͯ‫ۦ͕ܥ‬ಈ͞Εిࢠঢ়ଶ͕ॳ‫ظ‬ͷฏߧঢ়ଶ͔

ͨిࢠʹൺ΂ͯ࣌ؒεέʔϧͷඇৗʹ஗͍ࣗ༝౓ͱ݁

ΒͣΕΔɻ͜ͷ࣌ɺϙϯϓޫͷ৚݅ (प೾਺ɺ‫ʹ )౓ڧ‬ͼͭ͘͜ͱͰ΋ޫ༠‫ى‬૬సҠ͕ੜ͡͏Δɻ஗͍ࣗ༝౓

ґଘͯ͠ɺҟͳΔϝΧχζϜʹΑͬͯిࢠ͕ྭ‫͞ى‬Εʹର͍͔ͯͭ͘͠ͷ҆ఆঢ়ଶ͕ଘࡏ͢Δͱɺ͜ΕΒͷΔɻϙϯϓ͕ऴΘΔͱɺޫΤωϧΪʔ͕೤ʹؐ‫͞ݩ‬ΕؒΛޫʹΑͬͯεΠονͰ͖Δͱ͍͏Θ͚Ͱ͋Δɻͯ͜‫ܥ‬શମʹ‫ڞ‬༗͞Εͨ೤Խঢ়ଶ΁޲͚ͯ؇࿨͕࢝·ΔɻΕΒ͸ Franck-Condon ඳ૾Ͱଊ͑ΒΕ [1]ɺͦͷ࣌ؒ͜ͷ؇࿨ʹ͸͍͔ͭ͘ͷஈ֊͕͋Δ͜ͱ͕஌ΒΕͯεέʔϧ͸্‫ه‬ͷࢄཚաఔʹൺ΂Δͱ஗͘ͳΔɻ͍Δɻ·ͣɺྭ‫ى‬௚‫Ͱޙ‬͸ɺprethermal ঢ়ଶͱ‫ݺ‬͹Εྭ‫ޙى‬ͷ؇࿨աఔͷ‫ڀݚ‬ͷଞʹɺϙϯϓޫͰྭ‫ى‬தΔঢ়ଶʹ‫͕ܥ‬Ұ౓ۙͮ͘৔߹͕͋Δɻ͜ͷ prethermalͷ෺ੑɺ͞Βʹ͸ϙϯϓޫΛ౰ͯଓ͚ͨ࣌ʹੜ͡Δ෺

ঢ়ଶ͸೤ฏߧঢ়ଶͱ͸ҟͳΔྭ‫ى‬ঢ়ଶͰ͋Δ͕ɺిࢠؒ

ੑͷ‫ڀݚ‬΋੝ΜͰ͋Δ [4, 27, 38]ɻͦ͜Ͱ‫׆‬༂͢Δͷ͕ઌͷ Floquet ΤϯδχΞϦϯάͱ͍͏ߟ͑ํͰ͋Δɻ

《講義ノート》

物性研究・電子版 Vol. 10, No. 1, 101215(2022年3月号)‫ܥ‬ͷੑ࣭͸༗ޮϋϛϧτχΞϯͷ‫ʹܗ‬Ճ͑ɺͲͷΑ͏(b) 松原Im tͳঢ়ଶ͕ຒ·͍ͬͯΔ͔͕ॏཁͰ͋Δɻಛʹɺ֎৔Λ

 Kadanoﬀ-Baym
 Keldysh

Im t
Im t
t
Re t
Re t

t

౰ͯଓ͚Δͱ‫Ͱܥཱݽ‬͸‫ܥ‬͸࠷ऴతʹແ‫ݶ‬Թ౓ʹߦ͘ͱߟ͑ΒΕΔ͕ [27]ɺ࣮ࡍʹ͸஫໨‫ܥ‬͸೤ཋͱ݁ͼͭ

−β

−β

ਤ4

͘ͷͰɺ֎৔ʹΑΔඇฏߧఆৗঢ়ଶΛߟ͑Δ৔߹͸։Re tάϦʔϯؔ਺Λఆٛ͢Δࡾछྨͷ࣌ؒ‫ܦ‬࿏ɻ์‫ܥ‬Λߟ͑Δ (ਤ 3(b)) [28, 38]ɻͳɺ࣮࣌ؒํ޲ͷ‫ܦ‬࿏Λ‫ؚ‬Ή࣌ؒ‫ܦ‬࿏͕औΒΕΔɻਤ 4

3 ૬ؔిࢠ‫ܥ‬ͷͨΊͷඇฏߧཧ࿦ख๏(b) ͸ɺদ‫ܦ࣮ؒ࣌ʹࣜܗݪ‬࿏ΛՃ͑ͨ΋ͷͰɺদ‫ܗݪ‬ࣜͰ४උ͞Εͨฏߧঢ়ଶΛॳ‫ظ‬ঢ়ଶͱ͢Δ࣌ؒൃలΛଟମ໰୊͸ฏߧͰ͢Βѻ͍͸༰қͰ͸ͳ͍ɻ·ͯ͠‫ه‬ड़͢Δɻ͜Ε͸͠͹͠͹ Kadanoﬀ-Baym ‫ݺͱࣜܗ‬΍ඇฏߧʹ͓͍ͯ͸ɺ͞Βʹख๏͕‫ݶ‬ΒΕ͓ͯΓɺ৽͠͹ΕΔɻҰํɺ‫޲ํؒ࣌ڏ‬Λ͋ΒΘʹ‫͍ͳ·ؚ‬Α͏ͳ͍ख๏ͷ։ൃ͕೔ʑਐΊΒΕ͍ͯΔɻ͜ͷষͰ͸ɺ૬࣌ؒ‫ܦ‬࿏΋औΒΕΔ (ਤ 4 (c))ɻ͜Ε͸ Keldysh ‫ࣜܗ‬ؔిࢠ‫ܥ‬ͷඇฏߧ‫ݱ‬৅ΛϛΫϩείϐοΫʹ‫ه‬ड़͢Δͱ‫ݺ‬͹ΕΔ΋ͷͰ͋Δ͕ɺi) ฏߧঢ়ଶ͔Βग़ൃ͢Δ໰

ख๏ͱͯ͠ɺඇฏߧάϦʔϯؔ਺๏Λ঺հ͢Δɻಛʹɺ୊΍ ii) ඇฏߧఆৗঢ়ଶΛ‫ه‬ड़͢Δ৔߹ʹ࢖ΘΕΔɻi)

Mott ઈԑମΛѻ͑Δಈతฏ‫ۉ‬৔ཧ࿦Λॏ఺తʹઆ໌ͷ৔߹͸ɺ૬‫͠ͳ༻࡞ޓ‬ͷ໰୊͔Βग़ൃ͠૬‫༻࡞ޓ‬Λ͠ɺͦͷ࠷ۙͷख๏։ൃͷऔΓ૊ΈΛ঺հ͢Δɻ·ͨɺஅ೤తʹՃ͑ΔͳͲͷૢ࡞Ͱฏߧঢ়ଶΛ༻ҙ͢Δඞཁ

ඇฏߧάϦʔϯؔ਺๏Ҏ֎ͷख๏ʹؔͯ͠΋͜ͷষͷ͕͋Δɻii) ͷ৔߹͸ɺ஫໨͍ͯ͠Δ‫͕ܥ‬೤ཋͱ݁߹͠

࠷‫ʹޙ‬؆୯ʹ঺հ͢Δɻ

͍ͯΔͱߟ͑ɺͦΕʹΑΓੜ͡Δఆৗঢ়ଶΛ‫ه‬ड़͢Δɻ

3.1 ඇฏߧάϦʔϯؔ਺๏ͷಋೖ͜ͷͱ͖ɺ೤ཋͱͷ݁߹ͷͨΊॳ‫ظ‬ঢ়ଶʹΑΒͳ͍ఆ·ͣɺΠϝʔδ͕͖ͭ΍͍͢Α͏ʹɺ૬ؔిࢠ‫ܥ‬ͷৗঢ়ଶʹୡ͢Δɻయ‫ܕ‬ϞσϧͰ͋Δ̍όϯυ Hubbard ໛‫ܕ‬Λಋೖ͢Δɻ
ϋϛϧτχΞϯ͸

Ĥ(t) = −vijeiqA(t)rij ĉ†iσ ĉjσ + UҰཻࢠάϦʔϯؔ਺͸ɺ૬‫͍ͳ͕༻࡞ޓ‬৔߹ (ࣗ༝ͳ৔߹) ͸‫ٻʹີݫ‬ΊΒΕΔɻ͜ΕΛ G0 ͱද͢ͱɺ૬‫ޓ‬n̂i↑ n̂i↓(5)i࡞༻͕͋Δ৔߹͸ɺ࣍ͷ Dyson ํఔࣜΛղ͘͜ͱͰ‫ٻ‬ΊΒΕΔ;G(t, t ) = G0 (t, t )(7)
 
dt1 dt2 G0 (t, t1 )Σ(t1 , t2 )G(t2 , t ).
+Ͱ͋ΒΘ͞ΕΔɻ͜͜Ͱ σ ͸εϐϯࣗ༝౓ɺn̂iσ =ĉ†iσ ĉiσ ɺU ͸‫ॴہ‬తͳ Coulomb ૬‫͋Ͱ༻࡞ޓ‬ΔɻୈҰ߲͸ిࢠͷӡಈ߲Ͱ͋Γɺୈೋ߲͸૬‫͋Ͱ߲༻࡞ޓ‬Δɻ

CC

զʑͷ໨ඪ͸ɺ͜ͷΑ͏ʹ૬‫༻࡞ޓ‬Λ‫ؚ‬Μͩ‫ܥ‬ͷඇฏߧμΠφϛΫεΛ‫ه‬ड़͢Δ͜ͱͰ͋Δɻ

ΤωϧΪʔͱ‫ݺ‬͹ΕΔྔͰɺ૬‫ʹ༻࡞ޓ‬ΑΓੜ͡Δࣗ

C͸֤࣌ؒ‫ܦ‬࿏্ͷੵ෼Λද͢ɻ͜͜Ͱ Σ ͸ɺࣗ‫ݾ‬άϦʔϯؔ਺๏ͱ͸ɺ‫ܥ‬ͷ೾ಈؔ਺Λ‫ٻ‬ΊΔͷͰ͸༝ͳ৔߹͔Βͷิਖ਼Λද͢ɻࣗ‫ݾ‬ΤωϧΪʔ͸ɺ࣮͸ͳ͘ɺ͋ΔछͷԠ౴ؔ਺ʹয఺Λ౰ͯɺͦΕΛධՁ͢

G ΍ G0 ͷ൚ؔ਺ͱͯ͠දͤɺFeynman μΠΞάϥϜ

Δํ๏Λࢦ͢ɻ۩ମతʹ͸ɺҰཻࢠάϦʔϯؔ਺Λ༻͍ͯॻ͖Լ͢͜ͱ͕Ͱ͖Δɻ͔͠͠ͳ͕Βɺ͢΂

Gα,β (t, t ) ≡ −iTC ĉα (t)ĉ†β (t )

(6)೉Ͱ͋ΓɺԿ͔͠ΒͷۙࣅΛࢪ͢ඞཁ͕͋Δɻ௨ৗ͸ɺΛධՁ͢Δ͜ͱΛ໨తͱ͢Δɻ͜͜ͰɺTC ͸”࣌ؒ‫ܦ‬†ͯͷμΠΞάϥϜΛ‫ʹີݫ‬औΓࠐΉ͜ͱ͸Ұൠʹ͸ࠔ͋ΔϧʔϧʹैͬͯಛఆͷμΠΞάϥϜͷΈΛऔΓࠐΉ͜ͱͰࣗ‫ݾ‬ΤωϧΪʔΛۙࣅ͢Δ͕ɺҟͳΔμΠΞͰͷ”࣌ؒ‫ܦ‬࿏”(C) ʹԠͨ͡ Heisenberg දࣔͷԋࢉࢠάϥϜ͸෺ཧతʹҟͳΔϓϩηεΛද͢ɻͦͷͨΊɺΛද͢ [38]ɻͨͩ͠ɺ͜ͷ࣌ؒ‫ܦ‬࿏͸໰୊ʹԠͯ͡બͲͷΑ͏ʹࣗ‫ݾ‬ΤωϧΪʔΛධՁ (ۙࣅ) ͢Δ͔͕ɺά͹ΕΔɻҰཻࢠάϦʔϯؔ਺ͷҙຯ͸ɺ؆୯ʹ‫ͱ͏ݴ‬Ϧʔϯؔ਺๏ͷ؊ʹͳΔɻ

࿏”(C) ্Ͱͷ࣌ؒॱংԋࢉࢠΛද͠ɺĉ (t ) ͸࣌ࠁ t࣌ࠁ t ʹঢ়ଶ β ͷిࢠΛೖΕͨ࣌ʹɺ࣌ࠁ t Ͱిࢠ͕3.2 ඇฏߧಈతฏ‫ۉ‬৔ཧ࿦ঢ়ଶ α ʹ͍Δ֬཰ৼ෯Ͱ͋Δɻ‫·ؚ‬Ε͍ͯΔ৘ใྔͱಈ త ฏ ‫ ۉ‬৔ ཧ ࿦ (dynamical mean-ﬁeld theory;ͯ͠͸೾ಈؔ਺ΑΓ΋མͪΔ͕ɺిྲྀɺ࣓ԽɺҰཻࢠεDMFT) ͸ɺάϦʔϯؔ਺๏ͷҰͭͰ͋Γɺ‫ڧ‬૬ؔϖΫτϧͳͲͷҰମͷ෺ཧྔͷධՁʹ͸͜ΕͰे෼Ͱ‫ܥ‬ɺಛʹ Mott సҠɺΛѻ͏‫ͳྗڧ‬ཧ࿦ͱͯ͠ൃలͯ͠

͋Δɻ௨ৗɺ༗‫ݶ‬Թ౓ͷฏߧঢ়ଶΛѻ͏৔߹͸ɺ࣌ؒ‫ܦ‬͖ͨɻͦͷඇฏߧ΁ͷ֦ு൛Ͱ͋Δඇฏߧ DMFT ΋ɺ࿏ (C) ͸‫ʹ্ؒ࣌ڏ‬औΒΕΔ (ਤ 4 (a))ɻ͜ΕΛদ‫ܗݪ‬

Hubbard ໛‫Ͳͳܕ‬ͷ‫ڧ‬૬ؔ‫ʹܥ‬ద༻͞Εɺ͞·͟·ͳࣜͱ‫Ϳݺ‬ɻඇฏߧ໰୊Λѻ͏৔߹͸ɺਤ 4 (b,c) ͷΑ͏ඇฏߧ‫ݱ‬৅Λ໌Β͔ʹ͖ͯͨ͠ɻྫ͑͹ɺMott ઈԑ《講義ノート》

物性研究・電子版 Vol. 10, No. 1, 101215(2022年3月号)
Glat
ij


imp
Glat
[Δ](t, t )ii (t, t ) = G

格子模型

ܾ·Δɻ
Δ
有効不純物問題

DMFT Ͱ͸࣍ͷ৚͕݅ຬͨ͞ΕΔΑ͏ʹࣗ‫ݾ‬ແ಍ணJij

自己無撞着ループtʹܾఆ͢Δɻ͢ͳΘͪɺෆ७෺ʹର͢ΔάϦʔϯؔ਺t(Gimp ≡ −iTC d(t)d† (t )) ͕ɺ֨ࢠ໛‫ܕ‬ͷ‫ॴہ‬άϦʔΔ(t, t )Uਤ5༗ޮෆ७෺໰୊͸ࠞ੒ؔ਺ʹΑΓ‫ن‬ఆ͞ΕΔ͕ɺ

Σlat
ij


imp
Σlat
(t, t )ij (t, t ) = δij Σ†Σimp
Impurity solverಈతฏ‫ۉ‬৔ཧ࿦ (DMFT) ʹ͓͚Δɺ֨ࢠ໛‫ܕ‬ϯؔ਺ (Glatii ≡ −iTC ci (t)ci (t )ɺi ͸֨ࢠ໛‫ܕ‬ͷαΠ

τ) ͱ

imp
Glat
(t, t )ii (t, t ) = Gͱෆ७෺໰୊Λ८Δࣗ‫ݾ‬ແ಍ணϧʔϓͷ֓೦ਤɻࠞ
੒ؔ਺ Δ ʹ͍ͭͯ͸ɺ͜͜Ͱ͸Ұൠʹඇฏߧͷ৔߹(9)ͷΑ͏ʹҰக͠ɺ·ͨɺࣗ‫ݾ‬ΤωϧΪʔʢ૬‫༻࡞ޓ‬ΛೖʹରԠ͢ΔҾ਺ (t, t ) Ͱද͍ࣔͯ͠ΔɻΕͨ৔߹ͷάϦʔϯؔ਺΁ͷิਖ਼ʣ͕֨ࢠ໛‫ͱܕ‬ෆ७ମͷޫΩϟϦΞμΠφϛΫεɺΫΤϯν໰୊ (‫ܥ‬ͷύϥϝʔλΛಥવม͑ͨ࣌ʹԿ͕‫͖ى‬Δ͔ͱ͍͏໰୊)ɺ

Floquet ΤϯδχΞϦϯάͳͲ͕‫͛ڍ‬ΒΕΔɻ͞Βʹɺ෺໰୊ʹ͓͍ͯҰகͯ͠

imp
Σlat
(t, t )ij (t, t ) = δij Σ(10)DMFT ͸ిࢠϑΥϊϯ૬ؔ΍ଟόϯυޮՌ΋औΓࠐΉͱͳΔɺͱ͍͏৚݅Λ՝͢ɻ࣮ࡍʹ DMFT Λ࣮૷͢͜ͱ͕Ͱ͖ɺ࠷ۙͰ͸ɺDMFT Λ֦ு͠ɺඇ‫ॴہ‬૬‫ޓ‬Δ৔߹͸ɺਤ 5 ʹࣔ͢Α͏ʹɺద౰ͳࠞ੒ؔ਺ Δ(t, t )࡞༻΍εΫϦʔχϯάͷޮՌΛऔΓࠐΉͳͲ༷ʑͳํ

͔Βग़ൃ͠ɺෆ७෺໰୊Λղ͖ɺ֨ࢠ໛‫ʹܕ‬໭Γɺෆ޲΁ͷਐలΛΈ͍ͤͯΔɻ

७෺໰୊ʹର͢Δ Dyson ํఔࣜΛհͯ͠ߋ৽͞Εͨ

3.2.1 ‫ج‬ຊతͳΞΠσΟΞΔ(t, t ) Λ༻͍ͯෆ७෺໰୊Λղ͖௚͠ɺ࠶ͼ֨ࢠ໛‫ܕ‬DMFT ͸ɺ஫໨͢Δ֨ࢠ໛‫ܕ‬Λɺ҃ΔαΠτʢෆ७ʹ໭Γɺ͜ͷϧʔϓ͕ऩଋ͢Δ·Ͱஞ࣍୅ೖ͢Δʢͭ෺ͱ‫ݺ‬͹ΕΔʣʹରͯ͠ɺपғͷిࢠ‫ܥ‬Λฏ‫ۉ‬৔తʹ·Γࣗ‫ݾ‬ແ಍ணʹղ͘ʣ͜ͱͰɺDMFT Λղ͘ࣄ͕ग़

‫ͨ͠ۉ‬໛‫( ܕ‬૬‫༻࡞ޓ‬Λ࣋ͭαΠτͱͦΕͱ݁߹͢Δ

དྷΔɻ৚݅ (10) Λ‫ͯݟ‬෼͔ΔΑ͏ʹ DMFT Ͱ͸ࣗ‫ݾ‬ࣗ༝ిࢠ‫ܥ‬ʀෆ७෺໰୊) ʹஔ͖‫͑׵‬ɺࣗ‫ݾ‬ແ಍ணʹిΤωϧΪʔ͸ಉҰαΠτ্ͷΈͰϊϯθϩͰ͋ΔͱԾࢠάϦʔϯؔ਺Λ‫ٻ‬ΊΔํ๏Ͱ͋Δɻ࣮ۭؒͰͷฏ‫ۉ‬ఆ͠ (‫)ࣅۙॴہ‬ɺඇ‫ॴہ‬૬ؔΛແࢹ͢Δɻ৔ۙࣅʹରԠ͢Δ͕ɺ্࣌ؒ࣠ͷྔࢠ༳Β͗͸औΓࠐෆ७෺໰୊͸ɺ̍αΠτ໰୊ͱ͸͍͑ɺଟମ໰୊Ͱ͋

Ήɺͱ͍͏ͷ͕ಈతฏ‫ۉ‬৔ཧ࿦ͷ‫ج‬ຊతͳΞΠσΟΞΔ͔ΒɺԿΒ͔ͷํ๏ (impurity solver) Ͱղ͘ඞཁ͕

Ͱ͋Δɻ͜ͷ֓೦ਤΛਤ 5 ʹࣔ͢ɻDMFT ͸ɺ֨ࢠ໛

͋Δɻฏߧ‫ͱܥ‬ಉ͡Α͏ʹ਺஋తʹ‫ࢠྔͳີݫ‬Ϟϯς‫ܕ‬ͷۭؒ࣍‫͕ݩ‬ແ‫ݶ‬େͷ‫ͳʹີݫͰݶۃ‬Δ͜ͱ͕஌ΒΧϧϩ๏Λ༻͍ͯղ͘͜ͱ΋Ͱ͖Δ͕ɺෛූ߸໰୊ʹΕ͍ͯΔ [42, 43]ɻෆ७෺ʹର͢Δ࡞༻͸ిࢠཋ (ෆ७ΑΓ௕࣌ؒͷγϛϡϨʔγϣϯ͕೉͍͠ɻͦ͜Ͱɺ‫ڧ‬෺αΠτͱ݁߹͢Δࣗ༝ిࢠ‫ )ܥ‬தͷిࢠΛઌʹੵ෼݁߹ల։ (Non-crossing approximation, one-crossing

ͯ͠͠·͏͜ͱʹΑΓɺapproximation, etc.) [44, 45] ΍ऑ݁߹ల։ (iterative 
Simp =

C



C

dtdt



perturbation theory, Migdal theory, etc.) [38, 46, 47]dˆ†σ (t)Δ(t, t )dˆσ (t ) + Ĥloc

σ(8)Λ༻͍ͯෆ७෺໰୊ΛධՁ͢Δ͜ͱ͕ଟ͍ɻ

3.2.2 ࣮‫ͱݧ‬ͷൺֱDMFT ͸ɺ‫ڧ‬૬ؔిࢠ‫ܥ‬Λѻ͏‫ͳྗڧ‬ख๏Ͱ͋Δ

ͱද͞ΕΔɻ͜͜Ͱɺdˆ†σ ͸ෆ७෺αΠτʹεϐϯ σ ͷ͕ɺͦ͜ʹ͸༷ʑͳۙࣅ͕‫·ؚ‬ΕΔɻͦͷͨΊɺ࣮ࡍిࢠΛੜ੒͢ΔԋࢉࢠɺΔ(t, t ) ͸ࠞ੒ؔ਺ (hybridiza-ͷ࣮‫Ͱݧ‬ੜ͡Δ‫ݱ‬৅ΛͲΕ΄Ͳ‫ه‬ड़Ͱ͖Δ͔ͱ͍͏ٙ

tion function) ͱ‫ݺ‬͹Εɺిࢠ͕ෆ७෺αΠτ͔Βిࢠ໰͕༙͍ͯ͘Δɻ͜ͷΑ͏ͳഎ‫͔ܠ‬Βɺۙ೥ɺඇฏߧ

ཋʹඈͼग़͠ɺ·ͨ໭ͬͯ͘Δৼ෯Λද͢ɻ௨ৗͷʢฏDMFT ͱ࣮‫ݧ‬ͷൺֱ΋ਐΜͰ͍Δɻ

ߧʹର͢ΔʣDMFT Ͱ͸ɺࠞ੒ؔ਺͸ Δ(t) ͷΑ͏ʹྫ͑͹ɺྫྷ٫‫ؾࢠݪ‬ମͱͷൺֱʹ͓͍ͯ͸ɺप‫֎ظ‬Ұͭͷ࣌ؒͷΈͷؔ਺Ͱ͋Δ͕ɺඇฏߧʹ͓͍ͯ͸ೋ৔தͷ Hubbard ໛‫ܕ‬ͷμΠφϛΫεͷൺֱ͕ͳ͞Εͭͷ࣌ؒͷؔ਺ͱͳΔɻHloc ͸ෆ७෺αΠτʹ͓͚Δ

ͨ [29]ɻ͜͜Ͱ͸ɺ࣮‫͔ݧ‬ΒܾΊͨύϥϝʔλͷΈΛ૬‫༻࡞ޓ‬Λද͠ɺѻ͏໛‫Ͱܕ‬ͷ̍αΠτʹର͢Δ΋ͷ࢖༻ͯ͠ඇฏߧ DMFT Λ࣮ߦ͍ͯ͠Δɻਤ 6 ͸ɺઌʹ

ͱͳΔɻྫ͑͹ɺHubbard ໛‫Ͱܕ‬͸ Ĥloc = U n̂↑ n̂↓ ͱ࿩ͨ͠ಈత‫ࡏہ‬ͷޮՌΛௐ΂݁ՌΛ͍ࣔͯ͠Δɻ͢ͳ

ͳΔɻ࡞༻Λ‫ٻ‬ΊΔࡍͷ࣌ؒੵ෼ͷ‫ܦ‬࿏ C ͸ɺઌʹड़Θͪɺ‫͍ڧ‬प‫֎ظ‬৔ΛՃ͑Δ͜ͱͰཻࢠ͕‫͠ࡏہ‬ɺͦΕ΂ͨΑ͏ʹ໰୊ (ฏߧ͔ͲͷΑ͏ͳඇฏߧ͔) ʹԠͯ͡ʹ൐͍ɺ‫ॴہ‬૬‫ ༻࡞ޓ‬U ͷޮՌ͕૬ରతʹ‫·ڧ‬Γɺμ《講義ノート》

(a) Experiment

物性研究・電子版 Vol. 10, No. 1, 101215(2022年3月号)

ά΍༳Β͗΋มԽ͠ɺඇฏߧμΠφϛΫεʹӨ‫ڹ‬Λ‫ٴ‬

Theory:DMFT΅͢ɻ͜ͷΑ͏ͳ‫ݱ‬৅Λ‫ه‬ड़͢ΔͨΊʹ͸ɺDMFT Λ௒͑ͨ࿮૊Έ͕ඞཁͱͳΔɻ

DMFT Λ֦ுͯ͠ඇ‫ॴہ‬૬ؔΛऔΓࠐΉํ๏͸ɺେ
impurity

potential(a)

lattice(b)͖͘෼͚ͯ̎ͭ͋ΔɻҰͭͷํ޲͸ɺ̍αΠτ DMFT
ΛΫϥελʔʹ֦ுͯ͠୹‫཭ڑ‬૬ؔΛঃʑʹऔΓࠐΉ΋ͷͰɺ࣮ۭؒͰΫϥελʔΛߟ͑Δ৔߹ͷ cellular

DMFT [49, 50] ͱɺӡಈྔ্ۭؒͷϝογϡ఺্ͷӡಈྔΛߟ͑Δ dynamical cluster approximation [51]͕͋Δɻඇฏߧʹ͓͍ͯ΋ɺ͢Ͱʹ࣮૷͕ߦΘΕɺεϐϯ༳Β͗ͱྭ‫ిى‬ՙͷࢄཚͳͲ͕ௐ΂ΒΕ͍ͯΔ
ਤ6प‫֎ظ‬৔தͷ 3 ࣍‫ ݩ‬Hubbard ໛‫ܕ‬ͷྫྷ٫‫ݪ‬[52, 53]ɻ΋͏Ұͭͷํ޲͸ɺFeynman μΠΞάϥϜʹ

ࢠ‫ؾ‬ମͷ࣮‫ͱݧ‬ཧ࿦ͷൺֱɻ(a) ࣮‫ͱݧ‬ཧ࿦ͷ֓೦͓͍ͯඇ‫ॴہ‬తͳ૬ؔΛද߲͢Λ෇͚Ճ͍͑ͯ͘ํ๏

ਤɻ(b) प‫֎ظ‬৔ (प೾਺ Ω) தͷμϒϩϯ਺ͷมԽɻ
tramp ͷؒ֎৔Λ͔͚ɺμϒϩϯΛଌఆ͍ͯ͠Δɻoﬀresonant ͸ Ω > U ͷ৔߹ɺresonant ͸ Ω = U ͷ৔߹ɺͦͯ͠ɺundriven ͸ Floqeut ϋϛϧτχΞϯ

Ͱ͋ΔɻͲͷΑ͏ͳ߲ΛՃ͑Δ͔Ͱɺٞ࿦Ͱ͖Δ෺ཧ͕มΘͬͯ͘Δɻฏߧ‫Ͱܥ‬͸ɺ༷ʑͳ֦ு๏͕ఏҊ͞Ε࣮૷͞Ε͍ͯΔ͕ɺඇฏߧͷจ຺Ͱ͸ GW + ֦ுΛ༻͍ͨ࣌ؒൃలΛ͍ࣔͯ͠Δɻ·ͨɺ˔ͷσʔλ͸DMFT ͱ‫ݺ‬͹ΕΔख๏͕࣮૷͞Ε͍ͯΔ [54, 55]ɻ؆

ྭ‫͠ͳى‬ͷ৔߹Λද͢ɻจ‫[ ݙ‬29] ΑΓɻ୯ʹ‫͑ݴ‬͹ɺ͜ͷख๏͸ɺDMFT ͱεΫϦʔχϯάޮՌΛΑ͘‫ه‬ड़͢Δ GW ๏Λ Feynman μΠΞάϥϜϨ

ϒϩϯີ౓͕খ͘͞ͳΔɻͨͩ͠ɺμϒϩϯͱ͸̎ͭϕϧͰ૊Έ߹ΘͤΔํ๏Ͱ͋Δɻ͜ΕʹΑΓɺMott ઈͷཻࢠ͕ಉαΠτʹ͋Δঢ়ଶΛද͕͢ɺ͜ͷঢ়ଶ͸‫ہ‬ԑମ΍ిՙҠಈઈԑମʹ͓͚ΔɺޫΩϟϦΞͷಛҟͳॴ૬‫༻࡞ޓ‬ͷͨΊΤωϧΪʔతʹෆརͰ͋Δɻಈత‫ہ‬؇࿨‫ݱ‬৅ [54] ΍εΫϦʔχϯά΍ࢄཚաఔͷมԽͷΑࡏ͕‫ظ‬଴͞ΕΔඇ‫ڞ‬໐ྭ‫Ͱى‬͸ɺμϒϩϯີ౓ͷมԽΔεϖΫτϧ‫ܗ‬ঢ়ͷมԽ [55] ͕໌Β͔ʹͳ͍ͬͯΔɻྔ΍มԽͷ࣌ؒεέʔϧ (∼ 1 ms) ͕ɺ࣮‫ ͱݧ‬DMFT

3.3 ඇฏߧάϦʔϯؔ਺๏ͷ࠷ۙͷൃలͷؒͰɺϑΟοςΟϯάύϥϝʔλͳ͠ʹɺϦʔζφඇฏߧάϦʔϯؔ਺๏Λ࣮૷͢Δ্Ͱͷओͳ཯଎͸ɺϒϧͳҰகΛࣔ͢͜ͱ͕Θ͔Δɻྭ‫ى‬प೾਺ Ω ͕૬‫ޓ‬άϦʔϯؔ਺ͷอଘͱ Dyson ํఔࣜͷධՁʹ͋Δɻͨ

࡞༻ U ͱҰக͢Δ‫ڞ‬໐ྭ‫Ͱى‬͸ɺμϒϩϯ͕௚઀ྭ‫ى‬ͱ͑͹҆௚ʹߟ͑ΔͱάϦʔϯؔ਺ͷอଘʹඞཁͳϝ͞ΕΔͷͰಈత‫ࡏہ‬ͷޮՌ͸‫ͳ͘ͳ͑ݟ‬Δ͕ɺͦͷ༷ϞϦ͸ O(Nt2 ) ͰɺDyson ํఔࣜͷධՁ͸ O(Nt3 ) ࣌ࢠ΋࠶‫͍͖ͯͰݱ‬Δɻ͜ͷ݁Ռ͸ɺྫྷ٫‫ؾࢠݪ‬ମͷඇؒඞཁʹͳΔɻ͜͜Ͱ Nt ͸‫ؒ࣌͏࢖Ͱࢉܭ‬ͷ཭ࢄάฏߧ࣮‫ݧ‬ͷਫ਼౓ͱ DMFT ͷ৴པੑΛ‫୲ʹ͍ޓ‬อͯ͠Ϧουͷ਺Ͱ͋ΔɻͦͷͨΊɺ௕࣌ؒͷγϛϡϨʔγϣ

͍Δɻϯ͸ࠔ೉Ͱ͋Δ (࣮ࡍʹ͸ߴʑ 100fs ͙Β͍·Ͱ)ɻ͜

·ͨɺ࣮ࡍͷ෺࣭ͷޫ༠‫ى‬μΠφϛΫεͱͷൺֱ΋ͳ͞Ε͍ͯΔɻ1T -TaS2 ͷྫͰ͸ɺޫྭ‫ى‬ΩϟϦΞͷ

ͷ‫ࢉܭ‬ίετͷεέʔϦϯάΛվળ͠Α͏ͱ͍͏ࢼΈ͕ଓ͚ΒΕ͍ͯΔɻ

μΠφϛΫεͷৼΔ෣͍͕ɺDMFT ʹΑΓΑ͘આ໌Ͱྫ͑͹ɺάϦʔϯؔ਺ͷ‫͋ʹܗ‬ΔछͷԾఆ (Gener-͖Δ͜ͱ͕ใࠂ͞Ε͍ͯΔ [48]ɻಛʹɺMott ঢ়ଶۙ๣alized Kadanoﬀ-Baym Ansatz: GKBA) Λઃ͚Δ͜

Ͱ͸ɺฏߧঢ়ଶʹ͓͚Δυʔϓྔ͕ޫΩϟϦΞͷμΠ

ͱͰɺඞཁϝϞϦ͸ O(Nt )ɺDyson ํఔࣜͷධՁ͸φϛΫεʹେ͖͘Ө‫͢ڹ‬Δ͜ͱ͕ࣔ͞Εͨɻ

O(Nt2 ) ∼ O(Nt ) ʹͳΔͱใࠂ͞Ε͍ͯΔ [56, 57]ɻ3.2.3

GKBA ͸‫͘ݹ‬ΑΓଘࡏ͕ͨ͠ɺۙ೥Ͱ΋৽͍࣮͠૷DMFT Λ௒͑ͯDMFT ͸‫ॴہ‬తͳ૬‫͔༻࡞ޓ‬Βੜ͡Δ‫ॴہ‬తͳ૬ؔ๏ [57, 58] ΍ిࢠ֨ࢠ‫΁ܥ‬ͷ֦ு [59] ͕ߦΘΕɺ։ൃͷΈΛऔΓࠐΉ͜ͱ͕Ͱ͖Δ͕ɺ௕‫཭ڑ‬૬‫͔༻࡞ޓ‬Β͕ଓ͍͍ͯΔɻͨͩ͠ɺ͜ͷख๏͸ిࢠ૬͕ؔൺֱత

ੜ͡ΔޮՌ΍ඇ‫ॴہ‬తͳ૬ؔ͸ѻ͑ͳ͍ɻ͔͠͠ɺ‫ݱ‬ऑ͍৔߹ʹͷΈ༗ޮͰ͋Δ͜ͱ͕஌ΒΕ͓ͯΓɺ‫ڧ‬૬࣮తʹ͸‫ॴہ‬૬‫͔༻࡞ޓ‬Β΋࣓‫ؾ‬సҠ෇ۙͰͷεϐϯؔ‫΁ܥ‬ͷద༻͸ඇࣗ໌ͳ໰୊Ͱ͋Δɻ༳Β͗ͷΑ͏ͳඇ‫ॴہ‬తͳ૬ؔ΋ੜ͡Δ͠ɺ௕‫཭ڑ‬૬ҰํɺάϦʔϯؔ਺ G(t, t ) ΍ࣗ‫ݾ‬ΤωϧΪʔ Σ(t, t )‫͔༻࡞ޓ‬Βੜ͡ΔϓϥζϞϯͳͲͷूஂྭ‫ى‬΍ɺ૬‫ޓ‬ΛΑ͘ோΊΔͱ |t − t | ͕େ͖͘ͳΔͱ஋ࣗମ͕খ͞

࡞༻ͷεΫϦʔχϯάͷޮՌ΋͋Δɻޫྭ‫ʹى‬Αͬͯɺ

͘ͳΔ৔߹͕ଟ͍͜ͱ͕Θ͔Δɻ͜ͷੑ࣭Λ༻͍ͯɺฏߧͱ͸ҟͳΔిՙ෼෍͕ੜ͡ΔͨΊɺεΫϦʔχϯ

《講義ノート》

Dyson ํఔࣜͷੵ෼ൣғΛର֯෇ۙ (t  t ) ʹ‫ݶ‬ఆ͢
Δํ๏΋ఏҊ͞Ε͓ͯΓɺNc ΛͦͷΧοτΦϑͷେ͖

物性研究・電子版 Vol. 10, No. 1, 101215(2022年3月号)

4 ࠷ۙͷ‫͔ڀݚ‬Β

͞ͱ͢Δͱ‫ࢉܭ‬ίετ͸ϝϞϦ͕ O(Nc Nt )ɺ‫ؒ࣌ࢉܭ‬͜ͷষͰ͸ɺʮ‫ݟ‬Δʯͦͯ͠ʮૢΔʯͱ͍͏‫͔఺؍‬͕ O(Nt2 Nc ) ͱͳΔ [60]ɻ͜ͷํ๏͸ɺDMFT ʹ͢ͰΒɺͦΕͧΕ࠷ۙͷ‫ڀݚ‬Λ঺հ͢Δɻʮ‫ݟ‬Δʯͱ͍͏‫؍‬ʹద༻͞Εɺ‫ࢉܭ‬Մೳ͕࣌ؒେ෯ʹ૿͑Δ͜ͱ͕ใࠂ͞

఺͔Β͸ɺ‫ڧ‬૬ؔిࢠ‫͚͓ʹܥ‬Δߴ࣍ߴௐ೾ൃੜͷཧΕ͍ͯΔɻ·ͨɺԠ༻਺ֶͷ஌ࣝΛ༻͍ͯඞཁϝϞϦ͕࿦‫ڀݚ‬Λ঺հ͢Δɻ·ͨɺʮૢΔʯͱ͍͏‫͔఺؍‬Β͸ɺ

O(Nt ln Nt ) ͰɺDyson ํఔࣜͷධՁ͸ O(Nt2 ln Nt )Prethermal ͳঢ়ଶʹ͓͚Δடং૬ͷग़‫ݱ‬ͷՄೳੑɺ͞

·ͰԼ͛ΒΕΔͱ͍͏͜ͱ΋࠷ۙใࠂ͞Εͨ [61]ɻ͜

Βʹ͸؇࿨աఔʹ͓͚ΔϘτϧωοΫʹ༝དྷ͢Δடং

ΕΒͷ։ൃʹΑΓɺ୯७ͳ Hubbard ໛‫ܕ‬Λ௒͑ͨΑΓ૬ग़‫ݱ‬ͷՄೳੑʹؔ͢Δ‫ڀݚ‬Λ঺հ͢Δɻ‫࣮ݱ‬తͳ໛‫ܕ‬ͷղੳ΁ͷ‫ػ‬ӡ͕ߴ·͍ͬͯΔɻ

4.1 ‫ڧ‬૬ؔిࢠ‫͚͓ʹܥ‬Δߴ࣍ߴௐ೾ൃੜ

͞ΒʹɺDyson ํఔࣜͷ਺஋ղ๏ͳͲඇฏߧάϦʔϯઌʹ΋ड़΂ͨΑ͏ʹۙ೥ɺ‫ݻ‬ମʹ͓͚Δߴ࣍ߴௐ೾

ؔ਺๏ͷ֩෦෼Λఏ‫͢ڙ‬ΔΦʔϓϯϥΠϒϥϦʔ Nessi

ൃੜ (HHG) ͕‫؍‬ଌ͞Ε࿩୊Λ‫ݺ‬ΜͰ͍Δɻ‫ݻ‬ମதͷ

(Non-Equilibrium Systems Simulation Library) ΋ެ

HHG ͸ɺ·ͣ൒ಋମʹ͓͍ͯ‫؍‬ଌ͞Ε [12]ɺϨʔβʔ։͞Ε [62]ɺඇฏߧάϦʔϯؔ਺๏ͷ։ൃ‫͕ڥ؀‬੔͍ͷप೾਺ม‫׵‬๏ͱͯ͠ͷԠ༻ͷՄೳੑ΋͞Δ͜ͱͳ͕ͭͭ͋ΔɻΒɺϕϦʔ‫཰ۂ‬΍ΤωϧΪʔ෼ࢄͳͲͷόϯυߏ଄‫؍‬

3.4 ͦͷଞͷख๏ଌ΁ͷԠ༻΋‫ظ‬଴͞Ε͍ͯΔ [68, 69]ɻ͞Βʹ࠷ۙͰඇฏߧάϦʔϯؔ਺๏Ͱ͸ɺ௨ৗࣗ‫ݾ‬ΤωϧΪʔ Σ͸ɺάϥϑΣϯͳͲͷ൒ۚଐ [70]ɺप‫ظ‬ੑΛ࣋ͨͳ͍ʹԿ͔͠ΒͷۙࣅΛߦ͏ɻۙࣅʹґଘͯ͠ҟͳΔࢄཚ

ΞϞϧϑΝε‫[ ܥ‬69] ͞Βʹ͸ӷମ [71] Ͱ΋ HHG ͕΍༳Β͗ͷޮՌ͕ೖΔ͕ɺඞͣ͠΋શͯͷॏཁͳޮՌ

‫؍‬ଌ͞Εɺ‫ٸ‬଎ͳ޿͕ΓΛ‫͍ͯͤݟ‬ΔɻͦͷΑ͏ͳதΛऔΓೖΕΒΕ͍ͯΔͱ͸‫ݶ‬Βͳ͍ɻ·ͨɺҰ࣍‫ݩ‬ͷͰ΋ɺ‫ڧ‬૬ؔిࢠ‫ܥ‬͸ HHG Λ‫͢ڀݚ‬Δ෣୆ͱͯ͠‫ڵ‬໰୊ͳͲɺͦ΋ͦ΋ඇฏߧάϦʔϯؔ਺๏Ͱ͸ѻ͍ʹຯਂ͍ɻྫ͑͹ɺҰ࣍‫ݩ‬ͷ‫ڧ‬૬ؔిࢠ‫Ͱܥ‬͸ɺ‫ ͍ڧ‬3͍͘‫ܥ‬΋͋Δɻ͜ΕΒͷ໰୊Λѻ͏ʹ͸ଞͷख๏ͷྗ࣍ͷඇઢ‫ܗ‬ੑ͕ใࠂ͞Ε͍ͯΔ [72]ɻ͞Βʹɺ൒ಋମΛआΓΔඞཁ͕ग़ͯ͘ΔɻͰͷిՙͷ୲͍ख͸ిࢠ΍ϗʔϧͱ͍ͬͨ௨ৗͷཻࢠ͠͹͠͹࢖ΘΕΔख๏ͱͯ͠͸ɺ‫ີݫ‬ର֯Խ [63]ɺ̍

Ͱ͕͋ͬͨɺ‫ڧ‬૬ؔిࢠ‫Ͱܥ‬͸ɺHubbard ໛‫Ͱܕ‬͸μ࣍‫ʹݩ‬ಛԽͨ͠ DMRG ΍ iTEBD ͳͲͷख๏ [64,65]ɺ

ϒϩϯ (̍αΠτʹిࢠ͕ೋ‫͋ݸ‬Δঢ়ଶ) ΍ϗϩϯ (̍࣌ؒґଘม෼๏ [66, 67]ɺୈҰ‫ݪ‬ཧతͳ࣌ؒґଘີ౓൚αΠτʹిࢠ͕ͳ͍ঢ়ଶ)ɺμΠϚʔ Mott ‫Ͱܥ‬͸Ωϯؔ਺๏΍։์‫ܥ‬Λѻ͏ Lindblad ‫͛ڍ͕Ͳͳࣜܗ‬ΒΕΫ΍ΞϯνΩϯΫͱ͍ͬͨ [73]ɺ௨ৗͷཻࢠͰ͸ͳ͍Δɻ͜ΕΒͷख๏ʹ͸ͲΕ΋Ұ௕Ұ୹͕͋Γɺฏߧͷࣗ༝౓͕ిՙҠಈΛ୲͏ɻ͜ͷΑ͏ͳࣄ৘͔Βɺ‫ڧ‬૬৔߹ͱಉ༷ɺଞͷख๏ͱ૬ิతͰ͋Δɻؔిࢠ‫Ͱܥ‬͸ͲͷΑ͏ͳ HHG ͕ൃੜ͢Δͷ͔ɺͦͷྫ ͑ ͹ ɺ‫ ີ ݫ‬ର ֯ Խ ͸ จ ࣈ ௨ Γ ɺ೾ ಈ ؔ ਺ Λ‫ݯى‬͸Կ͔ɺͦ͜ʹ͸ͲͷΑ͏ͳ৘ใ͕٧·͍ͬͯͯ

Schrödinger ํఔ͔ࣜΒ‫ٻʹີݫ‬ΊΔํ๏Ͱ͋Γɺ‫ີݫ‬

HHG Λ‫؍‬ଌ๏ͱͯ͠࢖͑Δͷ͔ɺ൒ಋମͱ͸Ͳ͏ҟͳͰ͋Γ৘ใྔ͸ଟ͘ɺෳࡶͳ૬‫༻࡞ޓ‬΋ѻ͏͜ͱ͕Ͱ

Δͷ͔ɺͱ͍༷ͬͨʑͳ‫ڵ‬ຯਂ͍໰୊͕༙͍ͯ͘Δɻ

͖Δɻ͔͠͠ɺ‫ܥ‬ͷେ͖͞ʹରͯ͠೾ಈؔ਺ͷ࣍‫͕ݩ‬͜͜Ͱ͸ɺ‫ڧ‬૬ؔిࢠ‫ܥ‬Λ‫ه‬ड़͢Δ࠷΋‫ج‬ຊతͳ໛ࢦ਺తʹେ͖͘ͳΓ͋·Γେ͖ͳ‫( ܥ‬Hubbard ໛‫Ͱܕ‬‫͋Ͱܕ‬Δ̍όϯυ Hubbard ໛‫ʹܕ‬஫໨͠ɺແ‫ݩ࣍ݶ‬20 αΠτఔ౓) ͸ѻ͑ͳ͍ɻ·ͨɺiTEBD ͸Ұ࣍‫ʹݩ‬ͱ̍࣍‫ܥݩ‬ͷ HHG Λ୳͍ͬͯ͘ [45, 74, 75]ɻ·ͣ͸ಛԽ͠ɺແ‫ܥݶ‬Λਫ਼౓Α͘ѻ͑Δ͕ɺྭ‫ޙى‬ͷ࣌ؒൃແ‫ݩ࣍ݶ‬Λߟ͑Δɻཱํ֨ࢠͷߴ࣍‫ݩ‬൛Ͱ͋Δ hyper-లΛ‫ه‬ड़͢Δࡍʹඞཁͳ‫ࢉܭ‬ίετ͕࣌ؒʹରͯ͠ࢦ

cubic ֨ࢠΛߟ͑ɺDMFT Λద༻͢Δ [45]ɻͨͩ͠ɺ

਺తʹ૿Ճ͢Δ͜ͱ͕஌ΒΕ͍ͯΔɻ‫ࢉܭ‬ਫ਼౓ͷߴ͍

HHG ͸प‫֎ظ‬৔ʹΑͬͯ‫ۦ͕ܥ‬ಈ͞Εɺप‫ظ‬తʹৼΔ‫ࢉܭ‬ख๏͸ɺҰൠʹͲ͔͜ʹࢦ਺తͳࠔ೉͕ೖ͍ͬͯ

෣͏͜ͱͰൃੜ͢Δɻ͜ͷ෦෼Λଊ͑ΔͨΊɺप‫ظ‬తͳ

Δɻ࣌ؒґଘม෼๏͸ɺ೾ಈؔ਺ͷ‫ܗ‬ΛԾఆͯͦ͠ͷඇฏߧঢ়ଶΛѻ͏ Floquet DMFT Λ༻͍Δ [38, 45]ɻதͰ࣌ؒൃలΛ͢Δ΋ͷͰ͋Γɺ‫ີݫ‬ର֯ԽΑΓେ͖࣮‫ͱݧ‬ಉ༷ʹΪϟοϓʹൺ΂े෼খ͍͞प೾਺ͷ֎৔ͳ‫ܥ‬Λѻ͑ɺ࠷ۙͰ͸χϡʔϥϧωοτϫʔΫΛ༻͍ΛՃ͑Δɻਤ 7(a)(b) ʹ HHG ͷ‫౓ڧ‬෼෍Λࣔ͢ɻͨ͜ख๏ [67] ΋։ൃ͞Ε͍ͯΔɻ

͜Ͱ͸ɺ֎৔ͱͯ͠ి৔ E(t) = E0 sin(Ωt) Λ͔͚ͯ
͍Δɻ֎৔͕ϗοϐϯάΤωϧΪʔ v ʹൺ΂ͯখ͍͞৔߹ͱେ͖͍৔߹ͰఆੑతͳৼΔ෣͍͕ҟͳΔ͜ͱ͕

《講義ノート》

物性研究・電子版 Vol. 10, No. 1, 101215(2022年3月号)(a) E0  v

Aloc = −ImGR
locਤ8(a) ࣌ؒฏ‫͞ۉ‬Εͨ local ͳҰཻࢠεϖΫτϧ

 A(k , ω) Ihh(ω)


(b) E0  v(c)nΩ = U + E0nΩ = U − E0nΩ = U + 2E0ͷྭ‫౓ڧى‬ґଘੑɻE0 = 0 Ͱ͸ɺUpper Hubbard
όϯυͱ Lower Hubbard όϯυ͕‫ݟ‬ΒΕΔ͕ɺE0

E0 


͕‫ͳ͘ڧ‬ΔͱαΠυόϯυ (WS ladder) ͕‫ݱ‬ΕΔɻ(b) ฏߧঢ়ଶʹ͓͚Δ෼ࢄԽؔ܎ɻͨͩ͠ɺDMFT
ͷࣄ৘͔Βԣ࣠͸ӡಈྔͰ͸ͳࣗ͘༝ిࢠͷΤωϧΪʔͱͳ͍ͬͯΔɻ·ͨɺന͍఺ઢ͸൒ಋମ໛‫( ܕ‬΋͘͠͸ Hubbar I ۙࣅ) ͷ෼ࢄؔ܎ɻ‫ࢉܭ‬৚݅ͱҾ༻͸ਤ 7 ͱಉ͡ɻ


ਤ 7

(a)(b)Floquet DMFT ʹΑΔ HHG ͷ‫౓ڧ‬෼෍ɻ˜ ͸ΧοτΦϑप೾਺Λࣔ͢ɻ(c)Floquet
DMFT ʹΑΔ HHG ͷ‫౓ڧ‬෼෍ͷ·ͱΊɻ˓͸ΧοτΦϑप೾਺Λࣔ͠ɺ௚ઢ͸ΧοτΦϑप೾਺ͷ E0ґଘੑΛࣔ͢ɻͨͩ͠ɺ͜͜Ͱ U = 8.0ɺࣗ༝ిࢠͷঢ়ଶີ౓͸ D() = √1πv exp(−2 /v 2 )ɺྭ‫ى‬प೾਺Ω = 0.5 Λ༻͍͍ͯΔɻจ‫[ ݙ‬45] ΑΓɻͦͷ౴͑͸ɺ֎৔ʹґଘͨ͠μϒϩϯͱϗϩϯͷӡಈͷҧ͍ʹ͋Δɻ֎৔͕ϗοϐϯάΤωϧΪʔ v ʹൺ΂ऑ͍৔߹͸ɺμϒϩϯͱϗϩϯ͸αΠτؒΛࣗ༝ʹҠಈͰ͖֎৔ʹΑΔӡಈΤωϧΪʔΛ஝͑Δ͜ͱ͕Ͱ͖ɺ
ͦΕΛରফ໓ͷ࣌ʹు͖ग़͢ͱߟ͑ΒΕΔɻҰํɺ֎৔͕‫ͱ͍ڧ‬μϒϩϯͱϗϩϯͷαΠτؒͷҠಈ͸֎৔ͷϙςϯγϟϧΤωϧΪʔʹΑΓ્֐͞ΕΔɻͦͷͨΊɺμϒϩϯͱϗϩϯ͕͋Δ֨ࢠִؒ na Λ͚͋ͯ‫ࡏہ‬͢Δ (a ͸αΠτؒ‫)཭ڑ‬ɻ͜ͷϖΞ͕ରফ໓Λ‫͢͜ى‬Θ͔Δɻখ͍࣌͞͸ɺ௨ৗͷΨε‫ܥ‬ͷΑ͏ʹɺ̍ͭͷͱ U + nE0 a ͷΤωϧΪʔ͕์ग़͞ΕΔɻ࣮ࡍʹɺ֎ϓϥτʔͷΈ͕ੜ͡Δͷʹର͠ɺ֎৔͕‫ͱ͍ڧ‬ෳ਺ͷ৔͕‫͍ڧ‬৔߹ͷΧοτΦϑप೾਺͸ U + nE0 a ͰมԽϓϥτʔ͕ੜ͍ͯ͡Δ͜ͱ͕Θ͔Δɻ͜ͷϓϥτʔͷ

͍ͯ͠Δ͜ͱ͕Θ͔Δɻ·ͨɺిՙͷ‫ࡏہ‬ੑ͸ local ͳऴΘΓ (ΧοτΦϑप೾਺) ͱ‫౓ڧ‬Λਤ 7(c) ʹ·ͱΊҰཻࢠεϖΫτϧΛΈΔ͜ͱͰ΋Θ͔Γɺ֎৔͕‫͍ڧ‬Δɻ͍ͣΕͷྖҬʹ͓͍ͯ΋ΧοτΦϑप೾਺͸֎৔ྖҬͰ͸ Wannier Stark(WS) ladder ͱ‫ݴ‬ΘΕΔ‫ࡏہ‬ʹରͯ͠ઢ‫͍ͯ͑૿ʹܗ‬Δ͜ͱ͕Θ͔ΔɻΨε‫Ͱܥ‬͸ੑΛද͢ߏ଄͕‫ݱ‬ΕΔ͜ͱ͕Θ͔Δ (ਤ 8(a))ɻ

HHG ͸ϙϯσϩϞʔςΟϒΤωϧΪʔΛ൓ө͓ͯ͠Ҏ্ͷ݁Ռ͸ɺMott ઈԑମʹ͓͍ͯ΋൒ಋମͷࣅ2ΓɺΧοτΦϑप೾਺͸ E ͰৼΔ෣͏͜ͱ͕஌ΒΕͨ‫ Ͱݯى‬HHG ͕‫͖ى‬Δ͜ͱΛ͍ࣔࠦͯ͠Δɻ͔͠͠ɺ

͍ͯΔɻҰํɺ൒ಋମͰ͸ΧοτΦϑप೾਺͕ E Ͱৼిՙͷ୲͍ख͕μϒϩϯ΍ϗϩϯͱ͍͏௨ৗͷཻࢠͰ

Δ෣͏͜ͱ͕ใࠂ͞Ε͓ͯΓ [12, 14]ɺMott ઈԑମͷ͸ͳ͍͜ͱʹΑΓɺHHG ͱҰཻࢠεϖΫτϧͷؔ܎ʹ৔߹΋൒ಋମͱಉ༷ͷ E ґଘੑ͕͋Δ͜ͱ͕͜ͷ‫ࢉܭ‬େ͖ͳҧ͍͕ੜ·ΕΔɻ൒ಋମͷ৔߹͸ిՙͷಈ͖͸͔ΒΘ͔Δɻ

ҰཻࢠεϖΫτϧͷ৘ใͰ‫ه‬ड़Ͱ͖͕ͨɺMott ઈԑମͰ͸ɺHHG ͷ‫ݯى‬͸ͳΜͰ͋Ζ͏͔ʁ Hubbard ໛Ͱ͸ͦ͏Ͱͳ͍ɻDMFT ʹΑΔ Mott ઈԑମͷҰཻࢠ‫Ͱܕ‬ද͞ΕΔ Mott ઈԑମͷ৔߹ɺిྲྀ͸μϒϩϯ΍

εϖΫτϧ͸ਤ 8(b) ͷΑ͏ʹͳΓɺUpper Hubbard

ϗϩϯ͕୲͍ͬͯΔɻ͜ͷͱ͖ɺ͜ΕΒͷ”ཻࢠ”͕୯

band ͱ Lower Hubbard band ͕ฏߦʹͳΔɻͦ͜Ͱɺ

ʹҠಈ͢ΔͨΊʹྲྀΕΔిྲྀ (jhop ) ͱিಥΛͯ͠ରফμϒϩϯ΍ϗϩϯΛ௨ৗͷཻࢠͱΈͳͯ͠ɺಉ༷ͷ෼໓Λ‫ͨ͢͜ى‬ΊʹྲྀΕΔిྲྀ (jrc ) ͷೋछྨ͕ߟ͑ΒΕࢄؔ܎Λ࣋ͭ൒ಋମ໛‫ܕ‬Λ࡞Δ͜ͱ͕Ͱ͖ΔɻHHG

Δɻ͜Ε͸ͪΐ͏Ͳɺ൒ಋମͷ৔߹ͷόϯυ಺ిྲྀͱΛ‫͢ࢉܭ‬Δͱਤ 9 ͷΑ͏ʹͳΓɺHHG ͷ‫౓ڧ‬෼෍͸όϯυؒిྲྀʹରԠ͍ͯ͠Δɻ͜ΕΒΛ෼͚ͯධՁ͠

େ෦෼Ͱऑ·ΓɺͦͷεϖΫτϧ͸ Mott ઈԑମͱ͸ͯΈΔͱ jrc ͔Βͷ‫د‬༩͕ࢧ഑తͰ͋ΔͱΘ͔Δɻͭ

ҟͳΔ΋ͷʹͳΔɻҰํɺಉ͡൒ಋମ໛‫ͬ੾͍ࢥͰܕ‬·Γ Mott ઈԑମͷ HHG ͸μϒϩϯͱϗϩϯͷରফͯ෼ࢄؔ܎Λ௚઀Ϊϟοϓ‫ʹܕ‬ม͑ΔͱɺҰཻࢠεϖ໓ʹ༝དྷ͢Δͱ͍͏͜ͱͰ͋ΔɻͰ͸ɺઌʹࣔ͞ΕͨΫτϧ͸߹Θͳ͍͕ɺMott ઈԑମͷ HHG εϖΫτϧ

ೋͭͷҟͳΔ HHG ͷৼΔ෣͍͸Կʹ༝དྷ͢Δͷ͔ʁΛఆੑతʹ࠶‫͢ݱ‬Δ͜ͱ͕Θ͔Δ (ৄࡉ͸࿦จ [45] ࢀ《講義ノート》

物性研究・電子版 Vol. 10, No. 1, 101215(2022年3月号)
   



ਤ9

(a) ൒ಋମ໛‫ ͱܕ‬Hubbard ໛‫ܕ‬ͷ HHG ‫౓ڧ‬෼෍ͷൺֱͷҰྫɻ·ͨɺHubbard I ۙࣅͷ݁Ռ΋৐͍ͤͯΔɻ(b) ൒ಋମ໛‫ܕ‬ͷ HHG ͷ‫౓ڧ‬෼෍·ͱΊɻͨͩ͠ɺ൒ಋମͷ෼ࢄؔ܎͸ਤ 8 ʹࣔ͞ΕΔΑ͏ʹؒ઀Ϊϟοϓ‫͍ͯͬͳʹܕ‬Δɻ‫ࢉܭ‬৚݅ͱҾ༻͸
ਤ 7 ͱಉ͡ɻ

ਤ 10

Ұ࣍‫ ݩ‬Hubbard ໛‫͚͓ʹܕ‬ΔαϒαΠΫϧ

εϖΫτϧɻ੺ઢ͸ɺBethe ԾઃʹΑΔμϒϩϯɾϗϩϯ෼ࢄΛ༻͍ͨ 3 εςοϓ໛‫ܕ‬ͷ݁Ռɻจ‫[ ݙ‬75]ΑΓɻর)ɻ͜ͷ‫ݱ‬৅ʹؔ͢Δਖ਼֬ͳཧղ͸·ͩಘΒΕ͍ͯͳ
͍͕ɺMott ઈԑମͰ͸ HHG ͸ҰཻࢠεϖΫτϧͰ͸ͳ͘௚઀ଟମͷྭ‫ߏى‬଄Λ൓ө͍ͯ͠Δ͜ͱͷ‫ݱ‬Εͩͱߟ͍͑ͯΔɻ࣮ࡍʹಉ༷ͷ‫ݱ‬৅͕࣍ʹ঺հ͢Δ̍࣍
‫͍͓ͯʹܥݩ‬΋‫ݟ‬ΒΕΔɻ
ແ‫ݩ࣍ݶ‬ͷ໰୊͸ DMFT Λ༻͍ͯௐ΂Δ͜ͱ͕Ͱ͖Δ͕ɺଟମͷྭ‫ߏى‬଄ͷৄࡉ৘ใΛಘΔ͜ͱ͕೉͠
͍ɻͦ͜Ͱɺ࣍ʹ͜ͷ৘ใ͕Α͘Θ͔Δ̍࣍‫ʹܥݩ‬য఺Λ౰ͯɺHHG ͱྭ‫ߏى‬଄ͷؔ܎Λߟ࡯͍ͯ͘͠ [75]ɻ
͜ͷղੳͰ͸ iTEBD Λ༻͍ɺΨ΢εύϧεΛরࣹͨ࣌͠ͷ࣮࣌ؒγϛϡϨʔγϣϯΛߦ͏ɻ͜ͷ݁ՌɺDMFT ͷ৔߹ͱಉ༷ʹɺHHG ͸μϒϩϯͱϗϩϯͷରফ໓ʹ༝དྷ͠ɺΧοτΦϑप೾਺΋֎৔ʹઢ‫ʹܗ‬ґଘ͢Δ͜ͱ͕Θ͔ΔɻHHG ͷৼΔ෣͍Λৄ͘͠ௐ΂ΔͨΊʹͦͷαϒαΠΫϧεϖΫτϧ (I(ω, tp )) Λ‫͢ࢉܭ‬ΔɻI(ω, tp ) ͸ɺj(ω, tp ) =ordteiωt j(t)Fwindow (t−tp )Λ༻͍ͯ I(ω, tp ) = |ωj(ω, tp )|2 ͱఆٛ͞Εɺ͋Δ࣌ࠁ

tp ෇ۙͰͲͷΑ͏ͳ์ࣹ͕ൃੜ͢Δ͔Λҙຯ͢Δɻ͜͜ͰɺFwindow (t) ͸ t = 0 த৺ͷ૭ؔ਺Ͱ͋Δɻ݁ՌΛਤ 10 ʹࣔ͢ɻ֤࣌ࠁͰ‫౓ڧ͍ڧ‬Λࣔ͢प೾਺͕प‫ظ‬తʹมԽ͢Δ༷ࢠ͕‫ͯݟ‬औΕΔɻ
൒ಋମͰ͸ɺ͜ͷαϒαΠΫϧεϖΫτϧͷߏ଄͕ɺՁిࢠόϯυͱ఻ಋిࢠόϯυͷ෼ࢄؔ܎Λ༻͍ͨ൒‫ݹ‬యతͳ 3 εςοϓ໛‫Ͱܕ‬Α͘‫ه‬ड़͞ΕΔ͜ͱ͕஌ΒΕ͍ͯΔ [76]ɻ͜͜Ͱ൒‫ݹ‬యతͳ 3 εςοϓ໛‫ͱܕ‬͸ɺ࣍ͷΑ͏ͳ΋ͷͰ͋Δɻ·ͣɺ֎৔ʹΑΓ͋Δ࣌ࠁ t0Ͱిࢠɾਖ਼޸ϖΞ͕ྭ‫͞ى‬ΕΔ (εςοϓ̍)ɻͦͷϖΞͷ‫( ཭ڑ‬xrel (t)) ͕࣍ͷํఔࣜͰ‫ه‬ड़͞ΕΔΑ͏ʹมԽ͢Δ (εςοϓ 2)ɻ

dxrel (t)dg (p) ,=dt
dp p=p(t)p(t) = p0 − A(t) + A(t0 ).

c (p)−v (p) Ͱ͋Γɺc (p)(v (p)) ͸Ձిࢠ (఻ಋిࢠ)όϯυͷ෼ࢄΛද͍ͯ͠Δɻॳ‫ظ‬৚݅͸ xrel (t0 ) = 0
ͱͳΔɻిࢠɾਖ਼޸ϖΞ͸֎৔தΛӡಈ͠ɺ͋Δ࣌ࠁ tͰಉ͡Ґஔʹ໭ͬͯ͘Δ xrel (t) = 0ɻ͜ͷ࣌ɺ࠶݁߹
͕‫͖ى‬ΤωϧΪʔ g (p(t)) ͕์ग़͞ΕΔ (εςοϓ 3)ɻ
͜ͷ໛‫ܕ‬ͷϙΠϯτ͸ిࢠɾਖ਼޸ϖΞͷΤωϧΪʔͱ૬ରӡಈ͕ҰཻࢠεϖΫτϧͷ৘ใ g (p) Ͱ‫ه‬ड़͞Ε
Δͱ͍͏఺ʹ͋ΔɻҰํɺ̍࣍‫ ݩ‬Hubbard ໛‫Ͱܕ‬͸ిՙࣗ༝౓ͷଟମૉྭ‫͋Ͱى‬Δμϒϩϯɺϗϩϯͷ෼ࢄؔ܎ (d (p)ɺh (p)) ͕ղੳతʹΘ͔͍ͬͯΔɻ࣮͸ɺ൒‫ݹ‬యతͳ 3 εςοϓ໛‫Ͱܕ‬ɺ ൒ಋମʹ͓͚Δ c ɺvͷ୅ΘΓʹɺ͜ͷଟମྭ‫ى‬ͷ෼ࢄؔ܎Λ༻͍Δͱ Mottઈԑମʹ͓͚Δ HHG ͷαϒαΠΫϧεϖΫτϧ͕Α͘࠶‫͖Ͱݱ‬Δ͜ͱ͕Θ͔Δ (ਤ 10)ɻͭ·Γɺ൒‫ݹ‬యత
ͳ 3 εςοϓ໛‫ܕ‬͸ɺMott ઈԑମͰ΋ଟମૉྭ‫ى‬ͷ෼ࢄؔ܎Λ༻͍Ε͹༗ޮͰɺHHG ͷաఔΛ͏·͘આ໌Ͱ͖Δͱ͍͏͜ͱͰ͋Δɻ
ଟମૉྭ‫ى‬ͷ෼ࢄ͸ɺҰཻࢠεϖΫτϧʹ‫ݱ‬ΕΔ
৔߹΋͋Δ͕ɺඞͣ͠΋ͦ͏ͱ͸‫ݶ‬Βͳ͍ɻ̍࣍‫ݩ‬

Hubbard ໛‫ܕ‬ͷ৔߹ɺҰཻࢠ͸μϒϩϯ (΋͘͠͸ϗϩϯ) ͱεϐϊϯͱ‫ݺ‬͹ΕΔૉྭ‫ʹى‬෼ղ͞ΕΔɻͦͷͨΊɺҰཻࢠεϖΫτϧ͸ೋͭͷૉྭ‫ى‬Λ૊Έ߹Θͤͨ΋ͷʹͳΔɻਤ 11(a)(b) ʹɺHubbard ໛‫ܕ‬Λ

Schrieﬀer-Wolﬀ(SW) ม‫ͯ͠׵‬ಘΒΕΔ༗ޮ໛‫ʹܕ‬ର͢ΔҰཻࢠεϖΫτϧΛࣔͨ͠ɻ(a) ͱ (b) ͷҧ͍͸εϐϯ഑ྻͷҧ͍Ͱɺ(a) ͸ Heisenberg ໛‫ܕ‬ͷ‫ج‬ఈঢ়ଶɺ(b) ͸൓‫࣓ڧ‬ੑεϐϯঢ়ଶΛऔ͍ͬͯΔɻ͜ͷ໛‫ʹܕ‬͓͍ͯ΋ɺҰཻࢠεϖΫτϧ͸ೋͭͷૉྭ‫ى‬ͷ૊Έ߹ΘͤʹͳΔ͕ɺεϐϯঢ়ଶͷҧ͍ͰҰཻࢠεϖΫτϧ

(11a)
(11b)

͕େ͖͘มΘΓɺಛʹ (b) ͷ৔߹͸Կ΋ߏ଄Λද͞ͳ͍ɻ͔͠͠ɺରԠ͢Δ HHG εϖΫτϧΛ‫ݟ‬Δͱ྆ऀͷ৔߹Ͱ‫ܗ‬ঢ়͸Ұக͢Δ͜ͱ͕Θ͔Δ (ਤ 11(c))ɻ͜

ͨͩ͠ɺp(t) ͸ిࢠͷ࣋ͭӡಈྔ (ిࢠͷ࣋ͭӡಈྔ +Ε͸ɺHHG ͕ɺҰཻࢠεϖΫτϧͰ͸Ӆ͞Εͯ͠·͏ϗʔϧͷ࣋ͭӡಈྔ=0 ͕੒Γཱ͍ͬͯΔ)ɺg (p) ≡μϒϩϯɾϗϩϯͷ෼ࢄΛ௚઀൓ө͢Δ͜ͱΛҙຯ͠

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(a)
(a)

(a)

ω

(a)
(b)

k

(c)

P

(b)

t
ਤ 12

(a) ൒ಋମ໛‫ܕ‬ͷ෼ࢄؔ܎ͱྭ‫ى‬प೾਺ͷؔ܎ਤɻྭ‫΁ࢠى‬ͷ‫ڞ‬໐ྭ‫ى‬Λߟ͑ΔͨΊྭ‫ى‬प೾਺͸όϯυΪϟοϓΑΓখ͍͞ɻ(b) ‫ڞ‬໐ྭ‫ʹى‬༠‫͞ى‬ΕΔ෼‫ ۃ‬P ͷμΠφϛΫεɻฏ‫ۉ‬৔ (MF) ͱ GKBA(ࢄਤ 11(a)(b) SW ม‫׵‬Λ༻͍ͨҰ࣍‫ ݩ‬Hubbard ໛‫ܕ‬ͷ༗ޮ໛‫͔ܕ‬Β‫ͨ͠ࢉܭ‬ҰཻࢠεϖΫτϧɻ(a) ͸Heisenberg ໛‫ʹܕ‬Αͬͯ‫ه‬ड़͞ΕΔεϐϯঢ়ଶΛɺ(b) ͸൓‫࣓ڧ‬ੑεϐϯঢ়ଶΛ༻͍͍ͯΔɻ(a) ͷ݁Ռ͕ɺ‫ݩ‬ͷҰ࣍‫ ݩ‬Hubbard ໛‫ܕ‬ͷ෼ࢄؔ܎ͱྨࣅͨ͠
݁Ռͱͳ͍ͬͯΔɻ(c) ༗ޮ໛‫͔ܕ‬Β‫ ͨ͠ࢉܭ‬HHG
ͷ݁Ռɻεϐϯঢ়ଶ͕ Heisenberg ໛‫هͰܕ‬ड़͞ΕΔ
৔߹ͱ൓‫࣓ڧ‬ੑεϐϯঢ়ଶͷೋͭͷ৔߹Ͱ‫ͯ͠ࢉܭ‬͍Δɻจ‫[ ݙ‬75] ΑΓɻ͍ͯΔɻ͜ΕΒແ‫ݩ࣍̍ͱݩ࣍ݶ‬ͷղੳͷ݁Ռ͸ɺMott ઈԑମʹ͓͚Δ HHG ͸ଟମͷྭ‫ߏى‬଄Λ௚઀ଊ͍͑ͯΔ
͜ͱΛ͍ࣔࠦͯ͠ΔɻͦͷͨΊɺHHG ͕൒ಋମʹ͓͍ͯిࢠͷ෼ࢄؔ܎ͷ‫؍‬ଌ΁Ԡ༻͞Ε͍ͯΔΑ͏ʹɺ‫ڧ‬૬ؔిࢠ‫͚͓ʹܥ‬Δଟମͷྭ‫ߏى‬଄ͷ‫؍‬ଌʹ HHG ͕
࢖͑ΔՄೳੑ͕े෼ʹ͋Δɻ·ͨɺࢴ໘ͷ౎߹্ৄࡉ
͸ল͕͘ɺMott ઈԑମͷಛ௃తͳࣗ༝౓ͱͯ͠εϐϯ͕͋ΓɺεϐϯͷμΠφϛΫε͕ HHG ‫ʹ౓ڧ‬େ͖ͳӨ‫ڹ‬Λ‫ͱ͜͢΅ٴ‬΋ཧ࿦‫͔ࢉܭ‬Βࣔࠦ͞Ε͍ͯΔɻͦͷͨΊɺεϐϯࣗ༝౓Λ࢖ͬͨ HHG ͷૢ࡞ͳͲ‫ڧ‬૬ؔ‫ܥ‬ಛ༗ͷ HHG ͷ෺ཧͷల։͕‫ظ‬଴͞ΕΔɻ͞ΒʹɺμΠϚʔ Mott ‫Ͱܥ‬͸μϒϩϯɾϗϩϯͱ͸·ͨҟͳͬͨૉྭ‫( ى‬ΩϯΫ΍ΞϯνΩϯΫ) ͷμΠφϛΫεʹΑ
Γ HHG ͕ੜ͡Δ͜ͱ͕ࢦఠ͞Ε͓ͯΓ [73]ɺ༷ʑͳ
छྨͷଟମͷૉྭ‫ ͱى‬HHG ͷؔ܎ͷཧղɺ͞Βʹ͸ͦΕΒΛ༻͍ͨྭ‫ߏى‬଄‫؍‬ଌ๏ͷཱ֬ͳͲ‫ڵ‬ຯਂ͍໰୊͕ଟ͘࢒͍ͬͯΔɻ

4.2 ඇฏߧʹ͓͚Δடং૬༠‫ى‬ཚ༗Γ) ͷ݁ՌΛ͍ࣔͯ͠Δɻ఺ઢ͸ɺόϯυΪϟο
ϓΛ Egap ͱ͠ɺsin(Egap t) Λ͓ࣔͯ͠ΓɺGKBA
΍ MF ͷৼಈ͕ Egap ΑΓ଎͍͜ͱΛද͢ɻ

͍ͯΔɻ

4.2.1 Prethermal ঢ়ଶʹ͓͚Δடং༠‫ى‬Prethermal ঢ়ଶͱ͸ɺిࢠؒࢄཚ΍ిࢠϑΥϊϯࢄཚ͕ຊ֨తʹ࢝·Δલʹɺ‫ۦ‬ಈ͞Εͨ‫͕ܥ‬ୡ͢Δঢ়ଶ
Ͱ͋Δɻ͜ͷΑ͏ͳঢ়ଶʹ͓͍ͯɺடং૬ͱΈͳͤΔ
ঢ়ଶ͕༠‫͖Ͱى‬ΔՄೳੑ͕͋Δɻ͜ͷΑ͏ͳ‫ݱ‬৅͕‫ݱ‬ΕΔྫͱͯ͠ɺ·ͣྭ‫ࢠى‬ϞʔυΛ‫ڞ‬໐తʹྭ‫ͨ͠ى‬൒ಋମ͕‫͛ڍ‬ΒΕΔ [80–82]ɻޫ
ྭ‫͞ى‬Εͨ൒ಋମͰ͸ɺిࢠͱϗʔϧ͕ੜ੒͞ΕΔɻ
͜ΕΒ͸·ͣόϯυ಺؇࿨Λ‫͕͢͜ى‬ɺిࢠͱϗʔϧͷԹ౓͕े෼௿͘ͳΕ͹ྭ‫ڽࢠى‬ॖ૬ (EI ૬) ͕࣮‫ݱ‬͞ΕΔՄೳੑ͕‫͔͘ݹ‬Βࢦఠ͞Ε͖ͯͨ [77–79]ɻ࣮
͸ɺಉ༷ͷঢ়ଶ͕ɺྭ‫ࢠى‬ϞʔυΛ‫ڞ‬໐తʹྭ‫ͨ͠ى‬৔߹ʹ (ա౉తͰ͸͋Δ͕) ࣮‫͞ݱ‬ΕΔ͜ͱ͕ɺ࠷ۙͷ࣮‫ͱݧ‬ཧ࿦ͷ‫Ͱڀݚ‬ใࠂ͞Ε͍ͯΔ [80–82]ɻ
͜ͷ͜ͱΛઆ໌͢ΔͨΊʹɺ؆୯ͳεϐϯϨε̎όϯυ໛‫ܕ‬Λಋೖ͢Δ [82]ɻϋϛϧτχΞϯ͸ Ĥ(t) =Ĥkin (t) + Ĥint + Ĥdip (t) ͷΑ͏ʹ‫ه‬ड़͞Εɺ֤߲͸

Ĥkin (t) = −va eiqA(t)rij ĉ†i,a ĉj,a +i,j,a=c,v

Ĥint = UĤdip (t) = −E(t) ·Δa ĉ†i,a ĉi,a ,i,an̂i,c n̂i,v ,

i



(12)

da ĉ†i,a ĉi,ā .

i,aඇฏߧঢ়ଶʹ͓͍ͯฏߧঢ়ଶͰ͸ಘ೉͍৽͍͠૬ΛͷΑ͏ʹͳ͍ͬͯΔͱ͢Δɻ͜͜Ͱɺa = c, v ͸όϯ༠‫͢ى‬Δ͜ͱ͸ɺඇฏߧ෺ੑ‫ڀݚ‬ͷҰͭͷ໨ඪͱͳͬ

υͷछྨΛද͠ɺΔa ͸όϯυ͝ͱͷΤωϧΪʔγϑ͍ͯΔɻFranck-Condon ඳ૾Ͱઆ໌͞ΕΔΑ͏ͳൺֱτΛද͢ɻ·ͨɺĤint ͸ҟͳΔόϯυؒʹಇ͘ੱྗ૬

తΏͬ͘ΓͳաఔʹΑ࣮ͬͯ‫͞ݱ‬ΕΔޫ༠‫ى‬૬సҠҎ‫༻࡞ޓ‬Λද͢ɻిࢠͱϗʔϧͷ‫ݴ‬༿ʹ௚͢ͱҾྗ૬‫ޓ‬֎ʹ΋ɺ༷ʑͳඇฏߧ૬సҠͷϝΧχζϜ͕ఏҊ͞Ε࡞༻ͱͳΓɺ൒ಋମʹ͓͚Δྭ‫ࢠى‬४Ґ͕ੜ੒͞ΕΔɻ

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Ĥdip (t) ͸ɺόϯυؒͷμΠϙʔϧྭ‫ى‬Λද߲͢ʹͳͬͳཧ༝͔Β) ΋཈͑ΒΕΔɻ্ͷྫ͸͜ͷೋͭͷཁૉ

͍ͯΔɻ֎৔͕ͳ͍৔߹͸ɺc-όϯυͱ v-όϯυͷిࢠΛ‫ؚ‬ΜͰ͍ΔɻҰํɺ૬‫͍ڧʹٯ͕༻࡞ޓ‬৔߹Ͱ͸ɺ‫ڧ‬਺͕֤ʑอଘ͠ɺĉ†c ĉv  ͕டংύϥϝʔλͱͯ͠ಇ͘ɻ
ͭ·Γɺରশੑ͕ഁΕ͍ͯͳ͍ঢ়ଶͰ͸ ĉ†c ĉv  = 0 ͱ݁߹ͷฏ‫ۉ‬৔తͳඳ૾͕Α͘ͳΓɺ্ͷऑ૬ؔͷ৔߹

ͳΔҰํͰɺରশੑͷഁΕͨ EI ૬͸ ĉ†c ĉv  = 0 ͱͳ

ΔՄೳੑ΋͋Δ͜ͱΛ࣍ʹ‫[ ͍ͯ͘ݟ‬83]ɻͦͷͨΊʹΓɺc-όϯυͱ v-όϯυͷࣗൃతͳࠞ੒͕ੜ͡Δɻμ͜͜Ͱ͸̎όϯυ Hubbard ໛‫ܕ‬Λߟ͑ΔɻϋϛϧτΠϙʔϧྭ‫ى‬ͷ߲͸ɺc-όϯυͱ v-όϯυͷࠞ੒Λ΋χΞϯ͸ Ĥ = Ĥkin + Ĥcf + Ĥint ͷΑ͏ʹද͞Εɺ֤ͨΒͨ͢ΊɺடংύϥϝʔλΛ֎෦͔Β‫੍ڧ‬తʹ‫ۦ‬ಈ߲͸ Ĥkin =͢Δ߲ͱΈͳͤΔɻ+ͱಉ͡Α͏ʹɺprethermal ͳঢ়‫Ͱگ‬டং૬͕༠‫͞ى‬Ε





†
†
i,j,σ [v0 ĉi1σ ĉj1σ − v0 ĉi2σ ĉj2σ + h.c.] † †i,j,σ [v0 ĉi1σ ĉj2σ + v0 ĉi2σ ĉj1σ + h.c.] ɺͦͯ͠ɺ

͜ͷΑ͏ͳ‫ܥ‬Λྭ‫ࢠى‬४Ґʹ‫ڞ‬໐͢Δप೾਺Ͱྭ‫ى‬
͢Δ৔߹Λߟ͑Δ (ਤ 12)ɻ͜͜Ͱ͸ɺ‫ܥ‬ͷμΠφϛΫ
εΛɺࢄཚޮՌͷೖΒͳ͍ฏ‫ۉ‬৔ཧ࿦ʢHartree-Fockۙࣅ) ͱࢄཚաఔ͕औΓࠐΊΔ GKBA ͷ݁ՌΛൺֱ͠ͳ͕Βɺߟ࡯͍ͯ͘͠ɻ‫ࢉܭ‬ͷ݁ՌΛ‫ݟ‬Δͱɺྭ‫ޙى‬Ĥcf = Δcf
Ĥint = U೾਺Ͱৼಈ͢Δ͸ͣͰ͋Δ͕ɺ࣮ࡍͷৼಈ਺͸όϯυ(n̂i1σ − n̂i2σ ),iσn̂ic↑ n̂ic↓ + U i,c=1,2+ (U  − J)΋େ͖ͳடংύϥϝʔλͷৼಈ͕࢒Δɻ͔͠΋ɺ୯ͳΔྭ‫ࢠى‬ͷৼಈͰ͋Ε͹όϯυΪϟοϓΑΓখ͞ͳप





(14)



n̂i1σ n̂i2σ̄

i,σn̂i1σ n̂i2σ ,

i,σ

+J

 † †
 † †
ĉi1σ ĉi2σ̄ ĉi1σ̄ ĉi2σ + I[ĉi1↑ ĉi1↓ ĉi2↓ ĉi2↑ + h.c.],i,σiΪϟοϓ (Egap ) ΑΓେ͖͍͜ͱ͕Θ͔Δɻ͜ͷ͜ͱͱද͞ΕΔɻ͜͜Ͱ ĉiασ ͸αΠτ iɺ‫ي‬ಓ αɺεϐϯ σ͸ɺBCS తͳ EI ૬͕ੜ͍ͯ͡Δ͜ͱΛҙຯ͢Δɻฏͷిࢠͷফ໓ԋࢉࢠͰ͋ΔɻĤcf ͸݁থ৔ (Δcf ) ʹΑ‫ۉ‬৔ඳ૾Ͱ͸டংύϥϝʔλͦͷৼಈ͸΄ͱΜͲ؇࿨Δ‫ي‬ಓͷ෼྾ɺĤint ͸‫ॴہ‬తͳ૬‫༻࡞ޓ‬Λද͠ɺΫʔ

ͤͣ࢒Δ͕ɺGKBA ʹΑΓࢄཚաఔΛಋೖ͢Δͱͦϩϯ૬‫ ༻࡞ޓ‬U ɺU  ɺϑϯτ݁߹ J ͦͯ͠ϖΞϗοϐ

ͷৼಈ͕ঃʑʹऑ·͍ͬͯ͘͜ͱ͕Θ͔Δɻͭ·Γɺϯά૬‫ ༻࡞ޓ‬I Ͱಛ௃෇͚ΒΕΔɻDMFT ʹΑΔฏେ͖ͳடংύϥϝʔλͷৼಈͷຊ࣭͸ฏ‫ۉ‬৔ඳ૾Ͱઆ

ߧঢ়ଶͷ૬ਤΛਤ 13(a) ʹࣔ͢ɻϑϯτ݁߹͕݁থ৔

໌Ͱ͖ɺࢄཚ͸ͦΕΛ્֐͢Δํ޲ʹಇ͘ͱ͍͏Θ͚ΑΓ‫͍࣌ڧ‬͸ߴεϐϯঢ়ଶ͕ɺ‫͋Ͱٯ‬Ε͹௿εϐϯঢ়

Ͱ͋ΔɻͰ͸ɺฏ‫ۉ‬৔తͳඳ૾ͷ͏ͪॏཁͳཁૉ͸Կଶ͕ΑΓ҆ఆʹͳΔɻͦͷதؒʹ͸ɺ͜ΕΒͷঢ়ଶ͕͔ʁͦΕ͸೾ಈؔ਺ͷ‫͕ܗ‬୯७ͳ‫͚͔ʹܗ‬ɺ͋Δҙຯ”ࣗൃతʹࠞ੒͢Δ͜ͱͰੜ͡Δடং૬͕ग़‫͢ݱ‬Δɻ͜

Թ౓͕௿͍ঢ়ଶ”͕อͨΕ͍ͯΔͱ͍͏ͱ͜Ζʹ͋Δɻͷঢ়ଶ͸‫ڧ‬૬ؔ‫͚͓ʹܥ‬Δ EI ૬ͱΈͳͤɺͦͷடংύ͢ͳΘͪɺฏ‫ۉ‬৔೾ಈؔ਺͸ɺk Λӡಈྔͱͯ͠

ϥϝʔλ͸ φλ =|ψ(t) =

(αk (t)ĉ†k,c + βk (t)ĉ†k,v )|vac



†
λ

σσ  c1σ c2σ σσσ  ͱද͞ΕΔ [84]ɻ

ͨͩ͠ɺ͜͜Ͱ σ ͸ύ΢ϦߦྻΛද͠ɺX,Y ,Z ੒෼͸

(13)

k

‫ج‬ຊతʹ౳ՁͰ͋ΔɻҎԼͰ͸ɺY ੒෼ (φY ) ʹ஫໨ͯͦ͠ͷμΠφϛΫεΛ௥͍ͬͯ͘͜ͱʹ͢Δɻ

2

ͱ͔͚ɺ|αk (t)| + |βk (t)| = 1 ͕֤ k ͰอͨΕΔͱ࣍ʹ͜ͷ‫ܥ‬ͷΫΤϯν໰୊Λߟ͑ΔɻΫΤϯν໰୊͍͏͜ͱͰ͋Δɻ΋ͪΖΜɺࢄཚ͕࢝·Δͱ͜ͷঢ়ଶͱ͸‫ܥ‬ͷύϥϝʔλΛ‫ʹٸ‬ม͑ɺ‫ܥ‬Λฏߧ‫͔ܥ‬ΒͣΒ͸ঃʑʹྑ͍ۙࣅͰ͸ͳ͘ͳΔ͕ɺ͋Δ࣌ؒͷؒ͸ेͨ࣌͠ʹੜ͡ΔඇฏߧμΠφϛΫεʹ஫໨͢Δ໰୊ͷ෼ྑ͍ۙࣅͱͯ͠੒ΓཱͭɻҰํɺ͜ͷ‫ܗ‬ͷฏ‫ۉ‬৔೾͜ͱΛࢦ͢ɻY ੒෼ͷடংͷ༳Β͗ (”छ”) ͕͋Δͱ͠ಈؔ਺͸ઈରྵ౓ͷ௒఻ಋ BCS ཧ࿦Ͱग़ͯ͘Δ΋ͷͯɺͦͷൃలΛ௥͍ͬͯ͘͜ͱʹ͢Δɻࠓͷ৔߹͸ɺฏʹྨࣅ͢Δɻ༗‫ݶ‬Թ౓ঢ়ଶ͸͜ͷΑ͏ͳ‫ܗ‬ͷ೾ಈؔ਺ߧঢ়ଶͰ࣮‫͢ݱ‬ΔసҠԹ౓ Tc ͷ࠷େ஋ΑΓߴԹͷ௿εͷ Mixed ঢ়ଶͱͯ͠‫ه‬ड़͞ΕΔɻ͜ͷҙຯͰࠓͷঢ়ଶϐϯঢ়ଶ͔Βग़ൃ͠ɺ݁থ৔ΛมԽͤ͞Δ (ਤ 13(a))ɻ

͸”Թ౓͕௿͍ঢ়ଶ”Ͱ͋Γଓ͚Δɻ͜ͷͨΊɺޫྭ‫ى‬·ͣɺྭ‫͞ى‬Εͨঢ়ଶ͕࠷ऴతʹߦ͖ண͘ઌΛߟ͑ͯ͞Εͨடং͸ฏ‫ۉ‬৔ͷൣғͰ͸่Εʹ͘͘ɺͦͷදΕΈΑ͏ɻࠓɺ‫ܥ‬͸ด͍ͯ͡ΔͷͰΤωϧΪʔ E ͕อଘͱͯ͠େ͖ͳৼಈ͕࢒ΔͱཧղͰ͖Δɻ͢ΔɻΫΤϯν‫ޙ‬ͷͦΕΛ Ef ͱ͠ɺ͋Δ೤ฏߧঢ়ଶʹ2͜͜Ͱ‫ݱͨݟ‬৅͸ɺࢄཚޮՌ͕ೖΔલͰ͋Ε͹ฏ‫ۉ‬ߦ͖ண͘ͱԾఆ͢Δͱ Ef = Tr[e−βf Hf Hf ]/Zf ͱͳ৔తͳඳ૾͕Α͘ɺͦΕʹΑͬͯ‫ܥ‬͸͋Δҙຯ”Թ౓͕Δ βf ʹߦ͖ண͘ͱߟ͑ΒΕΔɻ͜ͷ༧૝͞ΕΔ࠷ऴ

௿͍ঢ়ଶ”ʹͳΓɺடং૬ͷΑ͏ͳঢ়ଶ͕ग़‫͢ݱ‬Δͱ͍Թ౓ (Tf = β1f ) Λਤʹࣔ͢ͱɺ͋ΔྖҬͰ͸Թ౓͕ॳ͏΋ͷͰ͋ͬͨɻࢄཚޮՌ͸ɺͦ΋ͦ΋૬‫͕༻࡞ޓ‬ऑ‫ظ‬ঢ়ଶΑΓ΋Լ͕Γ EI ૬͕࣮‫͞ݱ‬ΕΔ͜ͱ͕Θ͔Δɻ

͚Ε͹཈͑Δ͜ͱ͕Ͱ͖Δ͠ɺྭ‫ى‬৚݅Ͱ (Ґ૬ۭؒతͰ͸ɺ୹࣌ؒͷৼΔ෣͍͸Ͳ͏ͳΔͷ͔ʁɹਤ 13(b)《講義ノート》

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ʹடংύϥϝʔλͷ୹࣌ؒͷৼΔ෣͍Λ͍ࣔͯ͠Δɻ

(a)
(a)

 
 

0.20

࠷ऴతʹ EI ૬ʹߦ͘ͱ‫ظ‬଴Ͱ͖Δ৔߹͸ɺ‫ظ‬଴͞ΕΔ



0.15
0.10

ʹߦ͔ͳ͍৔߹Ͱ΋ɺա౉తʹྭ‫ࢠى‬டং͕ൃల͠ɺେ

0.05

͖ͳ஋ΛऔΔͱಉ࣌ʹɺઌͷ৔߹ͱҟͳΔ‫ي‬ಓΛࣔ͢

1.1ϥϝʔλ͸‫਺ڏ‬஋ΛͱΔͷʹର͠ɺ‫ऀޙ‬ͷա౉తͳྭ

0.5‫ࢠى‬டংͰ͸େ͖ͳ࣮੒෼Λ࣋ͪɺฏߧͱ͸ҟͳΔৼ

0.3

0.4

μΠφϛΫεʹ‫·ؚ‬Ε͍ͯΔ͜ͱ͕Θ͔Δɻ‫ྖ߹݁ڧ‬ମతʹ͸ɺ̍ͭͷ௿εϐϯঢ়ଶͱ̏ͭͷߴεϐϯঢ়ଶ͔Βग़ൃ͢Δઁಈ࿦Λల։͢Δɻ͞Βʹɺ௿εϐϯঢ়
low-spininsulator



1.2

1.3

1.4
Δcf

1.5

(b) DMFT

1.6

0.7

(b)

0.6

Δcf=1.4

1.7(c)  + (c)Δcf=1.39

0.5
1.45

1.35

0.4

1.3

0.3

0.2
0.1

1.5

0

1.275

1.68
1.345

0.2
0.1

1.25

-0.1

1.74
1.325

0
-0.1

-0.2

1.30

-0.2
-0.4

-0.2

0

0.2

0.4

Re φY

ҬͰͷ༗ޮ໛‫ܕ‬͸ Hubbard ໛‫͔ܕ‬Β Heisenberg ໛‫ܕ‬Λಋग़͢ΔΑ͏ʹઁಈ࿦Λ࢖͏͜ͱͰಋग़Ͱ͖Δɻ۩



τy

Im φY

࣮͸͜ͷ‫ݱ‬৅͸ɺ‫ྖ߹݁ڧ‬ҬͰͷ༗ޮ໛‫ܕ‬ͷฏ‫ۉ‬৔EI
high-spin
insulator0͜ͱ͕Θ͔Δ (Δcf = 1.3)ɻಛʹɺฏߧঢ়ଶͰ͸டংύ

Δ෣͍Λࣔ͢͜ͱ͕‫ڵ‬ຯਂ͍ɻ’metal’Tடংύϥϝʔλ஋ͷपΓΛճΔɻҰํɺ࠷ऴతʹ EI ૬



ਤ 13

-1 -0.8 -0.6 -0.4 -0.2 0

0.2 0.4 0.6 0.8τx(a) DMFT ʹΑΔ̎όϯυ Hubbard ໛‫ܕ‬ͷ૬ਤɻͨͩ͠ɺU = 6, U  = 4, J = I = 1ɺόϯ√υ෯ W = 4 2 Λ༻͍͍ͯΔɻਤதʹ͸ΫΤϯν
લͷॳ‫ظ‬ঢ়ଶͱΫΤϯν‫ޙ‬ͷ؇࿨Ͱ࠷ऴతʹߦ͖ணଶ͔ΒͷΫΤϯνΛߟ͍͑ͯΔࠓͷঢ়‫Ͱگ‬͸ɺ‫ܥ‬ͷμ͘Թ౓ (˔) Λද͢ɻ(b)DMFT ʹΑΔΫΤϯν‫ޙ‬ͷΠφϛΫε͸̍ͭͷ௿εϐϯঢ়ଶͱ 1 ͭͷߴεϐϯঢ়டংύϥϝʔλ φY ͷ‫ي‬ಓɻΫΤϯν‫ޙ‬ͷ Δcf ͷ஋ଶͷ̎ࣗ༝౓͕αΠτ͝ͱʹ͋Δ໰୊ʹஔ͖‫͑׵‬Δ͜
ͱ͕Ͱ͖Δɻ͜ͷ 2 ࣗ༝౓Λ༻͍ٖͨεϐϯΛఆٛͯ͠໛‫ܕ‬Λॻ͖Լ͢ͱ

Heﬀ = hz



z
τ̂Y,i
+ Jx

i,j

i

+ Jy





y
y
τ̂Y,i
τ̂Y,j
+ Jz

i,j

x
x
τ̂Y,i
τY,j



(15)

͸ਤதʹ͋Γɺ˔͸࠷ऴతͳடংύϥϝʔλͷ஋Ͱ͋Δɻ(c) ༗ޮ໛‫ܕ‬ͷฏ‫ۉ‬৔μΠφϛΫεɻͨͩ͠ɺ

√√τ x  2ReφY ɺτ y  2ImφY ͱͳ͍ͬͯΔɻจ‫[ ݙ‬83] ΑΓɻ

DMFT Λ༻͍ͨ͜ͱͰɺ‫ॴہ‬తͳ૬‫͔༻࡞ޓ‬Β͘ΔࢄཚޮՌ͸औΓࠐΊ͍ͯΔɻͦΕʹ΋͔͔ΘΒͣɺό

z
z
τ̂Y,i
τ̂Y,j

i,j

ϯυΪϟοϓ͕େ͖͘ͳ͍தؒ݁߹ྖҬۙ๣Ͱ΋ɺ্‫ه‬ͷΑ͏ͳฏ‫ۉ‬৔తͳμΠφϛΫε͕ੜ͖࢒͍ͬͯΔͷΑ͏ʹεϐϯ 12 ͷ XYZ ໛‫ͳͱܕ‬Δɻͨͩ͠ɺτ̂Y ͸
εϐϯ 12 ͷٖεϐϯΛද͠ɺ܎਺͸‫ݩ‬ͷϋϛϧτχΞ

ա౉ঢ়ଶͷҰཻࢠεϖΫτϧ΍ిࢠԹ౓ͳͲ΋ධՁͰϯͷύϥϝʔλͰܾ·Δɻ௿εϐϯঢ়ଶ͸ Z ํ޲ʹ͖ɺա౉తͳடং૬ͷಛ௃తͳεϖΫτϧߏ଄΋Θ͔Լ޲͖ͷঢ়ଶʹରԠ͢ΔɻΫΤϯνʹΑΓ XYZ ໛‫ܕ‬Δ (ৄ͘͠͸࿦จ [83] Λࢀর)ɻ͞Βʹɺ͜͜Ͱѻͬͨͷύϥϝʔλ͕มԽ͠ɺεϐϯӡಈ͕࢝·Δɻฏ‫ۉ‬৔

EI ૬͸ɺཧ࿦తʹ͸௒఻ಋͱྨࣅ͓ͯ͠Γɺ࣮ࡍʹ͋ཧ࿦ʹΑΔμΠφϛΫεΛਤ 13(c) ʹࣔ͢ɻఆੑతʹΔϢχλϦม‫׵‬Λ͢Δ͜ͱͰɺ্Ͱٞ࿦ͨ̎͠όϯυΑ͘ DMFT ͷ݁ՌΛ࠶‫͍ͯ͠ݱ‬Δ͜ͱ͕Θ͔Δɻಛ

Hubbard ໛‫͚͓ʹܕ‬Δ EI ૬͸εϐϯ triplet ͷ௒఻ʹɺடংύϥϝʔλͷ‫ي‬ಓͷมԽͱ‫ʹ࣠ڏ‬৳ͼͨடংಋʹϚοϓ͞Εɺ݁থ৔ͷΫΤϯν͸ཻࢠ਺ͷมԽʹύϥϝʔλͷൃల͕࠶‫͍͖ͯͰݱ‬Δɻ࣮͸͜ͷ‫ݱ‬৅͸ɺϚοϓ͞ΕΔɻཻࢠ਺ͷมԽ͸ɺଞͷόϯυ͔Βͷྭฏ‫ۉ‬৔ͷඳ૾Ͱ͸ɺٖεϐϯϊϧϜͷอଘ (‫ॴہ‬Τϯ‫ʹى‬Αͬͯ΋࣮‫͖Ͱݱ‬ΔͷͰɺ্Ͱٞ࿦ͨ͠Α͏ͳ͜τϩϐʔͷอଘ) ͱΤωϧΪʔอଘ͔Β͏·͘આ໌Ͱͱ͕௒఻ಋͰ΋࣮‫͢ݱ‬ΔՄೳੑ͸े෼ʹ͋Δɻ

͜ͱ͸஫໨ʹ஋͢Δɻ·ͨɺDMFT Λ༻͍Δ͜ͱͰɺ͖Δ͜ͱ͕Θ͔ΔɻಛʹɺલऀͷٖεϐϯϊϧϜͷอҎ্ͷ‫ڀݚ‬͸ɺऑ૬ؔͱ‫ڧ‬૬ؔͷ྆ଆʹ͓͍ͯɺࢄཚ

ଘ͸ฏ‫ۉ‬৔ಛ༗ͷ΋ͷͰ͋Δɻฏߧͷฏ‫ۉ‬৔ͷൣғͰޮՌ͕ຊ֨తʹ࢝·Δલͷ Prethermal ͳঢ়ଶͰɺඇࣗ

͸ɺ௿ԹͰ͸εϐϯϊϧϜ͸ 1/2 ʹ͕ۙͮ͘ɺԹ౓Λ໌ͳடং͕ա౉తʹൃల͠͏Δ͜ͱΛ͍ࣔͯ͠Δɻ্͛͜ΔͱϊϧϜ͸খ͘͞ͳΔɻ͔͠͠ɺฏ‫ۉ‬৔μΠφͷΑ͏ͳ‫ݱ‬৅͸͢Ͱʹใࠂ͞Ε͍ͯΔඇฏߧ‫ݱ‬৅ɺྫϛΫεͰ͸εϐϯϊϧϜ͸อଘ͢ΔͨΊɺ͋Δҙຯ௿͑͹ޫ༠‫ى‬௒఻ಋɺͱؔΘ͍ͬͯΔ͔΋͠Εͣɺࠓ‫ޙ‬Թͷঢ়ଶ͕อͨΕ͍ͯΔɻ͜Ε͸·͞ʹऑ݁߹৔߹ͷͷٞ࿦ͷൃల͕‫ظ‬଴͞ΕΔɻ

Eq. (13) ͷ೾ಈؔ਺‫ܗ‬ͷอଘʹରԠ͍ͯ͠Δɻ͜ͷ”

4.2.2 ϘτϧωοΫʹ༝དྷ͢Δடং૬ग़‫ݱ‬ͷՄೳੑ௿Թ”ͷঢ়ଶͷͨΊʹɺա౉తʹେ͖ͳடংύϥϝʔλ௨ৗɺిࢠؒࢄཚ΍ిࢠϑΥϊϯࢄཚ͕࢝·Δͱɺ͕࣮‫͏͠ݱ‬Δ͜ͱ΋Θ͔ΔɻΑΓৄ͍ٞ͠࿦͸ɺθϛPrethermal ͳঢ়ଶ͸่Εɺ‫ܥ‬͸֎৔ʹΑͬͯՃ͑ΒΕதʹߦ͏༧ఆͰ͋Δɻ

ͨΤωϧΪʔʹैͬͨ೤ฏߧঢ়ଶʹ͍͍ۙͮͯ͘ɻ͠

1

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











ਤ 14

൒ಋମʹ͓͚Δ෼‫ ۃ‬P(டংύϥϝʔλ) ͷྭ‫ޙى‬ͷ࣌ؒൃలɻg ͸ిࢠ֨ࢠ૬‫༻࡞ޓ‬Λࣔ͠ɺθϩͰͳ͍஋ͷ৔߹ϑΥϊϯʹΑΔిࢠͱਖ਼޸ͷόϯυ಺ྫྷ٫͕ੜ͡Δɻ‫ʹࢉܭ‬͸ GKBA ͕࢖ΘΕ͍ͯΔɻ

จ‫[ ݙ‬82] ΑΓɻਤ 15ޫυʔϓ͞Εͨେ͖ͳΪϟοϓΛ࣋ͭ Mott͔͠ɺࢄཚ΍؇࿨ͷϓϩηεͷ͏ͪɺԿΒ͔ͷཧ༝Ͱઈԑମʹ͓͚Δඇฏߧ૬ਤɻ ԣ࣠͸ޫυʔϓྔΛࣔͱͯ΋Ώͬ͘Γͳ΋ͷ͕͋ͬͨΒͲ͏ͳΔͩΖ͏͔ʁ͢μϒϩϯͷαΠτ͋ͨΓͷ਺ɻॎ࣠͸༗ޮ‫ٯ‬Թ౓࢒ΓͷϓϩηεͷΈ͕ಇ͖ɺ·ͣ͸೤ฏߧঢ়ଶͱ͸ҟΛࣔ͢ɻඇฏߧఆৗঢ়ଶʹର͢Δ DMFT ͕࢖ΘΕ͍ͯΔɻจ‫[ ݙ‬87] ΑΓɻͳΔঢ়ଶʹۙͮ͘͸ͣͰ͋Δɻ͜ͷΑ͏ͳ໰୊͸ɺޫྭ‫͞ى‬Εͨ൒ಋମͰΑ͘ௐ΂͏ͳ௒఻ಋঢ়ଶ (η ϖΞϦϯά௒఻ಋ [86, 91, 92]) ͷ࣮ΒΕ͖ͯͨ [56, 79, 85]ɻ൒ಋମͰ͸ޫྭ‫ޙى‬ɺిࢠͱ‫͕ݱ‬༧‫͞ݴ‬Ε͍ͯΔ [87]ɻզʑ΋ɺ࠷ۙɺ༗ޮతͳฏ

ਖ਼޸͕ੜ੒͞ΕΔ͕ɺͦΕΒͷ࠶݁߹ͷ࣌ؒεέʔϧߧঢ়ଶͷ໰୊ʹඇฏߧ໰୊Λஔ͖‫͑׵‬Δ͜ͱͰɺ؇࿨

͸ɺిࢠͱਖ਼޸͕֤ʑ఻ಋόϯυͱՁిࢠόϯυ಺ͰͷϘτϧωοΫ͕͋Δ৔߹ʹ༠‫͞ى‬ΕΔঢ়ଶͷ‫ه‬ड़Λ؇࿨͢Δ࣌ؒεέʔϧΑΓे෼௕͍ɻͦͷͨΊɺ·ͣߦ͏‫ڀݚ‬Λߦ͓ͬͯΓɺ௨ৗͷ൒ಋମͱ͸ҟͳΔ‫ڧ‬૬͸ిࢠͱਖ਼޸ͷ਺͕อͨΕͨ··ɺ͋Δछͷ೤ฏߧঢ়ؔ‫ܥ‬ಛ༗ͷঢ়ଶͷग़‫͍ͯ͑ݟ͕ݱ‬Δɻ‫ݱ‬ஈ֊Ͱ͸ະൃଶʹۙͮ͘ͱ‫ظ‬଴͞ΕΔ (͜Ε͸ਅͷ೤ฏߧঢ়ଶͰ͸දͳͷͰৄ͘͠͸ड़΂ͳ͍͕ɺՆͷֶߍ౰೔·Ͱʹ͸ͳ͍)ɻ͜ͷΑ͏ͳޫྭ‫͞ى‬Εͨ൒ಋମʹ͓͍ͯɺͲͷൃද͢ΔͷͰৄ͘͠͸ͦͷ࣌ʹ঺հ͍ͨ͠ɻ

Α͏ͳঢ়ଶ͕ग़‫͢ݱ‬Δ͔͸ɺ‫͔͘ݹ‬Β‫ڵ‬ຯΛ࣋ͨΕ͓ͯΓɺిࢠਖ਼޸ϓϥζϚɺྭ‫ࢠى‬Ψεͦͯ͠ྭ‫ڽࢠى‬ॖ

5 ͓ΘΓʹ

ͳͲ༷ʑͳঢ়ଶ͕ग़‫͢ݱ‬Δ͜ͱ͕‫ظ‬଴͞Ε͍ͯΔ [79]ɻ

ຊߘͰ͸ɺ૬ؔిࢠ‫ܥ‬ͷඇฏߧ෺ੑͷ࠷ۙͷ‫ؔʹڀݚ‬࣮ࡍʹɺઌͷ൒ಋମͷཧ࿦‫͍͓ͯʹࢉܭ‬΋ɺిࢠɾϑΥͯ͠ɺʮ‫ݟ‬Δʯͦͯ͠ʮૢΔʯͱ͍͏‫͔఺؍‬Β঺հ͖ͯ͠

ϊϯࢄཚΛߟྀͨ͠‫ࢉܭ‬Λߦ͏ͱɺࢄཚʹ൐͏؇࿨Ͱ

ͨɻಛʹɺʮ‫ݟ‬Δʯʹؔͯ͠͸‫ڧ‬૬ؔిࢠ‫͚͓ʹܥ‬Δߴ

Ұ౓ফ͑ͨྭ‫ࢠى‬டং͕͋Δ࣌ؒ‫ͱͭܦ‬෮‫͢׆‬Δ༷ࢠ

࣍ߴௐ೾ൃੜɺͦͯ͠ʮૢΔʯʹؔͯ͠͸ prethremal΋‫ͯݟ‬औΕΔ (ਤ 14) [82]ɻ

ঢ়ଶ΍؇࿨ͷ Bottleneck ޮՌʹ༝དྷ͢Δඇฏߧடং

‫ڧ‬૬ؔిࢠ‫͍͓ͯʹܥ‬͸ɺ͜ͷΑ͏ͳ‫ݱ‬৅͸ੜ͡Δ

૬ɺʹؔ͢Δ࠷ۙͷ‫ڀݚ‬Λ঺հͨ͠ɻ·ͨɺඇฏߧ෺ੑ

ͷͩΖ͏͔ʁɹੜ͡Δͱͨ͠ΒͲͷΑ͏ͳঢ়ଶ͕ग़‫ݱ‬ͷཧ࿦‫Ͱڀݚ‬͸ɺిࢠ૬ؔΛਖ਼͘͠ѻ͑Δཧ࿦ख๏ͷ͢ΔͷͩΖ͏͔ʁɹ͜ͷΑ͏ͳಈ‫ڀݚ͍ͨͮجʹػ‬΋ଘࡏ͕ॏཁʹͳΔɻຊߘͰ͸ɺ༗ྗͳཧ࿦ख๏ͷҰͭ࠷ۙਐΊΒΕ͍ͯΔ [86, 87]ɻMott ઈԑମͰ͸ɺྭ‫ى‬ͱͯ͠ɺඇฏߧάϦʔϯؔ਺๏ɺಛʹಈతฏ‫ۉ‬৔ཧ࿦ΛʹΑΓμϒϩϯͱϗϩϯ͕ੜ੒͞ΕΔɻ࣮͸͜ΕΒͷ঺հ͠ɺ͜ΕΒͷख๏ͷ࠷ۙͷਐలΛ঺հͨ͠ɻۙ೥ɺ

࠶݁߹ϓϩηε͸ɺόϯυΪϟοϓ͕େ͖͍৔߹͸ࢦޮ཰ͷྑ͍࣮૷ํ๏͕‫͓͔ͯͬͭݟ‬ΓɺΑΓෳࡶͦ͠਺తʹ཈੍͞ΕΔ͜ͱ͕஌ΒΕ͍ͯΔ [88–90]ɻͭ·ͯ‫࣮ݱ‬తͳ໛‫ܕ‬ͷղੳ΁ͷ‫ظ‬଴͕ߴ·͍ͬͯΔɻฏߧΓɺμϒϩϯͱϗϩϯͷ਺͕ۙࣅతʹอଘ͞ΕΔɻ͜

ঢ়ଶʹରͯ͠ߦΘΕ͍ͯΔΑ͏ͳୈҰ‫ݪ‬ཧతͳख๏ͱ

ͷঢ়‫گ‬͸ͪΐ͏Ͳ൒ಋମʹ͓͍ͯిࢠͱϗʔϧ͕อଘͷ૊Έ߹Θ͕ͤՄೳʹͳΔ೔΋ͦ͏ԕ͘ͳ͍͔΋͠Ε͞ΕΔঢ়‫͍ͯࣅʹگ‬Δɻͨͩ͠ɺμϒϩϯͱϗϩϯ͸ͳ͍ɻ‫؍‬ଌٕज़΍ཧ࿦ख๏͸೔ʑൃల͠ɺͦΕΛ௨͠୯ͳΔిࢠ΍ϗʔϧͱ͸ҟͳΓɺ୯७ͳϑΣϧϛΦϯ

ͯ৽͍͠ඇฏߧ‫ݱ‬৅ͷใࠂ΍ཧ࿦ఏҊ͕ͳ͞Ε͍ͯΔɻ

΍ϘκϯͰ͸ͳ͍ɻ͞Βʹɺvirtual ͳ࠶݁߹ϓϩηε͜ΕΒͷ‫ڀݚ‬Λ௨ͯ͠ɺࠓ‫ޙ‬ɺ෺ੑͷ஍ฏ͕͞Βʹ޿Λ௨ͯ͜͠ΕΒͷؒʹ૬‫͕༻࡞ޓ‬ੜ·ΕΔɻͦͷͨΊɺ͕Δ͜ͱ͕‫ظ‬଴͞ΕΔɻ

൒ಋମͷ৔߹ͱ͸ҟͳΔ૬͕ग़‫͢ݱ‬ΔՄೳੑ͕͋Δɻ࠷‫ʹޙ‬ɺඇฏߧ‫ݱ‬৅ͷ‫ڀݚ‬͸ଟ‫ʹذ‬Θ͓ͨͬͯΓɺ͜

࣮ࡍɺHubbard ໛‫ʹܕ‬೤ཋΛՃ͑ͨඇฏߧఆৗঢ়ଶ

͜Ͱ঺հͰ͖ͨ΋ͷ͸ɺ๲େͳ‫ڀݚ‬ͷҰ෦Ͱ͋Δɻ಺ͷ DMFT ͷղੳͰ͸ɺฏߧঢ়ଶͰ͸࣮‫͍ͳ͖Ͱݱ‬Α༰΋ɺภΓ͕ͳ͍΋ͷʹͳ͍ͬͯͳ͍͜ͱ͸͝༰͍ࣻ

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͖͍ͨͩͨɻ·ͨɺຊߘΛ௨ͯ͠ඇฏߧ෺ੑͷ෼໺ʹ‫ڵ‬ຯΛ͍͚࣋ͬͯͨͩͨͳΒ੣ʹ޾͍Ͱ͋Δɻ

(2018).
[25] L. Stojchevska et al., Science 344, 177 (2014).
[26] T. Mikami et al.,ࢀߟจ‫ݙ‬

Phys. Rev. B 93, 144307(2016).

[1] K. Yonemitsu and K. Nasu, Physics Reports[27] T. Mori, T. Kuwahara, and K. Saito, Phys. Rev.
Lett. 116, 120401 (2016).

465, 1 (2008).
[2] C. Giannetti et al., Advances in Physics 65, 58[28] N. Tsuji, T. Oka, and H. Aoki, Phys. Rev. B
78, 235124 (2008).

(2016).
[3] D. N. Basov, R. D. Averitt, and D. Hsieh, Nature Materials 16, 1077 (2017).
[4] T. Oka and S. Kitamura, Annual Review ofCondensed Matter Physics 10, 387 (2019).
[5] A. de la Torre et al., 2021, 2103.14888.
[6] R. Shimano and N. Tsuji, Annual Review ofCondensed Matter Physics 11, 103 (2020).
[7] R. Matsunaga et al.,Phys. Rev. Lett. 111,[29] K. Sandholzer et al.,Phys. Rev. Lett. 123,

193602 (2019).
[30] T. Ishikawa et al., Nature Communications 5,5528 (2014).
[31] T. Oka and H. Aoki, Phys. Rev. B 79, 081406(2009).
[32] T. Kitagawa, T. Oka, A. Brataas, L. Fu, and
E. Demler, Phys. Rev. B 84, 235108 (2011).
[33] J. H. Mentink, K. Balzer, and M. Eckstein, Na-

057002 (2013).
[8] R. Matsunaga et al., Science 345, 1145 (2014).

ture Communications 6, 6708 (2015).

[9] K. Katsumi et al., Phys. Rev. Lett. 120, 117001[34] G. Jotzu et al., Nature 515, 237 (2014).
[35] F. Görg et al., Nature 553, 481 (2018).

(2018).
[10] P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993).
[11] M. Lewenstein, P. Balcou, M. Y. Ivanov,
A. L’Huillier, and P. B. Corkum, Phys. Rev.
A 49, 2117 (1994).

[36] J. W. McIver et al.,Nature Physics 16, 38(2020).
[37] S. A. Sato et al.,Phys. Rev. B 99, 214302(2019).

[12] S. Ghimire et al., Nat. Phys. 7, 138 (2011).

[38] H. Aoki et al., Rev. Mod. Phys. 86, 779 (2014).

[13] S. Ghimire and D. A. Reis, Nat. Phys. 15, 10[39] M. Eckstein, M. Kollar, and P. Werner, Phys.
Rev. Lett. 103, 056403 (2009).

(2019).
[14] G. Vampa et al., Phys. Rev. Lett. 113, 073901

Rev. B 84, 054304 (2011).

(2014).
[15] U. Huttner, M. Kira, and S. W. Koch, Las.
Photon. Rev. 11, 1700049 (2017).
[16] S. Hellmann et al., Nature Communications 3,1069 EP (2012), Article.
[17] D. Fausti et al., Science 331, 189 (2011).
[18] S. Iwai et al.,Phys. Rev. Lett. 91, 057401(2003).

[41] N. Tsuji, M. Eckstein, and P. Werner, Phys.
Rev. Lett. 110, 136404 (2013).
[42] A. Georges, G. Kotliar, W. Krauth, and M. J.
Rozenberg, Rev. Mod. Phys. 68, 13 (1996).
[43] W. Metzner and D. Vollhardt, Phys. Rev. Lett.
62, 324 (1989).
[44] M. Eckstein and P. Werner, Phys. Rev. B 82,[19] H. Okamoto et al., Phys. Rev. B 82, 060513(2010).

115115 (2010).
[45] Y. Murakami, M. Eckstein, and P. Werner,[20] M. Rini et al., Nature 449, 72 (2007).
[21] S. Kaiser et al., Phys. Rev. B 89, 184516 (2014).
[22] W. Hu, S. Kaiser, D. Nicoletti, and C. Hunt,
Nature materials 13, 705 (2014).
[23] M. Mitrano et al., Nature 530, 461 (2016), Let-

Phys. Rev. Lett. 121, 057405 (2018).
[46] N. Tsuji and P. Werner,

Phys. Rev. B 88,

165115 (2013).
[47] Y. Murakami, P. Werner, N. Tsuji, and H. Aoki,
Phys. Rev. B 91, 045128 (2015).
[48] M. Ligges et al., Phys. Rev. Lett. 120, 166401

ter.
[24] A. Cavalleri,[40] M. Kollar, F. A. Wolf, and M. Eckstein, Phys.

Contemporary Physics 59, 31(2018).

《講義ノート》

物性研究・電子版 Vol. 10, No. 1, 101215(2022年3月号)[49] A. I. Lichtenstein and M. I. Katsnelson, Phys.

[70] N. Yoshikawa, T. Tamaya, and K. Tanaka, Science 356, 736 (2017).

Rev. B 62, R9283 (2000).
[50] G. Kotliar, S. Y. Savrasov, G. Pálsson, and[71] T. T. Luu et al., Nat. Comm. 9, 3723 (2018).

G. Biroli, Phys. Rev. Lett. 87, 186401 (2001).

[72] H. Kishida et al., Nature (London) 405, 929[51] M. H. Hettler, A. N. Tahvildar-Zadeh, M. Jarrell, T. Pruschke, and H. R. Krishnamurthy,
Phys. Rev. B 58, R7475 (1998).

(2000).
[73] S. Imai, A. Ono, and S. Ishihara, Phys. Rev.
Lett. 124, 157404 (2020).

[52] N. Tsuji, P. Barmettler, H. Aoki, and P. Werner,
Phys. Rev. B 90, 075117 (2014).

[74] Y. Murakami and P. Werner, Phys. Rev. B 98,
075102 (2018).

[53] M. Eckstein and P. Werner, 6, 21235 EP (2016),
Article.

[75] Y. Murakami, S. Takayoshi, A. Koga, and
P. Werner, Phys. Rev. B 103, 035110 (2021).

[54] D. Golež, L. Boehnke, H. U. R. Strand, M. Eck-[76] G. Vampa, C. R. McDonald, G. Orlando, P. B.

stein, and P. Werner, Phys. Rev. Lett. 118,

Corkum, and T. Brabec,

246402 (2017).

064302 (2015).

[55] D. Golez, L. Boehnke, M. Eckstein, and
P. Werner, Phys. Rev. B 100, 041111 (2019).
[56] H. Haug and S. W. Koch, Quantum Theory of
the Optical and Electronic Properties of Semi-

Phys. Rev. B 91,[77] L. V. Keldish and Y. V. Kopaev, Sov. Phys.
Solid State 6, 2219 (1965).
[78] D. Jérome, T. M. Rice, and W. Kohn, Phys.
Rev. 158, 462 (1967).
[79] ઙ໺‫ݐ‬Ұ, ిࢠਖ਼޸‫ܥ‬ͷ෺ཧ (෺ੑ‫ڀݚ‬ɾిࢠ൛

conductors (World Scientiﬁc, 1990).
[57] N. Schlünzen, J.-P. Joost, and M. Bonitz, Phys.
Rev. Lett. 124, 076601 (2020).

Vol.3ɺNo.1, 031207).
[80] Y. Murotani et al.,[58] D. Karlsson, R. van Leeuwen, E. Perfetto, andPhys. Rev. Lett. 123,

197401 (2019).

G. Stefanucci, Phys. Rev. B 98, 115148 (2018).

[81] E. Perfetto, D. Sangalli, A. Marini, and G. Ste-[59] D. Karlsson, R. van Leeuwen, Y. Pavlyukh,

fanucci, Phys. Rev. Materials 3, 124601 (2019).

E.

Perfetto,

andG.

Stefanucci,

arxiv:2006.14965.
[60] M. Schüler, M. Eckstein, and P. Werner, Phys.
Rev. B 97, 245129 (2018).
[61] J. Kaye and D. Golez, SciPost Phys. 10, 91(2021).
[62] M. Schüler et al., Computer Physics Communications 257, 107484 (2020).
[63] E. Dagotto, Rev. Mod. Phys. 66, 763 (1994).
[64] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003).
[65] J. A. Kjäll, M. P. Zaletel, R. S. K. Mong, J. H.
Bardarson, and F. Pollmann, Phys. Rev. B 87,
235106 (2013).
[66] K. Ido, T. Ohgoe, and M. Imada, Phys. Rev. B
92, 245106 (2015).
[67] M. Schmitt and M. Heyl, Phys. Rev. Lett. 125,
100503 (2020).
[68] T. T. Luu et al., Nature (London) 521, 498(2015).
[69] T. T. Luu and H. J. Wörner, Nat. Comm. 9,916 (2018).

[82] Y. Murakami, M. Schüler, S. Takayoshi, and
P. Werner, Phys. Rev. B 101, 035203 (2020).
[83] P. Werner and Y. Murakami, Phys. Rev. B 102,
241103 (2020).
[84] J. Kuneš, Journal of Physics: Condensed Matter 27, 333201 (2015).
[85] L. V. Keldysh, Contemporary Physics 27, 395(1986).
[86] A. Rosch, D. Rasch, B. Binz, and M. Vojta,
Phys. Rev. Lett. 101, 265301 (2008).
[87] J. Li, D. Golez, P. Werner, and M. Eckstein,Phys. Rev. B 102, 165136 (2020).
[88] N. Strohmaier et al.,Phys. Rev. Lett. 104,

080401 (2010).
[89] R. Sensarma et al., Phys. Rev. B 82, 224302(2010).
[90] M. Eckstein and P. Werner, Phys. Rev. B 84,
035122 (2011).
[91] C. N. Yang, Phys. Rev. Lett. 63, 2144 (1989).
[92] T. Kaneko, T. Shirakawa, S. Sorella, andS. Yunoki, Phys. Rev. Lett. 122, 077002 (2019).