早田智也 - Hiroshima UniversityT BZ dpxdt e x 2 Z Thouless, PRB 27, 6083 (1983) Nakajima...

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


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


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


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)


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



1

(2)

Z

TBZdp

x

dt ex 2 Z

Thouless, PRB 27, 6083 (1983)


ei = @piAt @tapi


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



1

(2)

Z

TBZdp

x

dt ex 2 Z

Thouless, PRB 27, 6083 (1983)


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)


è


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


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


=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)


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)


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)


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]


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()


可換è 非可換


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


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


• 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)



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


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


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


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



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



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)



カイラル磁気効果

断熱ポンピング

Nielsen-Ninomiya, PLB130, 389 (1983)


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


創発的対称性とトポロジー


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


@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


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)

早田智也 - Hiroshima UniversityT BZ dpxdt e x 2 Z Thouless, PRB 27, 6083 (1983) Nakajima...

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Transcript of 早田智也 - Hiroshima UniversityT BZ dpxdt e x 2 Z Thouless, PRB 27, 6083 (1983) Nakajima...

早田 智也 - Hiroshima UniversityT BZ dpxdt e x 2 Z Thouless, PRB 27, 6083 (1983) Nakajima...

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Transcript of 早田 智也 - Hiroshima UniversityT BZ dpxdt e x 2 Z Thouless, PRB 27, 6083 (1983) Nakajima...

早田智也 - Hiroshima UniversityT BZ dpxdt e x 2 Z Thouless, PRB 27, 6083 (1983) Nakajima...

Transcript of 早田智也 - Hiroshima UniversityT BZ dpxdt e x 2 Z Thouless, PRB 27, 6083 (1983) Nakajima...