Instantons and Berrys connections on quantum graph Inori Ueba - - PowerPoint PPT Presentation

Instantons and Berry’s connections


• n quantum graph

Inori Ueba 
 (Kobe Univ.)

2020/11/16 
 Strings and Fields 2020 Collaborator : Tomonori Inoue, Makoto Sakamoto (Kobe Univ.)

• No. 22

Introduction

Quantum graph：QM on the 1D graph with edges and vertices

• Quantum wire
• Scattering theory

Applications

⋯

Boundary conditions at vertices are important 1

Introduction

Quantum graph：QM on the 1D graph with edges and vertices Extra dimension = Quantum graph

• Quantum wire
• Scattering theory

Applications

⋯

Topology of boundary conditions

Fermion generation structure in the standard model Boundary conditions at vertices are important e.g. 1

(I. Ueba et al, J.Phys.A 52 (2019) 45, 455401)

Summary of this talk

Here we focus on the Berry’s connections Instantons appear as Berry’s connections 


• n the parameter space of Boundary conditions

Summary of this talk

We want to reveal the structure 


• f boundary conditions

2

Contents

• Quantum graph and extra dimensional model
• Introduction

3

• Berry’s connections and instantons

Contents

• Quantum graph and extra dimensional model
• Introduction

3

• Berry’s connections and instantons

Quantum graph and boundary condition

j1 + j2 = j3 + j4 j1 j2 j3 j4

Quantum graph：Quantum mechanics on the 
 1D graph with edges and vertices Conservation of probability currents Boundary conditions for
 wave functions 4

Rose quantum graph

cut the edges without the current flow

Rose graph is known as a master graph

rose graph

5

Rose quantum graph

cut the edges without the current flow

Rose graph is known as a master graph

rose graph

5

Rose quantum graph

cut the edges without the current flow

Rose graph is known as a master graph

rose graph

5

KK decomposition of Dirac field

⋯ ⋯

m1 m2

⋯ ⋯

mn ψ(i)

R,n(x)

ψ(i)

L,n(x)

+ ] Ψ(x, y) = ∑

i ∑ n [

ψ(i)

R,n(x)

ψ(i)

L,n(x)

f (i)

n (y)

g(i)

n (y)

⋯ ⋯ ⋯ ⋯

KK mode functions 4D chiral fields

n : mode number

i : degeneracy

new particles particles in SM

6

Zero modes on quantum graph

[iγμ∂μ+ ]Ψ(x, y) = 0 + ] Ψ(x, y) = ∑

i ∑ n [

ψ(i)

R,n(x)

ψ(i)

L,n(x)

f (i)

n (y)

g(i)

n (y)

γ5∂y + M

= 0 for zero modes

f (i)

0 (y) ∝

F(i)

1 e−My

(L0 < y < L1) ⋮ F(i)

N e−My

(LN−1 < y < LN) g(j)

0 (y) ∝

G(j)

1 e+My

(L0 < y < L1) ⋮ G(j)

N e+My

(LN−1 < y < LN)

L0 L1 L2 LN−1 LN

⋯ ⋯

y

L0 < y < L1 L1 < y < L2 LN−1 < y < LN L2 < y < L3

7

[iγμ∂μ+ ]Ψ(x, y) = 0 γ5∂y + M

= 0 for zero modes

L0 L1 L2 LN−1 LN

⋯ ⋯

y

f (i)

0 (y) ∝

F(i)

1 e−My

(L0 < y < L1) ⋮ F(i)

N e−My

(LN−1 < y < LN) g(j)

0 (y) ∝

G(j)

1 e+My

(L0 < y < L1) ⋮ G(j)

N e+My

(LN−1 < y < LN)

L0 < y < L1 L1 < y < L2 LN−1 < y < LN L2 < y < L3

Zero modes on quantum graph

⃗ F (i) = F(i)

1

⋮ F(i)

N

⃗ G( j) = G( j)

1

⋮ G( j)

N

# of independent vectors # of zero modes

7 + ] Ψ(x, y) = ∑

i ∑ n [

ψ(i)

R,n(x)

ψ(i)

L,n(x)

f (i)

n (y)

g(i)

n (y)

L0 < y < L1 L1 < y < L2 LN−1 < y < LN L2 < y < L3

Boundary condition on quantum graph

Current conservation j(y) = ¯ Ψ(x, y) iγ5 Ψ(x, y) j(La−1 + ε) =

N

∑

a=1

j(La − ε)

N

∑

a=1

BC for zero modes ⃗ ⃗

⃗

Δ( f )† ⃗ F (i) = 0 Δ(g)† ⃗ G(j) = 0

N × l complex matrix N × (N − K + l) complex matrix rank(Δ( f )) = l rank(Δ(g)) = N − K + l Δ( f ) : Δ(g) :

K = 0, 1, ⋯, 2N l = { 0, 1, ⋯, K (0 ≤ K ≤ N) K − N, ⋯, N (N < K ≤ 2N)

8

L0 < y < L1 L1 < y < L2 LN−1 < y < LN L2 < y < L3

Boundary condition on quantum graph

Current conservation j(y) = ¯ Ψ(x, y) iγ5 Ψ(x, y) j(La−1 + ε) =

N

∑

a=1

j(La − ε)

N

∑

a=1

BC for zero modes ⃗ ⃗

⃗

Δ( f )† ⃗ F (i) = 0 Δ(g)† ⃗ G(j) = 0

N × l complex matrix N × (N − K + l) complex matrix rank(Δ( f )) = l rank(Δ(g)) = N − K + l

K = 0, 1, ⋯, 2N

Δ( f ) : Δ(g) : ⃗ G( j) ⃗ F (i) N − l independent vectors

(i = 1, ⋯, N − l)

K − l independent vectors

(j = 1, ⋯, K − l)

g( j)

0 (y)

f (i)

0 (y)

(i = 1, ⋯, N − l) (j = 1, ⋯, K − l)

l = { 0, 1, ⋯, K (0 ≤ K ≤ N) K − N, ⋯, N (N < K ≤ 2N)

8

Generations of fermions in SM

# of # of

−

= N − K |N − K| chiral fermions

⋯ ⋯

mn ψ(i)

R,0(x)

ψ(i)

L,0(x)

⏟ ⏟

N − l K − l

K = N − 3 (N + 3)

for 3 generations K = 0, 1, ⋯, 2N l = { 0, 1, ⋯, K (0 ≤ K ≤ N) K − N, ⋯, N (N < K ≤ 2N) 9

Contents

• Introduction
• Berry’s connections and instantons

10

• Quantum graph and extra dimensional model

Contents

• Introduction

Topological quantities

monopole, instanton, ⋯

10

• Quantum graph and extra dimensional model
• Berry’s connections and instantons

Berry’s connection

H(R)|ϕ(R)⟩ = E(R)|ϕ(R)⟩ ≡ A γ = i∮C ⟨ϕ(R)|d|ϕ(R)⟩

Berry’s connection Berry’s phase

R : parameters in the system R

varies adiabatically depending on the time

|ϕ(R(t = 0))⟩ ⟶ ei∫T

0 dt E(R(t))+iγ|ϕ(R(t = T))⟩

R1 R2 R3 C R(t = T) = R(t = 0) 11

|ϕ(i)(R(t = 0))⟩ ⟶ ei∫T

0 dt E(R(t))

∑

j

(U(C))ij|ϕ(j)(R(t = T))⟩

Non-abelian Berry’s connection

H(R)|ϕ(i)(R)⟩ = E(R)|ϕ(i)(R)⟩ (A)ij = ⟨ϕ(i)(R)|d|ϕ(j)(R)⟩

non-abelian Berry’s connection

Degeneracy of the states

R1 R2 R3 C R(t = T) = R(t = 0)

U(C) = P exp (i∮C A)

R

varies adiabatically depending on the time

12

Berry’s connection on quantum graph

Berry’s connection for zero modes (A( f ))ij = ∫ dy f (i)†

0 (y) d f (j) 0 (y) =

⃗ F (i)†d ⃗ F (j)

A( f ) = F†dF Δ( f )†F = 0 F†F = 1N−l

F ≡ ( ⃗ F (1) ⋯ ⃗ F (N−l))

(i, j = 1, ⋯, N − l)

(A)ij = ⟨ϕ(i)(R)|d|ϕ(j)(R)⟩

For the quantum graph mode functions parameters in BC R1 R2 R3 C R(t = T) = R(t = 0)

If we define

13

Yang-Mills instanton

Fμν = ± ˜ Fμν

Self dual equation on ℝ4

Q = − 1 16π2 ∫ d4x Tr Fμν ˜ Fμν ∈ ℤ

( ˜ Fμν ≡ 1 2 ϵμνρλFρλ)

Instanton solutions

Instanton has a nontrivial topological charge

It is known how to construct the general instantons

14

ADHM construction for SU(n) instanton

Δ†

[2k]×[n+2k]V[n+2k]×[n](x) = 0 ,

V†V = 1n Aμ(x) = V†(x) ∂μV(x)

② Find matrix

(n + 2k) × n

V(x)

③ SU(n) instanton with

Q = k

Δ[n+2k]×[2k](x) = a[n+2k]×[2k] + b[n+2k]×[2k] ⋅ (xμeμ ⊗ 1k) rank(Δ) = 2k , [Δ†Δ , σi ⊗ 1k] = 0

eμ = (−iσi , 12)

(n + 2k) × 2k

complex matrix

ℝ4

coordinates of

(n + 2k) × 2k matrix Δ(x)

① Introduce

conditions :

ADHM data :

ADHM construction : method of constructing the general instantons

15

Berry’s connection and instanton

Δ( f )†[l]×[N] F[N]×[N−l] = 0 F†F = 1N−l A( f ) = F†dF Δ†

[2k]×[n+2k](x) V[n+2k]×[n](x) = 0

A(x) = V†(x) dV(x) V†V = 1n Berry’s connection ADHM construction

Δ( f)[N]×[l] = Δ[N]×[l](x) A( f) = Q = l/2

SU(N − l) instanton with

l : even

BC ADHM Berry’s connection Normalization Boundary condition Instanton Zero mode equation Normalization &

In the case that 16

Summary

Summary and discussion

Instantons in the parameter space


• f BC on quantum graph

ADHM 
 construction

Discussion

• Are there other topological structures in the parameter space of

boundary conditions ?

• Applications for topological insulator, quantum wire,

⋯

17

Back up

4D mass matrix

5D Yukawa interacton term mαβ = gY∫ dy ϕ(y)(f (0)′

α (y)) *g(0) β (y)

∫ d4x∫ dy gY ¯ Ψ′(x, y) Φ(x, y) Ψ(x, y) + h . c . ∫ d4x mαβ ¯ ψ′

R,α(x) ψL,β(x) + h . c .

4D mass term 4D mass matrix 4D mass matrix is given by 


• verlap integral

α, β : generations ϕ(y) : VEV of scalar field

f (0)′

1 (y)

g(0)

1 (y)

y

f (0)′

2 (y)

g(0)

2 (y)

⋯

m11 m12 m22

19

L0 < y < L1 L1 < y < L2 LN−1 < y < LN L2 < y < L3

Fermion generation structure

Overlap of localized modes Mass hierarchy (m11 ≪ m22) Overlap of different generations (m12) Flavor mixing Complex mode functions due to BC Origin of CP phase Topology of BC Generations

f (0)

1 (y)

g(0)

1 (y)

y

f (0)

2 (y)

g(0)

2 (y)

⋯

m11 m12 m22

20

mαβ = gY∫ dy ϕ(y)(f (0)′

α (y)) *g(0) β (y)

4D mass matrix α, β : generations ϕ(y) : VEV of scalar field