早田 智也 - hiroshima universityt bz dpxdt e x 2 z thouless, prb 27, 6083 (1983) nakajima...
TRANSCRIPT
References: TH-Hidaka, PRB 95 125137 (2017); PTEP 073I01 (2017),Ishizuka-TH-Ueda-Nagaosa, PRL 117 216601 (2016); PRB 95, 245211 (2017),TH-Kikuchi-Tanizaki, PRB 96, 085112 (2017)
ベリー曲率と異常輸送現象 早田 智也
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
絶縁体の低エネルギー有効理論
• フェルミ準位近傍に自由度無し
Qi-Hughes-Zhang , PRB 78, 195424 (2008)
トポロジーと輸送現象
生成汎関数は位相的場の理論を用いて現せる
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
fpx
py
= @px
apy
@py
apx
ベリー曲率
Jx
=e2
2~C1xyEy
• (磁気的)ブリュアンゾーンのトポロジー
ap = u†irpu
Thouless-Kohmoto-Nightingale-Nijs, PRL 49, 405 (1982)
C1 = 1
2
Zdp
x
dpy
fp
x
p
y
2 Z
量子ホール効果
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
SCS =e
2
2~C1
2
ZdtdxdyAµ
µ@A
Jµ =@SCS
@Aµ=
e2
2~C1µ@A
Zhang-Hansson-Kivelson, PRL 62, 82 (1989) Lopez-Fradkin, PRB 44, 5246 (1991)
チャーン-サイモンズ理論
Jx
=e2
2~C1xyEy
Thouless-Kohmoto-Nightingale-Nijs, PRL 49, 405 (1982)
C1 = 1
2
Zdp
x
dpy
fp
x
p
y
2 Z
量子ホール効果
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
周期駆動されたハミルトニアン サウレス(断熱)ポンピング Thouless, PRB 27, 6083 (1983)
H =X
i
(t+ ()i)a†i+1ai +()ia†iai + (h.c.)
Nakajima et.al., Nat. Phys. 12, 296 (2016)
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
サウレス(断熱)ポンピング e =
1
(2)
Z
TBZdp
x
dt ex 2 Z
Thouless, PRB 27, 6083 (1983)
Nakajima et.al., Nat. Phys. 12, 296 (2016)
ベリー曲率 At = u†i@tu
ei = @piAt @tapi
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
サウレス(断熱)ポンピング e =
1
(2)
Z
TBZdp
x
dt ex 2 Z
Thouless, PRB 27, 6083 (1983)
ベリー曲率 At = u†i@tu
ei = @piAt @tapi
Nakajima et.al., Nat. Phys. 12, 296 (2016)
Lohse et.al., Nat. Phys. 12, 350 (2016)
冷却原子系を用いた実験的観測
t/Tx
G
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
サウレス(断熱)ポンピング e =
1
(2)
Z
TBZdp
x
dt ex 2 Z
Thouless, PRB 27, 6083 (1983)
ベリー曲率 At = u†i@tu
ei = @piAt @tapi
チャーン-サイモンズ理論
SCS =1
4
Zdtdxdp
x
A
µ
µ
@
A
Aµ
= (At
, Ax
, ap
x
)
Jx =@SCS
@Ax
=1
2ex
Qi-Hughes-Zhang , PRB 78, 195424 (2008)
TH-Hidaka, PRB 95 125137 (2017)
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
è
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
b±(p) rp a±(p) = p/2p3
ベリー曲率 a±p = u±(p)†irpu
±(p)
ワイルハミルトニアン
運動量空間上の“磁気” モノポール
エネルギー固有値
p · u±p = ±|p|u±
p
H = †p p · p
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
磁場に依存しないホール効果
異常ホール効果Sundaram- Niu, PRB 59, 14915 (1999) Jungwirth-Niu-MacDonald, PRL 88, 207208 (2002)
時間反転対称性の破れた物質強磁性金属、強磁性半導体、トポロジカル絶縁体の薄膜…
Jx
=e2
2~ C1xyEy
C1 = 1
2
Zdp
x
dpy
fp
x
p
y
nF.D.
/2 Z
Lee et.al., Science 303, 1647 (2004)
xy
= R0H + sxy
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
=1
2
Z
F.S.d2S · b 2 Z
j = e2µ5B/22c
モノポール電荷
カイラル磁気効果 磁場に比例する電流 Nielsen-Ninomiya, PLB130, 389 (1983)
Fukushima-Kharzeev-Warringa, PRD 78, 074033 (2008)
= e2
42cµR,LB
2µ5 = µR µLカイラル化学ポテンシャル
j = e2
cB
Zd3p
(2)3"b ·rpnF.D.
Stephanov- Yin, PRL 109, 162001 (2012)
Son- Yamamoto, PRL 109, 181602 (2012)
Chen-Pu-Wang-Wang, PRL 110, 262301 (2013)
TH-Kikuchi-Tanizaki, PRB 96, 085112 (2017)
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
j = e2µ5B/22c
負性磁気抵抗効果
カイラル磁気効果 磁場に比例する電流 Nielsen-Ninomiya, PLB130, 389 (1983)
Fukushima-Kharzeev-Warringa, PRD 78, 074033 (2008)
2µ5 = µR µLカイラル化学ポテンシャル
µ5 E ·B
Li, et.al., Nat. Phys. 12, 550 (2016)
LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS3648
0.0
0.5
1.0
1.5
2.0
2.5a
(mΩ
cm
)ρ
0.0
0.000.02
a
0.040.06
0.5
1.0
1.5
2.0
(mΩ
cm
)ρ
−9 −6 −3 0
150 K100 K
70 K
60 K
40 K
20 K
5 K
3 6 9B (T)
−9 −6 −3 0 3 6 9B (T)
0 40 80T (K)
T = 20 K
b
Figure 2 | Magnetoresistance in magnetic fields parallel to the current (B ka) in ZrTe5. a, Magnetoresistance at various temperatures. For clarity, theresistivity curves have been shifted by 1.5 m cm (150 K), 0.9 m cm (100 K), 0.2 m cm (70 K) and 0.2 m cm (5 K). b, Magnetoresistance at 20 K(red symbols) fitted with the CME curve (blue line); inset: temperature dependence of the parameter a(T) in units of S cm1 T2 for B and E along thea-axis (red solid symbols), and 18 o the a-axis within the a–c plane (open circles). The line represents the dependence predicted by equation (8).
magnetoresistance21. It shows a semimetallic electronic structurewith extremely small and light ellipsoidal Fermi surface(s), centredat the centre (0 point) of the bulk Brillouin zone (BZ). Thecalculations predict a small direct gap at 0(=50meV), but previoustransport studies show semimetallic behaviour, with quantumoscillations indicating a tiny but finite Fermi surface22–25. Quantumoscillations show that the eective mass in the chain direction(m
a/me = 0.03) is comparable to that in a prototypical 3D Diracsemimetal, Cd3As2 (refs 24,26).
Figure 1a shows the temperature dependence of resistivity alongthe chain direction (a) in a magnetic field perpendicular to thea–c plane. The zero-field transport shows a characteristic peak in re-sistivity at T =60K, significantly lower than in earlier studies, prob-ably due to a much lower impurity concentration in our samples20.InBkb, the peak shifts to higher temperatures andwe observe a verylarge positive magnetoresistance in the whole temperature range,consistent with previous studies21. The inset shows the resistivitywith the magnetic field applied along the three major crystal axes.
Figure 1c,d shows the magnetoresistance measured at 20K forseveral angles of the applied magnetic field with respect to thecurrent along the chain direction. The angle rotates from the b-axisto the a-axis, so that at ' = 90 the field is parallel to the current(B k a)—the so-called Lorentz-force-free configuration. When themagnetic field is aligned along the b-axis (' =0), the magnetore-sistance is positive and quadratic in low fields, and tends to saturatein high fields. The similar angular dependence is also observed forthe field in the c–a plane. When the magnetic field is rotated awayfrom the b- or c-axis, the positive magnetoresistance drops withcos , as expected for the Lorentz force component. However, inthe Lorentz-force-free configuration (Bka), we see a large negativemagnetoresistance, a clear indication of CME in this material.
Figure 2 shows the magnetoresistance at various temperatures ina magnetic field parallel to the current. At elevated temperatures,T 110K, the versus B curves show a small upward curvature, acontribution from an inevitable perpendicular field component dueto imperfect alignment between the current and the magnetic field.In fact, the small perpendicular field contribution to the observedresistivity can be fittedwith a simple quadratic term (SupplementaryInformation and Supplementary Fig. 1). This term is treated as abackground and subtracted from the parallel field component forall magnetoresistance curves recorded at T 100K.
A negative magnetoresistance is observed for T 100K,increasing in magnitude as temperature decreases. We found thatthe magnetic field dependence of the negative magnetoresistancecan be nicely fitted with the CME contribution to the electrical
conductivity, given by CME =0 +a(T )B2, where 0 represents thezero-field conductivity. The fitting is illustrated in Fig. 2b forT =20K, with an excellent agreement between the data and theCME fitting curve. At 4 T, the CME conductivity is about the sameas the zero-field conductivity. At 9 T, the total magnetoconductivityincreases by 400% over the zero-field value as a result of the CMEcontribution. This results in a negative magnetoresistance that ismuch stronger than any conventional magnetoresistance reportedat an equivalent magnetic field in a non-magnetic material. Todemonstrate that the eect is not limited to any specific crystaldirection, but locked only to the current direction, we have alsoperformed magnetoresistance measurements with the current stillin the a–c plane, but 18 o the chain direction (a-axis). In thisdirection, we observe the same /B2 dependence of CME when themagnetic field is parallel to the current.
To obtain a numerical theoretical estimate for CME, we takeT =20K, Ohm
= 1.2m cm indicated by our measurements atB=0, and assume that the rate of chirality-changing transitions1V =/~. =50meV and µ=9meV is determined from the fitto the data. The ARPES data indicate that v =1/300 c. Evaluatingequation (8), we reproduce the correct order of magnitude of thecoecient a(T ) thatwe used to fit the quadratic dependence ofmag-netoresistance onmagnetic field. The inset in Fig. 2b shows the tem-perature dependence of parameter a(T ) for B and E along the a-axis(red solid symbols) and 18 o the a-axis, in the a–c plane (open cir-cles), together with the dependence (line) predicted by equation (8)in which the eective value of µ is found to be µ9meV.
At very low field, the data show a small cusp-like feature. The ori-gin of this feature is not completely understood, but it probably indi-cates some formofweak anti-localization (WAL). This, and the largepositive magnetoresistance for magnetic field perpendicular to cur-rent, seem to be common features in all the materials that were sub-sequently found to exhibit the CME in parallel fields27–32. Althoughthe reason for a simultaneous observation of CME in parallel fieldsand large magnetoresistance in perpendicular fields is not fully un-derstood yet, we note that the general requirements for the observa-tion of CME—that is, the low carrier concentration and nearly linearbands—seem to be also favourable for the observation of large pos-itive magnetoresistance in perpendicular fields. In these situations,the system is either very close to the quantum limit33 or, if it containstwo types of carriers, the charge compensation may lead to a largeresponse in theway that was recently observed inWTe2 (refs 34–36).
A necessary requirement for observation of the CME isthat a material has a 3D Dirac semimetal-like (zero gap), orsemiconductor-like (non-zero gap) electronic structure. Figure 3
552
© 2016 Macmillan Publishers Limited. All rights reserved
NATURE PHYSICS | VOL 12 | JUNE 2016 | www.nature.com/naturephysics
• ディラック半金属
• ワイル半金属
B (T)
(m
cm
)
Huang, et. al., PRX 5, 031023 (2015)
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
カイラル磁気効果荷電粒子の方位角相関
“Constraints on the chiral magnetic effect using charge-dependent azimuthal correlations in p-Pb and Pb-Pb collisions at the LHC,” CMS Collaboration, arXiv:1708.01602v1 [nucl-ex]
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
運動量空間における電磁誘導
駆動されたワイルノード 光電流
Ishizuka-TH-Ueda-Nagaosa, PRL 117 216601 (2016); PRB 95, 245211 (2017)
px
py
pz
rp e = @tb 4jm
e =
Z T
0dt
Zd3p
(2)3enF.D.
サウレス(断熱)ポンピング
ezR,L = ±4v20
15v2v2z(gD)
2! cos()
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
可換è 非可換
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
SU(2) ベリー曲率
ディラックハミルトニアン
• 二重縮退
↵ =
0 0
=
1 00 1
p
amn = u†mirpun
H = ↵ · p+ m
f = rp a ia a
"(p) = ±p
p2 +m2
f = m
2"(p)3
+
p · m2 +m"(p)
p
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
SU(2) ベリー曲率
ディラックハミルトニアン
• 二重縮退
↵ =
0 0
=
1 00 1
p
H = ↵ · p+ m"(p) = ±
pp2 +m2
p ! R(p,x, t) m ! m(p,x, t)
f ! fpt
, fpx
fxy
スピン電磁場
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
• スピン電場によるホール効果
異常ホール効果
J =e
2
Zd3p
(2)3
hEa ba
in(p)
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
異常ホール効果
J =e
2
Zd3p
(2)3
hEa ba
in(p)
カイラル磁気効果
J =e
2
Zdtd3p
(2)3
hBa(ba ·rp") + eB(ea · ba)
in(p)
• スピン磁場によるカイラル磁気効果• 磁場で向き制御された断熱ポンピング
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
2µ5
カイラル化学ポテンシャルのダイナミクス
スペクトラルフロー
µ
E ·B
散乱
@tn+rx
· j =e2
42cE ·B n
Son-Spivak, PRB 88, 104412 (2013)
b
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
2µ5µx
B
ea · ba
磁場で向き制御された断熱ポンピング 実空間上のスペクトラルフロー
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
スピンホール効果
ラッティンジャー模型
è非散逸的なスピン輸送Murakami-Nagaosa-Zhang, Science 301 1348 (2003)
Murakami-Nagaosa-Zhang, PRL 93 156804 (2004)
J as =
Zd3p
(2)3
heE ba
in(p)
H = "(p) +5X
i=1
da(p)a
a,b = 2ab
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
スピンホール効果
スピンカイラル磁気効果
J as =
Zd3p
(2)3
heE ba
in(p)
J as =
Zdtd3p
(2)3
heB(ba ·rp")
+3
10
Ba(eb · bb) + (bab) + (bba)
in(p)
• 磁場によるスピンカレントのカイラル磁気効果• スピン磁場で向き制御された断熱スピンポンピング
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
スピンホール効果
スピンカイラル磁気効果
J as =
Zd3p
(2)3
heE ba
in(p)
J as =
Zdtd3p
(2)3
heB(ba ·rp")
+3
10
Ba(eb · bb) + (bab) + (bba)
in(p)
断熱スピンポンピング
J as =
Zdtd3p
(2)3
hea
in(p)
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
異常ホール効果
カイラル磁気効果
断熱ポンピング
Nielsen-Ninomiya, PLB130, 389 (1983)
Fukushima-Kharzeev-Warringa, PRD 78, 074033 (2008)
Thouless, PRB 27, 6083 (1983)
j = e2Z
d3p
(2)3(bE)n
e =
Z T
0dt
Zd3p
(2)3en
Sundaram- Niu, PRB 59, 14915 (1999) Jungwirth-Niu-MacDonald, PRL 88, 207208 (2002)
j = e2B
Zd3p
(2)3"b ·rpn
= e2
42µR,LB
まとめ
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
創発的対称性とトポロジー
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
p = eE x e
cB
トポロジーとボルツマン方程式
• 運動量空間上のローレンツ力
@tn+ x ·rx
n+ p ·rp
n = I(n)
• 分布関数 n(t,x,p)
v = rp"
Sundaram- Niu, PRB 59, 14915 (1999)
Xiao-Chang-Niu, RMP 82, 1959 (2010)
x = v p b
ボルツマン方程式
運動方程式
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
@tn+ x ·rx
n+ p ·rp
n = I(n)
n(t,x,p)
v = rp"
Sundaram- Niu, PRB 59, 14915 (1999)
Xiao-Chang-Niu, RMP 82, 1959 (2010)
p!x = v + eE b+ (v · b) e
cB
p!p = eE v e
cB
eE · e
cBb
p! = 1 +
e
cB · b
トポロジーとボルツマン方程式 ボルツマン方程式
運動方程式
• 分布関数
• 不変測度
Tomoya Hayata 熱場の量子論とその応用@ YITP, 28 Aug, 2017
電流密度
生成汎関数
Pf(!) = a1···a2d!a1a2 · · ·!a2d1a2d/2dd!
=aba1···a2d2
(2)d2d1(d 1)!!a1a2 · · ·!a2d1a2d2!btn(t, )
ja = Pf(!)an(t, )/(2)d
SCS =
Zdtd2d
(2)d(d+ 1)!µ0...µ2dAµ0@µ1Aµ2 · · · @µ2d1Aµ2d
n(t, ) = 1
Bulmash-Hosur-Zhang-Qi, PRX 5, 021018 (2015)
位相空間上の位相的場の理論
TH-Hidaka, PRB 95 125137 (2017)