Õ mì ^ ; w K ° Í i Ñ £ ç Û ¦ ï Û w ¼ Æ Û « µmercury.yukawa.kyoto-u.ac.jp/~bussei.kenkyu/pdf/04/4/... · 2017-09-04 · 1 Ö 1.1 \ w ~ G å w , Hall % µ Ð ï ÷

$Page 1: Õ mì ^ ; w K ° Í i Ñ £ ç Û ¦ ï Û w ¼ Æ Û « µmercury.yukawa.kyoto-u.ac.jp/~bussei.kenkyu/pdf/04/4/... · 2017-09-04 · 1 Ö 1.1 \ w ~ G å w , Hall % µ Ð ï ÷$
∗

†

Majorana

, Kitaev . ,

infinite time evolving block decimation (iTEBD) ,

, ,

. , ,

. ,

( ) , .

, ,

.

. Bogoliubov

. , .

, Majorana ,

. Bogoliubov ,

.

∗ , .† E-mail: [email protected]

物性研究・電子版　Vol. 4, No. 4, 044601 （2015年11月号）《修士論文》

1 3

1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 10

2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Infinite time evolving block decimation . . . . . . . . . . . . . . . . . . . . . . 21

3 28

3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 . . . . . . . . . . . . . . . . . . . . . . . . 33

4 37

4.1 Bogoliubov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 45

A Schmidt 46

B AKLT 47

C Majorana 48

D 49

E Bogoliubov 50

2


1

1.1

, Hall ,

, . Landau

,

[1–8]. ,

[9–12].

, Li Haldane [13]

,

, . , Hall

. ,

[14–24]. ,

. 1 ,

[14].

, Valence Bond Solid (VBS)

,

[16, 17]. , ,

[20]. , AdS/CFT

( , [25] . [26]).

,

p Kitaev [27]. .

, Majorana

, . Majorana

, . Kitaev

[28, 29] , , Cooper

, .

, Majorana ,

. , .

Kitaev ,

[30]. ,

, .

. ,

Majorana , .

3


, .

, [31–33]. ,

Localizable entanglement [34] ,

. , ,

[35, 36].

.

. ,

( ) [37–39].

, ,

. Kibble–Zurek

[40]. ,

, Kibble–Zurek [41–43].

,

. , Majorana

. , ,

. ,

, .

: , ,

. 2 , . ,

Jordan–Wigner Majorana

, ( ) . ,

infinite time evolving block decimation (iTEBD) . 3

,

. 4 , .

,

. 5 .

.

1.2

(

[44, 45] ). , , ,

. ,

. ,

[46]. , Hall , Landau

.

, .

4


1.1 : A,B , A Lsub .

, .

, [47].

infinite time evolving block decimation (iTEBD) [48]

[49] ,

.

, .

A,B , ( 1.1). A

Lsub . , Schmidt ( A) ,

|ΨAB〉 A B :

|ΨAB〉 =χ∑

i=1

λi|φi〉A|ψi〉B . (1.1)

, {|φi〉A} {|ψi〉B} A B , {λi}∑χi=1 λ

2i = 1 . χ Schmidt , dimHα (α = A,B) α

Hilbert , χ ≤ min{dimHA,dimHB} .

, A ρA ,

ρA = TrB |ΨAB〉〈ΨAB | (1.2)

=

χ∑i=1

λ2i |φi〉A〈φi|A (1.3)

. , B , B .

A , B ,

ρA von Neumann , :

SA = −χ∑

i=1

λ2i lnλ

2i . (1.4)

, SA B ρB SB

. , A, B ,

. ,

. A, B

5


.|Ψ〉 = | ↑1〉| ↑2〉 (1.5)

χ = 1 , λ1 = 1 , 0 . ,

|Ψ〉 = 1√2(| ↑1〉| ↓2〉 − | ↓1〉| ↑2〉) (1.6)

χ = 2 , λ1 = λ2 = 1√2

, ln 2 .

,

.

[46].

,

. , Lsub ,

. , .

, , iTEBD

. , ,

[46, 50]. :

, , . ,

,

. ,

Ising

HTFIM = −N∑i=1

σxi σ

xi+1 − h

N∑i=1

σzi (1.7)

, h = 1 ( 1.2). ,

(1+1) , , A Lsub

SA(Lsub) ∼ c

3lnLsub (1.8)

[51]. c ,

. , c = 1, c = 12 . ,

Ising c = 12 , .

[13]. ρA

ρA =

χ∑i=1

λ2i |φi〉A〈φi|A (1.9)

= exp(−HE) (1.10)

, 1, HE

.

6


0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 50 100 150 200 250

SA

(Lsu

b)

Lsub

h=0.6h=0.8h=1.0

1.2 Ising

[46]: h = 0.6, 0.8 ,

. , h = 1

. 503, 249.

:

ξi = − lnλ2i . (1.11)

, {λi}. , Hall

, Li Haldane

[13]. , , Hall

. ,

, [14–24].

, .

( 1.3) . ,

. 1

Haldane [52, 53] , 12 ( B).

, [14]. 1.4

Haldane 1

H =N∑i=1

Si · Si+1 (1.12)

. , 2.1.1 Kitaev ,

Majorana .

(3.3 ).

. Haldane

[1] , Oα(L)

7


1.3 : ( ) , A,B ( )

. 1 Heisenberg (1.12) Haldane ,

.

0

2

4

6

8

10

12

14

16

18

20

0 5 10 15 20 25 30 35 40

ξ i

i

ξi

1.4 : Haldane 1

Heisenberg , iTEBD .

(2.3.1 ) χb 40 . . , 1.3

.

.

Oα = limL→∞

Oα(L), (1.13)

Oα(L) =

⟨Sα1

L−1∏k=2

exp(iπSαk )S

αL

⟩, for α = x, y, z. (1.14)

1 Heisenberg , iTEBD

, ( 1.5) . ,

,

[3]. , Z2 × Z2

. ,

.

8


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

20 40 60 80 100 120 140 160 180 200

Str

ing c

orr

ela

tion f

uncti

on

L

Ox

1.5 : Haldane 1 Heisenberg

(1.12) , iTEBD (χb = 40).

.

9


2

2.1

. 2.1.1

Kitaev , 2.1.2 Majorana

, . 2.1.3

. , N ,

, Pauli σαN+1 = σα

1 , σαN+2 = σα

2 (α = x, y, z), ,

σαN+1 = σα

N+2 = 0 (α = x, y, z) .

2.1.1

Ising HTFIM , Jordan–Wigner

ci =i−1∏j=1

(−σzj )σ

−i , c†i =

i−1∏j=1

(−σzj )σ

+i (2.1)

, Kitaev ( p ) HKitaev [27]:

HTFIM = −N∑i=1

(σxi σ

xi+1 + λσz

i ) (2.2)

= −N∑i=1

[(c†i − ci)(c

†i+1 + ci+1) + 2λc†i ci

](2.3)

= HKitaev. (2.4)

, cic†i+1 (c†i ci+1) , cici+1 (c†i c

†i+1) , c†i ci

. Kitaev . 2.1.2

, Majorana

. λ , . λ < 1

( ) , Ising . ,

Majorana , . ,

λ > 1 , Ising .

, Majorana , .

, HC

HC = −N∑i=1

JXZXσxi σ

zi+1σ

xi+2 (2.5)

[30,54]. Kitaev , Jordan–Wigner ,

10


:

HC = −N∑i=1

JXZXσxi σ

zi+1σ

xi+2 (2.6)

=N∑i=1

JXZX(c†i − ci)(c†i+2 + ci+2). (2.7)

( ) .

, Kitaev ,

p . ,

Majorana ,

.

2.1.2 Majorana

Kitaev Majorana . Majorana

c2i−1 = c†i + ci, c2i = i (ci − c†i ), i = 1, 2, , N (2.8)

, c , :

ci = c†i , {ci, cj} = 2δij . (2.9)

i Jordan–Wigner ci Majorana c2i−1, c2i

2.1 . Majorana Kitaev

HKitaev = −N∑i=1

[(c†i − ci)(c

†i+1 + ci+1) + 2λc†i ci

](2.10)

= −iN∑i=1

(c2ic2i+1 − λc2i−1c2i) (2.11)

. , Majorana 2.2 .

(c2i−1c2i), (c2ic2i+1) .

λ = 0 , c1 c2N , .

, Majorana . λ < 1 (

) , 2.2 ,

Majorana ( C).

, . , λ > 1 ( ) ,

2.2 , Majorana

.

11


2.1 Majorana : Jordan–Wigner ci Majorana

c2i−1, c2i .

2.2 Kitaev Majorana : Majorana

. (c2i−1c2i),

(c2ic2i+1) . (λ ),

.

, . (2.5)

Majorana {ci}

HC = i

N∑i=1

JXZX c2ic2i+3 (2.12)

. 2.3 . Majorana

. , 2N Majorana

Majorana . , ,

( ) Majorana

. Majorana (

) . 2.2.1 , Majorana

. , Majorana 2.4

12


2.3 Majorana : Majorana

. , Majorana

.

2.4 Majorana : Majorana c1

( ), c2N−2 ( ) (N = 101).

. Majorana .

3 Kitaev ,

. Ising ,

H =

N∑i=1

(−hσzi + JXXσx

i σxi+1 − JXZXσx

i σzi+1σ

xi+2 (2.13)

+ JY Y σyi σ

yi+1 + JY ZY σy

i σzi+1σ

yi+2)

. Majorana , :

H =iN∑i=1

(hc2i−1c2i + JXX c2ic2i+1 + JXZX c2ic2i+3

− JY Y c2i−1c2i+2 + JY ZY c2i−1c2i+4). (2.14)

Majorana , 2.5 .

Majorana ( ) .

13


2.5 Majorana : Majorana

. Majorana .

Ψ = (c1, c2, . . . , c2N−1, c2N )T ,

H =i

2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 h 0 −JY Y 0 JY ZY . . .−h 0 JXX 0 JXZX 0 . . .

0 −JXX 0 h. . .

JY Y 0 −h 0. . .

0 −JXZX . . .. . .

−JY ZY 0. . .

. . ....

...

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(2.15)

2N . , (2.15)

( C). , JY Y = JY ZY = 0 ,

[59]. , ,

.

2.1.3

2.1.2 , Majorana

. ,

. .

, L

OXZX(L) = (−1)L

⟨σx1σ

y2

⎛⎝L−2∏

j=3

σzj

⎞⎠σy

L−1σxL

⟩(2.16)

[54].

OXZX = limL→∞

OXZX(L) (2.17)

14


.

, Ising

H(λ) =N∑i=1

(−σxi σ

zi+1σ

xi+2 + λσy

i σyi+1) (2.18)

μzi = σx

i σxi+1, μx

i =i∏

j=1

σzj , σx

N+1 = 1 (2.19)

( D). , μ

OXZX(L) = (−1)L⟨μy1μ

yL−1

⟩(2.20)

. Haldane , ,

(2.16) , (2.20) .

z π

HC∗ =

N∑i=1

σyi σ

zi+1σ

yi+2 (2.21)

. ( ) .

,

OY ZY (L) =

⟨σy1σ

x2

⎛⎝L−2∏

j=3

σzj

⎞⎠σx

L−1σyL

⟩(2.22)

, .

2.2

2.2.1 , Lieb–Schultz–Mattis [55] , ,

. , , .

2.2.2 . 2.2.3 , Bogoliubov ,

.

2.2.1

, Jordan–Wigner

. ,

[46,55] .

, Jordan–Wigner :

ci =i−1∏j=1

(−σzj )σ

−i , c†i =

i−1∏j=1

(−σzj )σ

+i . (2.23)

15


,

σzi = 2c†i ci − 1, (2.24)

σxi σ

xi+1 = (c†i − ci)(c

†i+1 + ci+1), (2.25)

σyi σ

yi+1 = −(c†i + ci)(c

†i+1 − ci+1), (2.26)

σxi σ

zi+1σ

xi+2 = −(c†i − ci)(c

†i+2 + ci+2), (2.27)

σyi σ

zi+1σ

yi+2 = (c†i + ci)(c

†i+2 − ci+2) (2.28)

, Ising

HTFIM = −N∑i=1

(Jσxi σ

xi+1 + hσz

i ) (2.29)

XY

HXY =N∑i=1

(JXXσxi σ

xi+1 + JY Y σy

i σyi+1) (2.30)

HC = −N∑i=1

JXZXσxi σ

zi+1σ

xi+2 (2.31)

HC∗ =

N∑i=1

JY ZY σyi σ

zi+1σ

yi+2 (2.32)

. , N

:

H =

N∑i,j=1

[c†iAijcj +

1

2

(c†iBijc

†j + ciB

∗jicj

)]. (2.33)

, A N , B N . ,

. Bogoliubov

ηj =

N∑i=1

[φij + ψij

2ci +

φij − ψij

2c†i

](2.34)

:

H =

N∑μ=1

Eμ(η†μημ − 1

2), Eμ ≥ 0. (2.35)

, φμ, ψμ N φ,ψ μ , :

{Eμψi = (A+B)φμ, (2.36a)

Eμφi = (A−B)ψμ. (2.36b)

16


, |GS〉 Bogoliubov

ημ|GS〉 = 0, μ = 1, 2, , N (2.37)

.

, Majorana :

c2i−1 = c†i + ci, c2i = i (ci − c†i ), i = 1, 2, , N. (2.38)

, Bogoliubov ημ

ημ =1

2

N∑i=1

(φiμc2i−1 − iψiμc2i) (2.39)

, i μ Majorana

φiμ ψiμ . c

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

c2i−1 =N∑

μ=1

φiμ

(ημ + η†μ

), (2.40a)

c2i = i

N∑μ=1

ψiμ

(ημ − η†μ

)(2.40b)

. Bogoliubov , ημ|GS〉 = 0 (∀μ) , Majorana

〈c2i−1c2j−1〉 =N∑

μ=1

φiμφjμ = δij , (2.41)

〈c2ic2j〉 =N∑

μ=1

ψiμψjμ = δij , (2.42)

〈c2ic2j−1〉 = −〈c2j−1c2i〉 = iN∑

μ=1

ψiμφjμ (2.43)

. Majorana ,

:〈cicj〉 = δij + iΓA

ij . (2.44)

ΓA 2N . Wick ,

. , 2N A , 2N W ,

:

W †AW =

N⊕i=1

νi

(0 1−1 0

). (2.45)

17


, ΓA :

W †ΓAW =N⊕i=1

νi

(0 1−1 0

)= ΓC . (2.46)

ΓC , c2i−1 =∑N

j=1 W2i−1,j cj c2i =∑N

j=1 W2i,j cj

. , (Dirac )

:

di =1

2(c2i−1 + i c2i). (2.47)

d

〈d†idj〉 = δij1− νi

2(2.48)

. , N . ,

. Lsub

, ρ

ρ = ρ1 ⊗ ρ2 ⊗ ⊗ ρLsub(2.49)

,

S(Lsub) =

Lsub∑i=1

[−pi ln pi − (1− pi) ln (1− pi)] , pi =1− νi

2(2.50)

. {λi(Lsub)} 2Lsub , [p1 + (1− p1)] [pLsub+ (1−

pLsub)] .

λj(Lsub) =

Lsub∏i=1

p1−sii (1− pi)

si , si = 0, 1 (2.51)

. 2.6 .

2.2.2

, ,

. .

, .

[55].

, .

,

OXZX(L) = (−1)L

⟨σx1σ

y2

(L−2∏k=3

σzk

)σyL−1σ

xL

⟩(2.52)

18


2.6 .

[54]. , A B :

Ai = ci + c†i = c2i−1, (2.53)

Bi = ci − c†i = −ic2i. (2.54)

Wick ,

〈AiAj〉 = δij , (2.55)

〈BiBj〉 = −δij , (2.56)

〈BiAj〉 =N∑

μ=1

ψiμφjμ = D(i, j) (2.57)

. A B ,

OXZX(L) = (−1)L+1〈B1B2A3B3 AL−2BL−2AL−1AL〉 (2.58)

. Wick , Pfaffian

. , A (B), . , Pfaffian

OXZX(L) =

∣∣∣∣∣∣∣∣∣∣

D(1, L) D(1, 3) . . . D(1, L− 1)

D(2, L) D(2, 3) . . ....

......

. . ....

D(L− 2, L) . . . . . . D(L− 2, L− 1)

∣∣∣∣∣∣∣∣∣∣(2.59)

.

Ising . Jordan–Wigner

, . ,

LOXX(L) = 〈σx

1σxL〉 (2.60)

19


.

OXX(L) = 〈B1B2A2B2 BL−1AL−1AL〉 (2.61)

=

∣∣∣∣∣∣∣∣∣∣

D(1, 2) D(1, 3) . . . D(1, L)

D(2, 2) D(2, 3) . . ....

......

. . ....

D(L− 1, 2) . . . . . . D(L− 1, L)

∣∣∣∣∣∣∣∣∣∣(2.62)

. L− 1 .

2.2.3 Bogoliubov

, .

Jordan–Wigner c ,

.

H(t) =N∑

i,j=1

[c†iAij(t)cj +

1

2

(c†iBij(t)c

†j + ciB

∗ji(t)cj

)](2.63)

. , . , Heisenberg

Jordan–Wigner {ci,H(t)} Heisenberg

:

id

dtci,H(t) =

N∑j=1

(Aij(t)cj,H(t) +Bij(t)c

†j,H(t)

). (2.64)

tin H(tin) Bogoliubov ηinμ , φinμ ,

ψinμ ,

uinμ =

φinμ + ψin

μ

2, vinμ =

φinμ − ψin

μ

2(2.65)

. Heisenberg cH(t)

ci,H(t) =N∑

μ=1

(ui,μ(t)η

inμ + v∗i,μ(t)η

in†μ

)(2.66)

, uμ(t), vμ(t)

uμ(tin) = uinμ , vμ(tin) = vinμ (2.67)

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

id

dtui,μ(t) =

N∑j=1

(Aij(t)uj,μ(t) +Bij(t)vj,μ(t)) , (2.68a)

id

dtvi,μ(t) = −

N∑j=1

(Aij(t)vj,μ(t) +Bij(t)uj,μ(t)) (2.68b)

20


[56].

t . , Schrodinger Heisenberg

〈Ψ(t)|O(ci, c†i )|Ψ(t)〉 = 〈Ψ(tin)|O(ci,H(t), c

†i,H(t))|Ψ(tin)〉 (2.69)

. |Ψ(tin)〉 Bogoliubov , Wick

. uμ(t), vμ(t) , ,

uμ(t), vμ(t) .

,

. t Dirac ,

S(Lsub, t) {λi(Lsub, t)} .

2.3 Infinite time evolving block decimation

,

, infinite time evolving block decimation (iTEBD)

. ,

. , iTEBD ,

iTEBD .

2.3.1

|ψ〉 , Hilbert {|i〉}

|Ψ〉 =∑i

Ci|i〉 (2.70)

. (Hilbert ) ,

. , .

. iTEBD

.

|Ψ〉αLβR=

∑{mi},{α},{β}

AαL,β1(m1) Aαi,βi(mi) AαN ,βR(mN )|m1〉 |mi〉 |mN 〉 (2.71)

. , |mi〉 i Hilbert ( ) ,

, di . , {αi}, {βi} , ,

χb,i . di χb,i i , ,

i , d, χb . χb

, ,

. 2.7 .

(Matrix Product State, MPS) .

21


2.7 : A . ,

.

.

, 12 .

:

|Ψ〉 = 1√2(| ↑1〉| ↓2〉 − | ↓1〉| ↑2〉) (2.72)

. ,

|Ψ〉 =∑

i,j=↑,↓C1,i|i1〉C2,j |j2〉 (2.73)

{C} . ,

A(↑1) = (14√20), A(↓1) = (0

14√2), (2.74)

A(↑2) = (014√2)T, A(↓2) = (− 1

4√20)T (2.75)

, . ,

. ,

, Schmidt ( A)

|Ψ〉 =∑α

λα|ψα〉|φα〉 (2.76)

, . .

Schmidt [57]. , r − 1 r

Schmidt ( 2.8):

|Ψ〉 =χb∑α=1

λ[r−1]α |ψ[�r−1]

α 〉|ψ[r�]α 〉. (2.77)

, r−1 |ψ[�r−1]α 〉, r |ψ[r�]

α 〉 . λ[r−1]α

, . |ψ[r�]α 〉 r r + 1

22


2.8 Schmidt : |Ψ〉 r − 1 r Schmidt .

2.9 Schmidt : Γ[r] .

2.10 : {Γ} {λ} .

Schmidt :

|ψ[r�]α 〉 =

χb∑β=1

d∑mr=1

Γ[r]αβ(mr)|mr〉λ[r]

β |ψ[r+1�]β 〉. (2.78)

, r r Hilbert , Γ[r]

( 2.9). ,

|Ψ〉αLβR=

∑{mi},{α},{β}

Γ[1]αLβ1

(m1)λ[1]β1α2

Γ[i]αiβi

(mi)λ[i]βiαi+1

Γ[N ]αNβR

(mN )|m1〉 |mi〉 |mN 〉

(2.79)

, 2.10 . , {A} {Γ} {λ}. {λ} , ,

.

2.3.2

iTEBD . iTEBD H

. n n . iTEBD

23


, . n

{Γ} {λ} . iMPS (infinite matrix product state) .

iMPS , |Ψ0〉 .

, |Ψ〉 :

|Ψ〉 = limτ→∞

exp(−Hτ)|Ψ0〉| exp(−Hτ)|Ψ0〉| . (2.80)

iMPS . n ,

dtU = exp(−Hdτ), dτ 1 (2.81)

, –Trotter [30] . ,

, . iMPS Γ[A], λ[A],Γ[B], λ[B]

.

H = HA +HB =∑

even i

Hi,i+1 +∑odd i

Hi,i+1 (2.82)

. , –Trotter

expH = limn→∞

(exp(

1

nHA) exp(

1

nHB)

)n

(2.83)

, iMPS UA = exp(−HAdτ) UB = exp(−HBdτ) ( 2.11).

dτ . ,

, . , iMPS ,

–Trotter

expH = limn→∞

(exp(

1

2nHA) exp(

1

nHB) exp(

1

2nHA)

)n

(2.84)

. , n . , –Trotter1n , –Trotter 1

n2 .

iMPS . , UA .

. 2.12 .

step1

UA . UA Γ[A],Γ[B] .

step2

Γ[A], λ[A],Γ[B], λ[B], UA , M :

Mαijβ =

d∑k,l=1

χb∑γ,δ,ε,ζ=1

λ[B]αγ Γ

[A]γδ (k)λ

[A]δε Γ

[B]εζ (l)λ

[B]ζβ U

[A]ij;kl. (2.85)

.

step3

24


2.11 –Trotter : H U ,

HA, HB UA, UB

, M dχb × dχb ,

:

Mαi;βj =

dχb∑γ,δ=1

M[A]αi;γΛγδM

[B]δ;βj . (2.86)

Λ dχb . , dχb

χb . Λ

, λ[A] . , ( ) dχb M [A]

(M [B]) , .

X (Y ) .

step4

X 1 = λ[B]λ[B]−1, Y 1 = λ[B]−1λ[B] ,

λ[A] = λ[B]−1X, λ[B] = Y λ[B]−1 (2.87)

iMPS iMPS {Γ[A], λ[A], Γ[B], λ[B]} .

UA iMPS . UB . ,

, iMPS

.

2.3.3

|Ψ(0)〉 . ,

. t

|Ψ(t)〉 = exp(−iHt)|Ψ(0)〉 (2.88)

25


2.12 iMPS : iMPS iMPS

.

. , iMPS

. , , t

|Ψ(t)〉 = exp

(−i

∫ t

0

H(t)dt

)|Ψ(0)〉 (2.89)

.

2.3.4

, .

S {ξi}, iMPS :

S = −χb∑i=1

λ2i lnλ

2i , (2.90)

ξi = − lnλ2i . (2.91)

, . i O[i] . , Schmidt

〈Ψ|O[i]|Ψ〉 =∑

α,β,j,k

(λα)2Γ

[j]α,βO

[i]j,k(Γ

[k]α,β)

∗(λβ)2 (2.92)

( 2.13). i O[i] , iMPS

. , i j

( 2.14). , iMPS |i− j| ,

.

26


2.13 i .

2.14 i j .

27


3

2.1 , Kitaev ( ) , Majorana

. ,

,

HGC =N∑i=1

(−JXZXσxi σ

zi+1σ

xi+2 + JY Y σy

i σyi+1 + JY ZY σy

i σzi+1σ

yi+2) (3.1)

. 3.1 ,

, 3.2 , ( )

. 3.3 , .

3.1

Lieb–Schultz–Mattis [55] , .

, ,

Fourier Bogoliubov .

XY , :

H =

N∑i=1

(−hσzi + JXXσx

i σxi+1 − JXZXσx

i σzi+1σ

xi+2 + JY Y σy


i σzi+1σ

yi+2). (3.2)

, Jordan–Wigner , Fourier

H = 2∑

0≤k≤π

[εk(c

†kck + c†−kc−k) + iδk(c

†kc

†−k + ckc−k)

](3.3)

.

εk = (JXZX − JY ZY ) cos 2k + (JXX + JY Y ) cos k − h, (3.4)

δk = (JXZX + JY ZY ) sin 2k + (JXX − JY Y ) sin k (3.5)

. Bogoliubov .

ηk = cosθk2ck − i sin

θk2c†−k, (3.6)

{ηk, η†k′} = δkk′ , (3.7)

tan θk = −δkεk

(3.8)

28


H =∑

0≤k≤π

Δk(η†kηk + η†−kη−k), (3.9)

Δk = 2√

ε2k + δ2k (3.10)

( E). |GS〉 Bogoliubov

ηk|GS〉 = 0, ∀k (3.11)

, Bogoliubov ( ) η†k .

Δk , Δk = 0 k ,

. , (3.2) ,

.

: XY

HXY =N∑i=1

(σxi σ

xi+1 + JY Y σy

i σyi+1) (3.12)

. (3.2) h = 0, JXX = 1, JXZX =

0, JY ZY = 0 . ,

εk = (1 + JY Y ) cos k, (3.13)

δk = (1− JY Y ) sin k (3.14)

1. k = 0, π

δk = 0 , Δk = 0 JY Y = −1

2. k = π2

εk = 0 , Δk = 0 JY Y = 1

3. k �= 0, π2 , π

Δk = 0 k .

JY Y = ±1 ( ) .

2:

HGC =N∑i=1

(−JXZXσxi σ

zi+1σ

xi+2 + JY Y σy


i σzi+1σ

yi+2) (3.15)

. (3.2) h = 0, JXX = 0

. . JXZX = 1 ,

3.1 . , (a) JY Y /JXZX = 0, JY ZY /JXZX = 1 (b)

JY Y /JXZX = −1, JY ZY /JXZX = 0 3.2 .

.

29


3.1 (3.15) : .

3.3 . C , C∗

, AF , F , P .

.

3.2

Fourier (3.3) , Bogoliubov–de Gennes

HBdG(k) :

HBdG(k) =

(εk iδk

−iδk −εk

). (3.16)

εk = (JXZX − JY ZY ) cos 2k + (JXX + JY Y ) cos k − h, (3.17)

δk = (JXZX + JY ZY ) sin 2k + (JXX − JY Y ) sin k (3.18)

. , .

(3.16) . Anderson [58]

(pseudospin) d(k)d(k) = εkez + δkey (3.19)

,HBdG(k) = d(k) · τ (3.20)

30


3.2 : (a) JY Y /JXZX = 0, JY ZY /JXZX = 1, (b)

JY Y /JXZX = −1, JY ZY /JXZX = 0 .

. , ey, ez y, z , τ Pauli

. , Anderson

d(k) =d(k)

|d(k)| = cos θkez + sin θkey (3.21)

. k Brillouin ( ) , d(k) yz

. , .

W

W =

∫B.Z.

dθk2π

(3.22)

. , k Brillouin d(k) yz

, . ,

εk = δk = 0 , .

. ,

, . ,

. , Majorana

[59]. 3.1 . yz

31


3.3 : εk, δk .

. W .

3.1 (a)JY Y /JXZX = 0, JY ZY /JXZX = 0 ( ), (b)JJY Y /JXZX =

−2.0, JY ZY /JXZX = 0 ( ), (c)JY Y /JXZX = −1, JY ZY /JXZX = −1 ( )

.

3.3 . k −π π ,

3.3 . (a) , (b) , (c)

.

32


3.3

3.1 .

.

HGC =

N∑i=1

(−JXZXσxi σ

zi+1σ

xi+2 + JY Y σy


i σzi+1σ

yi+2) (3.23)

3.4 (JXZX = 1). . ,

iTEBD . , 3.4 (JY Y /JXZX ∈ [−1.5, 0],

JY ZY /JXZX = 0.5) . χb 60

OXZX = limL→∞

(−1)L

⟨σx1σ

y2

⎛⎝L−2∏

j=3

σzj

⎞⎠σy

L−1σxL

⟩, (3.24)

OXX = limL→∞

〈σx1σ

xL〉, (3.25)

OY Y = limL→∞

〈σy1σ

yL〉 (3.26)

. 3.5 . ,

. , L = 200 ,

.

.

, . , 3.4

. JY Y /JXZX = 0, JY ZY /JXZX = −0.5 ,

. JY Y /JXZX 0 ,

JY Y /JXZX = 0.6 ( 3.6). ,

. ,

. ,

. , (C ) (P ) .

. JY Y /JXZX

JY Y /JXZX = 1.5 , .

, (y )

(AF(y) ) .

, 3.4 . (C∗) ,

( 3.7).

, JY ZY /JXZX = 1

, .

33


3.4 (3.23) : JXZX = 1 . C

, C∗ , AF , F , P

. . iTEBD

( 3.5).

( 3.6, 3.7).

0

0.2

0.4

0.6

0.8

1

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

Ord

er

para

mete

rs

JYY

/JXZX

OXZX

OXX

OYY

3.5 : iTEBD , 3.4 (JY Y /JXZX ∈[−1.5, 0], JY ZY /JXZX = 0.5) (χb = 60).

OXZX , X OXX ,

Y OY Y .

34


0

1

2

3

4

5

0 0.5 1 1.5 2

ξ i

JYY

/JXZX

ξ1

ξ2

ξ3

ξ4

3.6 : 3.4 .

. (N =503 )

,

. , , ,

. , ,

. , ,

. , .

.

, ,

. 2.2, 2.3, 2.5 . ,

JXZX . ,

, ( 2.3). A B ,

.

.

, . 3.3

, , 2, ( ) 1, 0

. ,

.

. 3.4 (JY ZY /JXZX = JY Y /JXZX + 1)

k = 0 , (JY ZY = −JY Y + 1) k = ±π ,

(JY Y 2 − (1 − JY ZY 2)(1 + JY ZY ) = 0) k = arccos JY Y

2(1+JY ZY )

. , (JY Y /JXZX , JY ZY /JXZX) = (0,−1) k = ±π4 ,± 3π

4

.

. JY Y /JXZX = 0, JY ZY /JXZX = 1

. ( 3.8) ,

35


1

2

3

4

5

6

0.4 0.6 0.8 1 1.2 1.4 1.6

ξ i

JYZY

/JXZX

ξ1

ξ2

ξ3

ξ4

ξ5

3.7 : 3.4 .

(JY Y /JXZX , JY ZY /JXZX) = (0, 1) . (N =503

)

, 2 . , 3.2(a)

, Ising (4 × 12 ). ,

(JY Y /JXZX = 1, JY ZY /JXZX = 0) [54] ,32 . Ising (3× 1

2 ).

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 50 100 150 200 250

Lsub

SA(Lsub)2/3log(Lsub)+0.952

3.8 (JY Y /JXZX , JY ZY /JXZX) = (0, 1)

: , 2 .

(N =503, Lsub = 249 )

36


4

,

HGC =N∑i=1

(−JXZXσxi σ

zi+1σ

xi+2 + JY Y σy


i σzi+1σ

yi+2) (4.1)

. , 4.1 . 4.2

. ,

,

.

4.1 Bogoliubov

Bogoliubov ( ) ,

, .

. .

, ,

.

. , .

2L .

P2L = 〈Ψ1Ψ2 Ψ2L−1Ψ2L〉 (4.2)

Pfaffian . Pfaffian

Pij = 〈ΨiΨj〉 (4.3)

, :

P2L =∑

π∈S<2L

(−1)πPπ(1)π(2)Pπ(3)π(4) Pπ(2L−1)π(2L) (4.4)

=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

P1,2 P1,3 . . . P1,L P1,L+1 P1,L+2 . . . P1,2L

P2,3 . . . . . . P2,L+1 . . . . . . P2,2L

. . .... . . . . . .

.... . .

... . . . . . ....

PL,L+1 PL,L+2 . . . PL,2L

PL+1,L+2

.... . .

...P2L−1,2L

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

. (4.5)

37


, S<2L 2L S2L , :

π(2i− 1) < π(2i) π(2i− 1) < π(2j − 1), 1 ≤ i < j ≤ L. (4.6)

Pfaffian : (4.5) 2L

AdetA = (P2L)

2 (4.7)

. .

μ ημ

|k〉 = η†μ|0〉 (μ = 1, 2, . . . , N) (4.8)

. A B , (2.2.2

). , (2L− 2) A,B

〈ηkB1B2A3B3 AL−2BL−2AL−1ALη†k〉 (4.9)

. (2L− 2) .

〈AiAj〉 = −〈BiBj〉 = δij , (4.10)

〈AiBj〉 = −N∑

ν=1

φiνψjν , (4.11)

〈BiAj〉 =N∑

ν=1

ψiνφjν , (4.12)

〈ημAi〉 = 〈Aiη†μ〉 = φiμ, (4.13)

〈ημBi〉 = −〈Biη†μ〉 = −ψiμ, (4.14)

〈ημη†ν〉 = δμν (4.15)

. η . , ,

, 4.1 . (4.1)

(JY Y /JXZX , JY ZY /JXZX) = (0, 0.5) ,

( 4.2, 4.3). ,

, (η1, η2) 0

. , |0〉, η†1|0〉, η†2|0〉, η†1η†2|0〉 . ,

, Bogoliubov

( 4.3 ). 4.3 ( )

. ,

. ,

, .

, . 4.4 , (4.1)

(JY Y /JXZX , JY ZY /JXZX) = (0, 0.5) .

38


4.1 : .

0

1

0 10 20 30 40 50

OX

ZX(L

)

L

1st Bogolon excited state

4.2 Bogoliubov : C

(JY Y /JXZX , JY ZY /JXZX) = (0, 0.5) . N =101.

0

1

0 10 20 30 40 50

OX

ZX(L

)

L

1st Bogolon excited state3rd Bogolon excited state

4.3 : C

(JY Y /JXZX , JY ZY /JXZX) = (0, 0.5) . (

) Bogoliubov ( 4.2) .

,

. N =101.

k = 0,±π2 ,±π . , ,

.

4.2

,

. , , .

39


4.4 (JY Y /JXZX , JY ZY /JXZX) = (0, 0.5)

: k = 0,±π2,±π .

, . ,

,

.

(C ) .

,

H1(t) = −JXZXN∑i=1

σxi σ

zi+1σ

xi+2 + J(t)

N∑i=1

σyi σ

zi+1σ

yi+2, (4.16a)

J(t)/JXZX = 2t/τ, t ∈ [0, τ ] (4.16b)

. τ = 0 .

, t = 0 Bogoliubov . , C

(C∗ ) (JY Y /JXZX , JY ZY /JXZX) = (0, 1) t = 0.5τ

( 4.5, 4.6). , ,

,

. N =101 , .

, . t 0.5τ ,

Majorana , ,

( 4.7).

, Majorana , .

,

.

, . ,

25, 200 . , τ = 25 ,

( 4.8).

, .

.

40


4.5 (4.16)

:

.

4.6 :

t = 0.5τ , C C∗

.

, τ = 200 ( 4.9). ,

.

, .

, , τ = 25

. , τ = 200 4.3

. , τ

. , τ = 25 , ,

, .

, . ,

. 4.10 ,

.

.

41


4.7 (a)-(d) τ = 25, 50, 100, 200 .

. ,

.

0

0.5

1

0 10 20 30 40 50

OY

ZY(L

)

L

t=0.6τt=0.8τ

t=τ

4.8 τ = 25 , t

: ,

,

.

.

0

0.5

1

0 10 20 30 40 50

OY

ZY(L

)

L

t=0.6τt=0.8τ

t=τ

4.9 τ = 200 , t

: ,

.

42


1.5

2

2.5

3

3.5

4

0 10 20 30 40 50

SA

(L)

L

τ=25τ=50

τ=100τ=200

4.10 (t = τ)

:

.

4.11 .

. t = 0.5τ ,

. ,

. ,

. , ,

( 4.12).

1

2

3

4

5

0 0.25 0.5 0.75 1

SA

(t/τ

)

t/τ

τ=25τ=50

τ=100τ=200

4.11 : t = 0

(JY Y /JXZX , JY ZY /JXZX) = (0, 0), t = τ (JY Y /JXZX , JY ZY /JXZX) = (0, 2)

.

43


1.6 1.8

2 2.2 2.4 2.6 2.8

3 3.2 3.4

0.4 0.6 0.8 1 1.2 1.4 1.6

SA

(Lsu

b)

JYZY

SA(Lsub)

4.12 :

. N = 103, Lsub = 49.

44


5

, , p

Kitaev ,

. , iTEBD ,

. , ,

.

. ,

, .

.

.

,

,

, .

45


A Schmidt

|Ψ〉 , A,B ( Hilbert

HA,HB) . , |Ψ〉 A,B

|Ψ〉 =χ∑

α=1

λα|φα〉A|ψα〉B ∈ HA ⊗ HB (A.1)

. λα Schmidt , ,

χ∑α=1

λ2α = 1, χ ≤ min(dimHA, dimHB) (A.2)

. Schmidt .

1

|Ψ〉 = 1

2√2

(|00〉+

√3|01〉+

√3|10〉+ |11〉

)(A.3)

=

√3 + 1

2√2

|φ1〉A|ψ1〉B +

√3− 1

2√2

|φ2〉A|ψ2〉B . (A.4)

|φ1〉A =1√2(|0〉A + |1〉A) , |φ2〉A =

1√2(|0〉A − |1〉A) (A.5)

A ,

|ψ1〉B =1√2(|0〉B + |1〉B) , |ψ2〉B =

1√2(−|0〉B + |1〉B) (A.6)

B .

2

|Ψ〉 = 1√3|00〉+ 1√

6|01〉 − i√

3|10〉 − i√

6|11〉 (A.7)

=1√6(|0〉A − i|1〉A)(

√2|0〉B + |1〉B) (A.8)

.

{λα} .

S ,

S = −χ∑

α=1

λ2α lnλ2

α (A.9)

. S 0 . (maximally entangled

state), Schmidt , lnχ .

46


B AKLT

1 Haldane .

, AKLT (Affleck, Kennedy, Lieb, and Tasaki) [52,53] :

H =N∑i=1

[Si · Si+1 +

1

3(Si · Si+1)

2 +2

3

]. (B.1)

, i i+ 1 1 Stot = 2

P 2i,i+1 , AKLT Valence Bond Solid (VBS) .

AKLT 2 . , 2 12

. , . , |α〉|α〉 ,

2∑α=1

|α〉|α〉 =2∑

α,β

Rαβ |α〉|β〉 (B.2)

R . , 2 12 S = 1

.

A(iL, iR) =∑

α,β,mi

Bαβ(mi)|mi〉〈αβ| (B.3)

=∑

β,γ,mi

Aβγ(mi)|mi〉〈γβ|, (B.4)

B =1√2

( √2|1〉 |0〉|0〉 √

2| − 1〉)

(B.5)

. , VBS

Aαβ(i) =1√2

( |0〉 √2| − 1〉

−√2|1〉 −|0〉

)(B.6)

. iMPS (2.3.1 )

Γ(1) =

(0 0−1 0

), Γ(0) =

1√2

(1 00 −1

), (B.7)

Γ(−1) =

(0 10 0

), λ =

1√2

(1 00 1

)(B.8)

. , ln 2 ,

.

47


C Majorana

Kitaev ( Ising ) [27]

H =N∑i=1

[−(c†i ci+1 + c†i+1ci)− (c†i c

†i+1 + ci+1ci)− 2hc†i ci

](C.1)

. , Bogoliubov

ημ =N∑i=1

(φiμ + ψiμ

2ci +

φiμ − ψiμ

2c†i

)(C.2)

. , N φμ, ψμ N φ,ψ μ (μ =

1, 2, . . . , N) , (Heisenberg ) :

{Eμψμ = (A+B)φμ, (C.3a)

Eμφμ = (A−B)ψμ. (C.3b)

, A B N

A =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

−2h −1 0 . . . 0

−1 −2h −1. . .

...

0 −1. . .

. . ....

.... . .

. . .. . . −1

0 . . . . . . −1 −2h

⎞⎟⎟⎟⎟⎟⎟⎟⎠

, B =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

0 −1 0 . . . 0

1 0 −1. . .

...

0 1. . .

. . ....

.... . .

. . .. . . −1

0 . . . . . . 1 0

⎞⎟⎟⎟⎟⎟⎟⎟⎠

(C.4)

. , φ ψ :

φTφ = IN , ψTψ = IN . (C.5)

, . m Em = 0 , (C.3)

: {0 = (A+B)φm, (C.6a)

0 = (A−B)ψm. (C.6b)

, 0 ≤ h < 1

φm = (1,−h, (−h)2, . . . )T , ψm = (. . . , (−h)2,−h, 1)T (C.7)

, .

48


D

, Ising

H(λ) =N∑i=1

(−σxi σ

zi+1σ

xi+2 + λσy

i σyi+1) (D.1)

Ising (Kramers–Wannier ). ,

μzi = σx

i σxi+1, μx

i =

i∏j=1

σzj , σx

N+1 = 1 (D.2)

−σxi σ

zi+1σ

xi+2 = μy

i μyi+1, σy

i σyi+1 = −μx

i−1μziμ

xi+1, μx

0 = 1 (D.3)

. ,

Hdual(λ) = λH

(1

λ

)(D.4)

[54]. λ = 1 .

, (XZX) z π (Y ZY )

H(λ) =N∑i=1

(−σxi σ

zi+1σ

xi+2 + λσy

i σzi+1σ

yi+2) (D.5)

z π , .

49


E Bogoliubov

Ising Jordan–Wigner

HTFIM = −N∑i=1

(σxi σ

xi+1 + λσz

i ) (E.1)

= −N∑i=1

[(c†i − ci)(c

†i+1 + ci+1) + λ(2c†i ci − 1)

](E.2)

= −N∑i=1

[(c†i ci+1 + c†i+1ci) + (c†i c

†i+1 + ci+1ci) + 2λ(c†i ci −

1

2)

](E.3)

, p Kitaev :

HKitaev =

N∑i=1

[−t(c†i ci+1 + c†i+1ci)− (Δc†i c

†i+1 +Δ∗ci+1ci)− μc†i ci

], (E.4)

t = Δ = Δ∗ = 1, μ = 2λ. (E.5)

P = (−1)∑N

i=1 c†i ci (E.6)

. P 2 = 1 P = ±1 . ( ) (

) .

Ising . , Fourier

cj =1√2π

∫ π

−π

dk eikjck (E.7)

,

HTFIM = 2

∫ π

−π

dk (− cos k − λ)c†kck − 2i

∫ π

−π

dk (sin k ckc−k − sin k c†−kc†k) (E.8)

. , Bogoliubov {ηk = ck cos θk − i c†−k sin θk, (E.9a)

η†−k = −i ck sin θk + c†−k cos θk (E.9b)

.

{ηk, η†k′} = δkk′ , tan 2θk = −δkεk

(E.10)

. ,

ε(k) = 2

√(cos k + λ)2 + sin2 k (−π < k ≤ π) (E.11)

50


, |Ψ0〉 η- , BCS :

|Ψ0〉 =∏k>0

(cos θk + sin θkc†kc

†−k)|0〉c. (E.12)

|0〉c c- .

Bogoliubov . Bogoliubov UB

UB = exp

[−i∑k>0

θk(c−kck − c†−kc†k)

](E.13)

. Baker–Campbell–Hausdorff

eABe−A = B + [A,B] +1

2![A, [A,B]] + . . . (E.14)

[AB,C] = A{B,C} − {A,C}B (E.15)

, UB Bogoliubov :

U−1B ηkUB = ck. (E.16)

|GS〉 Bogoliubov , c |0〉c|GS〉 = UB|0〉c (E.17)

.

Bogoliubov WB . , φ ψ

W± . W± ,

φ = expW+, ψ = expW− (E.18)

. , Bogoliubov WB

WB = exp

⎡⎣12

N∑i,j=1

(W+ −W−

2

)ij

(cicj + c†i c†j) +

N∑i,j=1

(W+ +W−

2

)ij

c†i cj

⎤⎦ (E.19)

:W−1

B ηiWB = ci. (E.20)

|GS〉 Bogoliubov , c |0〉c|GS〉 = WB|0〉c (E.21)

.

51


,

. ,

. , , , ,

. ,

, , ,

. .

, ,

.

52


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54


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Õ mì ^ ; w K ° Í i Ñ £ ç Û ¦ ï Û w ¼ Æ Û « µmercury.yukawa.kyoto-u.ac.jp/~bussei.kenkyu/pdf/04/4/... · 2017-09-04 · 1 Ö 1.1 \ w ~ G å w , Hall % µ Ð ï ÷