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

Instantons and Berry’s connections 
 on 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 Applications Boundary conditions at vertices are important 1 • Quantum wire • Scattering theory ⋯

Introduction Quantum graph：QM on the 1D graph with edges and vertices Extra dimension = Quantum graph Applications 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) • Quantum wire • Scattering theory ⋯ Topology of boundary conditions

Summary of this talk Here we focus on the Berry’s connections Instantons appear as Berry’s connections 
 on the parameter space of Boundary conditions Summary of this talk We want to reveal the structure 
 of 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：Quantum mechanics on the 
 Quantum graph and boundary condition 1D graph with edges and vertices Conservation of probability currents Boundary conditions for 
 wave functions 4 j 2 j 3 j 1 + j 2 = j 3 + j 4 j 4 j 1

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 KK mode functions particles in SM new particles 4D chiral fields 6 Ψ ( x , y ) = ∑ n [ i ∑ ψ ( i ) f ( i ) R, n ( x ) n ( y ) ] n : mode number ψ ( i ) g ( i ) L, n ( x ) + n ( y ) i : degeneracy ψ ( i ) ψ ( i ) L, n ( x ) R, n ( x ) m n ⋯ ⋯ ⋯ ⋯ m 2 ⋯ ⋯ m 1 ⋯ ⋯ 0

Zero modes on quantum graph 7 L 0 < y < L 1 Ψ ( x , y ) = ∑ n [ ] i ∑ ψ ( i ) ψ ( i ) f ( i ) g ( i ) R, n ( x ) + L, n ( x ) n ( y ) n ( y ) L N − 1 < y < L N L 1 < y < L 2 [ i γ μ ∂ μ + γ 5 ∂ y + M ] Ψ ( x , y ) = 0 L 2 < y < L 3 = 0 for zero modes F ( i ) 1 e − My ( L 0 < y < L 1 ) f ( i ) 0 ( y ) ∝ ⋮ F ( i ) N e − My ( L N − 1 < y < L N ) ⋯ G ( j ) 1 e + My ( L 0 < y < L 1 ) g ( j ) 0 ( y ) ∝ ⋮ y ⋯ G ( j ) L 0 L 1 L 2 L N − 1 L N N e + My ( L N − 1 < y < L N )

Zero modes on quantum graph ⃗ ⃗ # of independent vectors # of zero modes 7 L 0 < y < L 1 Ψ ( x , y ) = ∑ n [ ] i ∑ ψ ( i ) ψ ( i ) f ( i ) g ( i ) R, n ( x ) + L, n ( x ) n ( y ) n ( y ) L N − 1 < y < L N L 1 < y < L 2 [ i γ μ ∂ μ + γ 5 ∂ y + M ] Ψ ( x , y ) = 0 L 2 < y < L 3 = 0 for zero modes F ( i ) G ( j ) 1 1 F ( i ) 1 e − My ( L 0 < y < L 1 ) F ( i ) = G ( j ) = ⋮ ⋮ f ( i ) 0 ( y ) ∝ ⋮ F ( i ) G ( j ) N N F ( i ) N e − My ( L N − 1 < y < L N ) ⋯ G ( j ) 1 e + My ( L 0 < y < L 1 ) g ( j ) 0 ( y ) ∝ ⋮ y ⋯ G ( j ) L 0 L 1 L 2 L N − 1 L N N e + My ( L N − 1 < y < L N )

8 BC for zero modes ⃗ ⃗ ⃗ ⃗ ⃗ Current conservation Boundary condition on quantum graph L 0 < y < L 1 L N − 1 < y < L N L 1 < y < L 2 L 2 < y < L 3 Ψ ( x , y ) i γ 5 Ψ ( x , y ) j ( y ) = ¯ N N ∑ ∑ j ( L a − ε ) j ( L a − 1 + ε ) = a =1 a =1 rank( Δ ( f ) ) = l F ( i ) = 0 G ( j ) = 0 Δ ( f )† Δ ( g )† rank( Δ ( g ) ) = N − K + l Δ ( f ) : Δ ( g ) : N × l complex matrix N × ( N − K + l ) complex matrix l = { 0, 1, ⋯ , K (0 ≤ K ≤ N ) K = 0, 1, ⋯ , 2 N K − N , ⋯ , N ( N < K ≤ 2 N )

8 BC for zero modes ⃗ ⃗ Boundary condition on quantum graph Current conservation ⃗ ⃗ ⃗ ⃗ ⃗ L 0 < y < L 1 L N − 1 < y < L N L 1 < y < L 2 N − l independent vectors F ( i ) ( i = 1, ⋯ , N − l ) L 2 < y < L 3 ( j = 1, ⋯ , K − l ) K − l independent vectors G ( j ) Ψ ( x , y ) i γ 5 Ψ ( x , y ) j ( y ) = ¯ f ( i ) 0 ( y ) ( i = 1, ⋯ , N − l ) N N ∑ ∑ j ( L a − ε ) j ( L a − 1 + ε ) = g ( j ) 0 ( y ) ( j = 1, ⋯ , K − l ) a =1 a =1 rank( Δ ( f ) ) = l F ( i ) = 0 G ( j ) = 0 Δ ( f )† Δ ( g )† rank( Δ ( g ) ) = N − K + l Δ ( f ) : Δ ( g ) : N × l complex matrix N × ( N − K + l ) complex matrix l = { 0, 1, ⋯ , K (0 ≤ K ≤ N ) K = 0, 1, ⋯ , 2 N K − N , ⋯ , N ( N < K ≤ 2 N )

Generations of fermions in SM # of 3 generations for 9 # of m n ψ ( i ) ψ ( i ) R,0 ( x ) L,0 ( x ) K = 0, 1, ⋯ , 2 N ⏟ ⏟ l = { ⋯ ⋯ 0, 1, ⋯ , K (0 ≤ K ≤ N ) 0 K − N , ⋯ , N ( N < K ≤ 2 N ) N − l K − l − = N − K | N − K | chiral fermions K = N − 3 ( N + 3)

Contents • Introduction • Berry’s connections and instantons 10 • Quantum graph and extra dimensional model

Contents • Introduction 10 • Quantum graph and extra dimensional model • Berry’s connections and instantons Topological quantities monopole, instanton, ⋯

Berry’s connection Berry’s connection depending on the time Berry’s phase varies adiabatically 11 R 3 R : parameters in the system H ( R ) | ϕ ( R ) ⟩ = E ( R ) | ϕ ( R ) ⟩ C R R 2 R ( t = T ) = R ( t = 0) R 1 | ϕ ( R ( t = 0)) ⟩ ⟶ e i ∫ T 0 dt E ( R ( t ))+ i γ | ϕ ( R ( t = T )) ⟩ γ = i ∮ C ⟨ ϕ ( R ) | d | ϕ ( R ) ⟩ ≡ A

12 depending on the time varies adiabatically Non-abelian Berry’s connection non-abelian Berry’s connection Degeneracy of the states R 3 H ( R ) | ϕ ( i ) ( R ) ⟩ = E ( R ) | ϕ ( i ) ( R ) ⟩ C R R 2 R ( t = T ) = R ( t = 0) R 1 | ϕ ( i ) ( R ( t = 0)) ⟩ ⟶ e i ∫ T ∑ 0 dt E ( R ( t )) ( U ( C ) ) ij | ϕ ( j ) ( R ( t = T )) ⟩ U ( C ) = P exp ( i ∮ C j A ) ( A ) ij = ⟨ ϕ ( i ) ( R ) | d | ϕ ( j ) ( R ) ⟩

Berry’s connection on quantum graph ⃗ If we define parameters in BC mode functions For the quantum graph ⃗ Berry’s connection for zero modes 13 ⃗ ⃗ R 3 ( A ) ij = ⟨ ϕ ( i ) ( R ) | d | ϕ ( j ) ( R ) ⟩ C R 2 R ( t = T ) = R ( t = 0) R 1 ( A ( f ) ) ij = ∫ dy f ( i )† 0 ( y ) d f ( j ) F ( i )† d F ( j ) 0 ( y ) = ( i , j = 1, ⋯ , N − l ) F ≡ ( Δ ( f )† F = 0 F † F = 1 N − l F ( N − l ) ) F (1) ⋯ A ( f ) = F † d F

Yang-Mills instanton Instanton solutions Instanton has a nontrivial topological charge It is known how to construct the general instantons 14 Self dual equation on ℝ 4 ( ˜ 2 ϵ μνρλ F ρλ ) F μν ≡ 1 F μν = ± ˜ F μν 16 π 2 ∫ d 4 x Tr F μν ˜ 1 Q = − F μν ∈ ℤ

ADHM construction for SU(n) instanton complex matrix ADHM construction : method of constructing the general instantons ADHM data : conditions : Introduce ① coordinates of 15 ③ SU(n) instanton with ② Find matrix ( n + 2 k ) × 2 k matrix Δ ( x ) Δ [ n +2 k ] × [2 k ] ( x ) = a [ n +2 k ] × [2 k ] + b [ n +2 k ] × [2 k ] ⋅ ( x μ e μ ⊗ 1 k ) e μ = ( − i σ i , 1 2 ) ( n + 2 k ) × 2 k ℝ 4 [ Δ † Δ , σ i ⊗ 1 k ] = 0 rank( Δ ) = 2 k , ( n + 2 k ) × n V ( x ) Δ † V † V = 1 n [2 k ] × [ n +2 k ] V [ n +2 k ] × [ n ] ( x ) = 0 , Q = k A μ ( x ) = V † ( x ) ∂ μ V ( x )

Berry’s connection and instanton BC In the case that & Normalization Zero mode equation Instanton Boundary condition Normalization Berry’s connection ADHM 16 Berry’s connection ADHM construction Δ † [2 k ] × [ n +2 k ] ( x ) V [ n +2 k ] × [ n ] ( x ) = 0 Δ ( f )†[ l ] × [ N ] F [ N ] × [ N − l ] = 0 F † F = 1 N − l V † V = 1 n A ( f ) = F † d F A ( x ) = V † ( x ) d V ( x ) Δ ( f )[ N ] × [ l ] = Δ [ N ] × [ l ] ( x ) l : even A ( f ) = SU ( N − l ) instanton with Q = l /2

Summary Summary and discussion Instantons in the parameter space 
 of BC on quantum graph ADHM 
 construction Discussion boundary conditions ? 17 • Are there other topological structures in the parameter space of ⋯ • Applications for topological insulator, quantum wire,

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4D mass matrix 4D mass matrix 19 5D Yukawa interacton term overlap integral 4D mass matrix is given by 
 4D mass term L 0 < y < L 1 L N − 1 < y < L N L 1 < y < L 2 ∫ d 4 x ∫ dy g Y ¯ Ψ′ � ( x , y ) Φ ( x , y ) Ψ ( x , y ) + h . c . L 2 < y < L 3 ∫ d 4 x m αβ ¯ ψ ′ � R , α ( x ) ψ L , β ( x ) + h . c . f (0) ′ � g (0) 2 ( y ) 2 ( y ) f (0) ′ � 1 ( y ) m 11 m 12 m 22 m αβ = g Y ∫ dy ϕ ( y ) ( f (0) ′ � * g (0) α ( y ) ) β ( y ) g (0) 1 ( y ) ⋯ α , β : generations ϕ ( y ) : VEV of scalar field y

Fermion generation structure Overlap of localized modes Mass hierarchy 4D mass matrix 20 Flavor mixing Complex mode functions due to BC Origin of CP phase Topology of BC Generations f (0) g (0) 2 ( y ) 2 ( y ) f (0) 1 ( y ) m 11 m 12 m 22 m αβ = g Y ∫ dy ϕ ( y ) ( f (0) ′ � * g (0) α ( y ) ) β ( y ) g (0) 1 ( y ) ⋯ α , β : generations ϕ ( y ) : VEV of scalar field y ( m 11 ≪ m 22 ) Overlap of different generations ( m 12 )