Classical analog for exotic quantum holonomy

(This presentation is in Japanese)
題目：新奇な量子ホロノミーの古典対応物
日本物理学会 2017年秋季大会 23pJ24-10 での講演スライド

Transcript

1. 1 / 15 ৽حͳྔࢠϗϩϊϛʔͷݹయରԠ෺ Classical analog for exotic quantum holonomy ాதಞ࢘ (ट౎େཧ޻)ɺશ୎थ (ߴ஌޻Պେ) Atushi Tanaka (TMU) and Taksu Cheon (KUT) 2017-09-23 ೔ຊ෺ཧֶձ 2017 ೥ळقେձ ؠखେֶʢ্ాΩϟϯύεʣ 23pJ24-10
2. 2 / 15 ໨࣍ ʮ৽حͳྔࢠϗϩϊϛʔʯͷݹయՄੵ෼ܥͰͷରԠ෺ɺͭ·Γɺ ʮτʔϥε͕அ೤αΠΫϧʹΑͬͯผͷτʔϥεʹҠΔʯ͜ͱʹ͍ͭͯ ೋͭͷࣄྫΛ঺հ͢Δɻ ͸͡Ίʹ ઢܗΩοΫυεϐϯͰͷݫີͳྫ ඇઢܗΩοΫυεϐϯͰͷ਺஋ྫ ·ͱΊ
3. ͸͡Ίʹ 3 / 15 ৽حͳྔࢠϗϩϊϛʔ அ೤αΠΫϧʹର͢Δ (ٖ) ݻ༗ΤωϧΪʔ΍ݻ༗ۭؒͷඇࣗ໌ͳԠ౴ɻ (1) அ೤αΠΫϧ C ʹԊͬͯ (ٖ) ݻ༗Τω ϧΪʔΛ௥੻ͨ݁͠Ռɺݩʹ໭Βͳ͍͜ͱɻ λ 0 2π E E1 E2 E3 ਤ. λ ͕ 2π पظͷ৔߹ (2) ͢Δͱɺఆৗঢ়ଶΛஅ೤ดܦ࿏ C ʹԊͬͯ࣌ؒൃలͤ͞Δͱ ⟨࢝ঢ়ଶ|ऴঢ়ଶ⟩ = 0 ͱͳΔɻ Ref. AT and TC, ෺ཧֶձࢽ 4 ݄߸ (2017)ɻ
4. ͸͡Ίʹ 4 / 15 அ೤ঢ়ଶͷมԽΛ൒ݹయతʹߟ͑Δͱ. . . ԾఆɿରԠ͢Δݹయܥ͕Մੵ෼ͩͱ͢Δɻ Մੵ෼ܥͰ͸൒ݹయྔࢠԽ৚͔݅Βݻ༗ঢ়ଶͱτʔϥε͕ରԠ͢Δɿ C | 1 | 0 q p q p ࢝ঢ়ଶͷτʔϥε͸அ೤αΠΫϧ C ʹΑͬͯผͷτʔϥεʹҠΔ͸ͣɻ
5. ઢܗΩοΫυεϐϯͰͷݫີͳྫ 5 / 15 ໨࣍ ͸͡Ίʹ ઢܗΩοΫυεϐϯͰͷݫີͳྫ ඇઢܗΩοΫυεϐϯͰͷ਺஋ྫ ·ͱΊ
6. ઢܗΩοΫυεϐϯͰͷݫີͳྫ 6 / 15 ݹయઢܗΩοΫυεϐϯ εϐϯ S ʹ੩࣓৔ͱύϧε࣓৔ (पظ T) ΛҹՃɿ H(t) = B(t)·S, ͜͜Ͱ B(t) ≡ ωe z +λn∑ n δ(t −nT). n (୯ҐϕΫτϧ) ͱ λ ͸ύϧε࣓৔ͷํ޲ͱڧ͞ɻϙΞϯΧϨࣸ૾͸ɿ S′ = Mλ S, Mλ ͸ 3x3 ௚ަߦྻͰ͋Γճసߦྻͷੵʹ෼ղͰ͖Δ: Mλ = R(e z,ωT)R(n,λ) ͜͜ͰɺR(n,λ) ͸ճస࣠ n ʹ͍ͭͯ֯౓ λ ͷճసɻ Mλ+2π = Mλ ͔Βɺλ Λ 0 ͔Β 2π ʹ૿Ճͤ͞Δดܦ࿏ C Λௐ΂Δɻ
7. ઢܗΩοΫυεϐϯͰͷݫີͳྫ 7 / 15 ݹయઢܗΩοΫυεϐϯɿ਺஋ྫ ดܦ࿏ C ʹԊͬͯ λ Λ 0 ͔Β 2π ʹஅ೤తʹ૿΍ͨ͠৔߹ɻ ਤͷԣ࣠͸ํҐ֯ q ≡ arctan(Sy /Sx ). ਖ਼४ڞ໾ͳӡಈྔ p ≡ Sz ͕ॎ࣠ɻ அ೤αΠΫϧ C ʹΑͬͯɺτʔϥε͸ผͷτʔϥεʹҠͬͨɻ
8. ઢܗΩοΫυεϐϯͰͷݫີͳྫ 8 / 15 આ໌ 1: ճస࣠ͱճస֯΁ͷ෼ղͷᐆດ͔͞Β ઢܗΩοΫυεϐϯಛ༗ͷॳ౳తͳઆ໌ɻ (1) ઢܗͳͷͰϙΞϯΧϨࣸ૾͸ Mλ = R(l λ ,∆λ ) ͱ Ͱ͖Δɻ͜͜Ͱஅ೤తͳճస࣠ l λ ͱճస֯ ∆λ Λಋ ೖͨ͠ɻ͜ΕΒ͸ λ ʹ͍ͭͯ࿈ଓɻ (2) ௚઀తͳܭࢉ͔Β l 2π = −l 0, ∆2π = 2π −∆0. ͭ ·Γɺ͜ΕΒ͸ λ ʹ͍ͭͯ 2π पظͰ͸ແ͍ɻ͜Ε ͸ Mλ+2π = Mλ ͱໃ६͠ͳ͍͜ͱʹ஫ҙ (ӈਤ)ɻ (3) M0 ͷݻఆ఺ l 0 ͸அ೤αΠΫϧ C ͷ݁Ռͱͯ͠ −l 0 ʹҠΔɻτʔϥεͷҠಈ΋ಉ༷ɻ ∆0 2π − ∆0
9. ઢܗΩοΫυεϐϯͰͷݫີͳྫ 9 / 15 આ໌ 2: spin-1 2 ͰͷݫີͳྔࢠݹయରԠ εϐϯ-1 2 ಛ༗ͷઆ໌ɻ ͜ͷͱ͖ྔࢠܥͷظ଴஋ͷӡಈํఔࣜͱݹయܥͷӡಈํఔࣜ͸ಉ͡ܗɿ d dt S = B(t)×S (͍ΘΏΔϒϩοϗํఔࣜ). ͜ΕͱɺྔࢠܥͰஅ೤αΠΫϧ C ͕৽حͳྔࢠϗϩϊϛʔΛى͜͢͜ͱ ͔ΒɺݹయܥͰ΋τʔϥε͕Ҡಈ͢Δɻ ͳ͓ɺྔࢠܥͷϑϩέ࡞༻ૉ͕ C Λดܦ࿏ʹ࣋ͭͨΊʹ͸ɺ͜ͷྫͷݹయܥͷ ϋϛϧτχΞϯʹʮ͓·͚ʯ͕ඞཁ (AT and Miyamoto 2007): Hྔࢠ(t) = Hݹయ(t)+ 1 2 ¯ hλ ∑ n δ(t −nT).
10. ઢܗΩοΫυεϐϯͰͷݫີͳྫ 10 / 15 આ໌ 3: Ґ૬زԿֶతͳઆ໌ʢγφϦΦʣ ͜ͷܥͷτʔϥεશͯ (༿૚ߏ଄) Λߟ͑Δɻ͋Δτʔϥε͔Βग़ൃͯ͠ɺ அ೤αΠΫϧͰͨͲΓ͖ͭಘΔͷ͸ɺ࡞༻ม਺ I = τʔϥε p dq ͷ஋͕ಉ ͡τʔϥεͷΈɻ ͪͳΈʹɺεϐϯͰ͸ I ∝ ʢτʔϥε͕ுΔཱମ֯ʣͱͰ͖Δɻ ઢܗΩοΫυεϐϯͰ͸ɺ͜ͷΑ͏ͳτʔϥε͸ࣗ෼ ΛؚΊೋͭͷΈ (ॖୀ͠ಘΔ͕ྫ֎త)ɻ ͜ͷτʔϥεͷ૊͸ඃ෴ۭؒΛͳ͢ɻ͢Δͱɺ৽حͳ ྔࢠϗϩϊϛʔͷҐ૬زԿֶతͳఆࣜԽ (AT and TC 2015) Λར༻Ͱ͖Δɻྫ͑͹Ұ఺ͱϗϞτϐοΫͳஅ ೤αΠΫϧ͸τʔϥεΛҠಈͤ͞Δ͜ͱ͸ෆՄೳɻ
11. ඇઢܗΩοΫυεϐϯͰͷ਺஋ྫ 11 / 15 ໨࣍ ͸͡Ίʹ ઢܗΩοΫυεϐϯͰͷݫີͳྫ ඇઢܗΩοΫυεϐϯͰͷ਺஋ྫ ·ͱΊ
12. ඇઢܗΩοΫυεϐϯͰͷ਺஋ྫ 12 / 15 ඇઢܗΩοΫυεϐϯ ۙՄੵ෼ܥͰͷྫΛ঺հ͢ΔɻۙՄੵ෼ܥʹ͸Ұൠతͳஅ೤ఆཧ͕ແ͍ ͜ͱʹ஫ҙɻ H(t) = B(t)·S ͱͯ͠ɺB(t) ͕ऑ͘ Sz ʹґଘ͢Δ৔߹Λߟ͑Δ: H(t) = [ ωSz + 1 2 kS2 z ] +λn·S∑ n δ(t −nT). cf. Haake, K´ us and Scharf 1987. ϙΞϯΧϨࣸ૾͸ඇઢܗʹͳΔ͕ɺλ ͷपظੑ͸มΘΒͳ͍ͷͰಉ͡அ ೤αΠΫϧ C Λ࢖͏ɻ
13. ඇઢܗΩοΫυεϐϯͰͷ਺஋ྫ 13 / 15 ඇઢܗ ΩοΫυεϐϯɿ਺஋ྫ (k = 0.1) C ʹԊͬͨஅ೤తͳ࣌ؒൃలɻ͜͜Ͱ (q,p) = (arctan(Sy /Sx ),Sz). அ೤αΠΫϧ C ʹΑͬͯɺτʔϥε͸ผͷτʔϥεͷۙ͘ʹҠͬͨɻ
14. ·ͱΊ 14 / 15 ·ͱΊ ▶ ৽حͳྔࢠϗϩϊϛʔͷݹయՄੵ෼ܥͰͷରԠ෺Λಘͨɻ͢ͳΘͪɺ அ೤αΠΫϧ C ʹΑͬͯɺݹయܥͷτʔϥε͕ผͷτʔϥεʹҠΔ Α͏ͳྫΛݟ͍ͩͨ͠ɻ ▶ ઢܗΩοΫυεϐϯͰͷݫີͳྫʹ͍ͭͯɺࡾͭͷղऍΛ༩͑ͨ (ճసߦྻ͔Βɺεϐϯ-1/2 ͷྔࢠݹయରԠɺҐ૬زԿֶతͳղऍ)ɻ ▶ ඇՄੵ෼ܥͷ਺஋ྫΛࣔͨ͠ɻ [͜Ε͸Մੵ෼ۙࣅͰղऍͰ͖ΔʢՄੵ෼ۙࣅʹ͍ͭͯ͸ྫ͑͹ɺ Hanada, Shudo and Ikeda, PRE 2015ʣ ɻ]
15. ·ͱΊ 15 / 15 ల๬ ۩ମྫΛ૿΍͢ Euler top, αΠΫϩτϩϯӡಈͷ֦ு౳ɻ Ґ૬زԿֶతͳղऍͷ੔උ ʮ༿૚ߏ଄ͷύϥϝʔλʔมܗʹର͢ΔϞϊυϩϛʔʯͱղऍͰ͖ Δͱ༧૝ɻ ࡞༻ม਺ͷ஋ΛมԽͤ͞Δஅ೤αΠΫϧ͸ଘࡏ͢Δ͔ʁ ৽حͳྔࢠϗϩϊϛʔ͔Β͸ଘࡏͯ͠ ΋ྑͦ͞͏͕ͩɺ͍·ͷͱ͜Ζଘ൱͸ ෆ໌ɻ C | 1 | 0 q p q p