Fermion bags, topology and index theorems Shailesh Chandrasekharan - - PowerPoint PPT Presentation

Fermion bags, topology and index theorems

Shailesh Chandrasekharan (Duke University)

Supported by: US Department of Energy, Nuclear Physics Division

Lattice 2016, Southampton UK

work done in collaboration with V. Ayyar

Summary

The concepts of topology and index theorems that arise in the context of QCD, have analogies in simple fermion lattice field theories with staggered fermions, when formulated in the fermion bag approach

Summary

The concepts of topology and index theorems that arise in the context of QCD, have analogies in simple fermion lattice field theories with staggered fermions, when formulated in the fermion bag approach

Summary

This connection gives a more complete perspective

• n fermion mass generation mechanisms,

including a mechanism where fermions acquire a mass through four-fermion condensates instead of fermion bilinear condensates.

The concepts of topology and index theorems that arise in the context of QCD, have analogies in simple fermion lattice field theories with staggered fermions, when formulated in the fermion bag approach

Summary

This connection gives a more complete perspective

• n fermion mass generation mechanisms,

including a mechanism where fermions acquire a mass through four-fermion condensates instead of fermion bilinear condensates. Can non-Abelian gauge theories also demonstrate this alternate mechanism of fermion mass generation?

QCD Partition Function

QCD Partition Function

Partition function of a non-Abelian gauge theory (formal, continuum, finite volume)

QCD Partition Function

Partition function of a non-Abelian gauge theory (formal, continuum, finite volume)

background gauge field integration weight of the background gauge field anti-Hermitian operator depends on the gauge field

D(A) = (γµ∂µ − iAµ)

Z = Z [dA] e−SG (A) Z [dψ dψ] e−ψ D(A) ψ

Fermion Bag Analogy

Fermion Bag Analogy

Partition function of staggered fermion lattice field theories in the fermion bag approach.

Fermion Bag Analogy

Partition function of staggered fermion lattice field theories in the fermion bag approach.

Z = X

B

e−S(B) Z [dψ dψ] e−ψ W (B) ψ

sum over fermion bag configurations weight of a fermion bag configuration anti-Hermitian “fermion bag” matrix depends fermion bag configuration

Topology and Index Theorem in QCD

Topology and Index Theorem in QCD

For every gauge field configuration we can define a topological charge:

Q

A

Topology and Index Theorem in QCD

For every gauge field configuration we can define a topological charge:

Q

A

γ5 D(A) = −D(A) γ5

Dirac operators satisfy: non-zero modes come in pairs

D(A) |z±i = 0, γ5 |z±i = ± |z±i

zero modes are eigenstates of number of modes

n± =

|z±i

D(A) |λi = iλ |λi

D(A) γ5 |λi = iλ γ5 |λi

γ5

The index of the Dirac operator : D(A)

(n+ − n−)

Topology and Index Theorem in QCD

For every gauge field configuration we can define a topological charge:

Q

A

γ5 D(A) = −D(A) γ5

Dirac operators satisfy: non-zero modes come in pairs

D(A) |z±i = 0, γ5 |z±i = ± |z±i

zero modes are eigenstates of number of modes

n± =

|z±i

D(A) |λi = iλ |λi

D(A) γ5 |λi = iλ γ5 |λi

γ5

The index of the Dirac operator : D(A)

(n+ − n−)

Index Theorem: Q = (n+ − n−)

Topology and Index Theorem in QCD

For every gauge field configuration we can define a topological charge:

Q

A

γ5 D(A) = −D(A) γ5

Dirac operators satisfy: non-zero modes come in pairs

D(A) |z±i = 0, γ5 |z±i = ± |z±i

zero modes are eigenstates of number of modes

n± =

|z±i

D(A) |λi = iλ |λi

D(A) γ5 |λi = iλ γ5 |λi

γ5

The index of the Dirac operator : D(A)

(n+ − n−)

Index Theorem: Q = (n+ − n−) D(A) has at least |Q| zero modes

Fermion Bag Analogy

Fermion Bag Analogy

For every fermion bag configuration we can define a topological charge:

Q

B

Fermion Bag Analogy

For every fermion bag configuration we can define a topological charge:

Q

B Fermion bag matrix satisfies: non-zero modes come in pairs zero modes are eigenstates of number of modes

n± =

|z±i

The index of the Dirac operator : (n+ − n−)

W (B) Ξ |λi = iλ Ξ |λi W (B) |λi = iλ |λi

W (B) |z±i = 0, Ξ |z±i = ± |z±i Ξ

Ξ W (B) = − W (B) Ξ

W (B)

Fermion Bag Analogy

For every fermion bag configuration we can define a topological charge:

Q

B Fermion bag matrix satisfies: non-zero modes come in pairs zero modes are eigenstates of number of modes

n± =

|z±i

The index of the Dirac operator : (n+ − n−)

W (B) Ξ |λi = iλ Ξ |λi W (B) |λi = iλ |λi

W (B) |z±i = 0, Ξ |z±i = ± |z±i Ξ

Ξ W (B) = − W (B) Ξ

W (B)

Index Theorem:

Q = (n+ − n−)

Fermion Bag Analogy

For every fermion bag configuration we can define a topological charge:

Q

B Fermion bag matrix satisfies: non-zero modes come in pairs zero modes are eigenstates of number of modes

n± =

|z±i

The index of the Dirac operator : (n+ − n−)

W (B) Ξ |λi = iλ Ξ |λi W (B) |λi = iλ |λi

W (B) |z±i = 0, Ξ |z±i = ± |z±i Ξ

Ξ W (B) = − W (B) Ξ

W (B)

W(B) has at least |Q| zero modes Index Theorem:

Q = (n+ − n−)

Example of a Fermion Bag Approach

Example of a Fermion Bag Approach

Consider free massive staggered fermions:

S = 1 2 X

x,α

ηx,α ⇣ ψxψx+α − ψx+αψx ⌘ + m X

x

ψxψx

X

x,y

ψxDxyψy

Example of a Fermion Bag Approach

Consider free massive staggered fermions:

S = 1 2 X

x,α

ηx,α ⇣ ψxψx+α − ψx+αψx ⌘ + m X

x

ψxψx

X

x,y

ψxDxyψy

D is an anti-Hermitian matrix of the form

D = ✓ C −C T ◆

even

• dd

even

• dd

Example of a Fermion Bag Approach

Consider free massive staggered fermions:

S = 1 2 X

x,α

ηx,α ⇣ ψxψx+α − ψx+αψx ⌘ + m X

x

ψxψx

X

x,y

ψxDxyψy

D is an anti-Hermitian matrix of the form

D = ✓ C −C T ◆

even

• dd

even

• dd

even

• dd

even

• dd

Ξ = ✓ 1 −1 ◆

Example of a Fermion Bag Approach

Consider free massive staggered fermions:

S = 1 2 X

x,α

ηx,α ⇣ ψxψx+α − ψx+αψx ⌘ + m X

x

ψxψx

X

x,y

ψxDxyψy

D is an anti-Hermitian matrix of the form

D = ✓ C −C T ◆

even

• dd

even

• dd

even

• dd

even

• dd

Ξ = ✓ 1 −1 ◆

DΞ = −ΞD

Partition function:

Z = Z [dψ dψ] e−S

Partition function:

Z = Z [dψ dψ] e−S Z = Z [dψ dψ] e−ψDψ e−m P

x ψxψx

Femion Bag Approach:

= Z [dψ dψ] e−ψDψ Y

x

• 1 − m ψxψx
• =

X

B

mk Z [dψ dψ] e−ψ W (B) ψ

Partition function:

Z = Z [dψ dψ] e−S Z = Z [dψ dψ] e−ψDψ e−m P

x ψxψx

Femion Bag Approach:

= Z [dψ dψ] e−ψDψ Y

x

• 1 − m ψxψx
• =

X

B

mk Z [dψ dψ] e−ψ W (B) ψ

configuration B space-time defects free fermion bags k = total number of defects B = B1 + B2 + …

Partition function:

Z = Z [dψ dψ] e−S Z = Z [dψ dψ] e−ψDψ e−m P

x ψxψx

Femion Bag Approach:

= Z [dψ dψ] e−ψDψ Y

x

• 1 − m ψxψx
• =

X

B

mk Z [dψ dψ] e−ψ W (B) ψ

Z = X

B

exp(−S(B)) Z [dψ dψ] e−ψ W (B) ψ

Defining

S(B) = −k log(m)

configuration B space-time defects free fermion bags k = total number of defects B = B1 + B2 + …

Partition function:

Z = Z [dψ dψ] e−S Z = Z [dψ dψ] e−ψDψ e−m P

x ψxψx

Femion Bag Approach:

= Z [dψ dψ] e−ψDψ Y

x

• 1 − m ψxψx
• =

X

B

mk Z [dψ dψ] e−ψ W (B) ψ

Z = X

B

exp(−S(B)) Z [dψ dψ] e−ψ W (B) ψ

Defining

S(B) = −k log(m)

W (B) = ✓ C(B) −C(B)T ◆

even

• dd

even

• dd

anti-Hermitian configuration B space-time defects free fermion bags k = total number of defects B = B1 + B2 + …

Topology and index theorem for a fermion bag

Topology and index theorem for a fermion bag

Topological charge of a fermion bag

Q = neven − nodd

Topology and index theorem for a fermion bag

Topological charge of a fermion bag

Q = neven − nodd

Fermion bag Dirac operator

W (B) = ✓ C(B) −C(B)T ◆

even

• dd

even

• dd

Topology and index theorem for a fermion bag

Topological charge of a fermion bag

Q = neven − nodd

Fermion bag Dirac operator

W (B) = ✓ C(B) −C(B)T ◆

even

• dd

even

• dd

even

• dd

even

• dd

ΞB = ✓ 1 −1 ◆

define

Topology and index theorem for a fermion bag

Topological charge of a fermion bag

Q = neven − nodd

Fermion bag Dirac operator

W (B) = ✓ C(B) −C(B)T ◆

even

• dd

even

• dd

even

• dd

even

• dd

ΞB = ✓ 1 −1 ◆

define

Let n± be the number of zero modes of W (B)

that are also eigenvalues of ΞB with eigenvalues ±1.

Topology and index theorem for a fermion bag

Topological charge of a fermion bag

Q = neven − nodd

Index theorem:

Q = n+ − n−

Fermion bag Dirac operator

W (B) = ✓ C(B) −C(B)T ◆

even

• dd

even

• dd

even

• dd

even

• dd

ΞB = ✓ 1 −1 ◆

define

Let n± be the number of zero modes of W (B)

that are also eigenvalues of ΞB with eigenvalues ±1.

Topology and index theorem for a fermion bag

Topological charge of a fermion bag

Q = neven − nodd

Index theorem:

Q = n+ − n−

Fermion bag Dirac operator

W (B) = ✓ C(B) −C(B)T ◆

even

• dd

even

• dd

has at least zero modes.

|Q|

W (B)

even

• dd

even

• dd

ΞB = ✓ 1 −1 ◆

define

Let n± be the number of zero modes of W (B)

that are also eigenvalues of ΞB with eigenvalues ±1.

Nf = 1 QCD Chiral Condensate

Nf = 1 QCD Chiral Condensate

Partition function gets contribution only from Q = 0 sector.

Z = Z [dAQ=0] e−SG (AQ=0) Det(D(AQ=0))

Nf = 1 QCD Chiral Condensate

Partition function gets contribution only from Q = 0 sector.

Z = Z [dAQ=0] e−SG (AQ=0) Det(D(AQ=0))

Chiral Condensate gets contribution from the sector with topological charge one.

hψψi = 1 Z Z [dA dψ dψ] e−S n 1 V Z d4x ψ(x)ψ(x)

• hψψi =

1 ZV Z [dAQ=1] e−SG (AQ=1) ⇣ Y

λ

λ(AQ=1) ⌘

Nf = 1 QCD Chiral Condensate

Partition function gets contribution only from Q = 0 sector.

Z = Z [dAQ=0] e−SG (AQ=0) Det(D(AQ=0))

Chiral Condensate gets contribution from the sector with topological charge one.

hψψi = 1 Z Z [dA dψ dψ] e−S n 1 V Z d4x ψ(x)ψ(x)

• hψψi =

1 ZV Z [dAQ=1] e−SG (AQ=1) ⇣ Y

λ

λ(AQ=1) ⌘

Anomalous chiral symmetry breaking!

Fermion Bag Analogy

Fermion Bag Analogy

S = 1 2 X

x,α

ηx,α ⇣ ψxψx+α − ψx+αψx ⌘ + m X

x

ψxψx Partition function gets contribution from Q = 0 sectors.

Z = X

BQ=0

ek log(m) Det(W (BQ=0))

Chiral condensate get contribution from Q=1 sectors!

hψψi = 1 ZV X

BQ=1

ek log(m) ⇣ Y

λ

λ(BQ=1) ⌘

Fermion Bag Analogy

S = 1 2 X

x,α

ηx,α ⇣ ψxψx+α − ψx+αψx ⌘ + m X

x

ψxψx Partition function gets contribution from Q = 0 sectors.

Z = X

BQ=0

ek log(m) Det(W (BQ=0))

Chiral condensate get contribution from Q=1 sectors!

hψψi = 1 ZV X

BQ=1

ek log(m) ⇣ Y

λ

λ(BQ=1) ⌘

fermion bag with Q = 1

Fermion Bag Analogy

S = 1 2 X

x,α

ηx,α ⇣ ψxψx+α − ψx+αψx ⌘ + m X

x

ψxψx Partition function gets contribution from Q = 0 sectors.

Z = X

BQ=0

ek log(m) Det(W (BQ=0))

Chiral condensate get contribution from Q=1 sectors!

hψψi = 1 ZV X

BQ=1

ek log(m) ⇣ Y

λ

λ(BQ=1) ⌘

fermion bag with Q = 1 Anomalous chiral symmetry breaking ∼ Explicit chiral symmetry breaking

Nf = 2 QCD

Nf = 2 QCD

Chiral condensate vanishes due to additional chiral symmetries! hψψi = 0

Nf = 2 QCD

Chiral susceptibility is non-zero and gets contribution from the sectors, Q = 0 and Q=1.

χ = 1 ZV Z [dA dψ dψ] e−S n Z d4xψ(x)ψ(x)

• n Z

d4yψ(y)ψ(y)

• χ =

1 ZV Z [dAQ=0] e−SG (AQ=0) Det(D(AQ=0)) n X

λ

1 λ2

• + …

Chiral condensate vanishes due to additional chiral symmetries! hψψi = 0

Nf = 2 QCD

Chiral susceptibility is non-zero and gets contribution from the sectors, Q = 0 and Q=1.

χ = 1 ZV Z [dA dψ dψ] e−S n Z d4xψ(x)ψ(x)

• n Z

d4yψ(y)ψ(y)

• χ =

1 ZV Z [dAQ=0] e−SG (AQ=0) Det(D(AQ=0)) n X

λ

1 λ2

• + …

Spontaneous chiral symmetry breaking requires χ ∼ Σ2V Chiral condensate vanishes due to additional chiral symmetries! hψψi = 0

Nf = 2 QCD

Chiral susceptibility is non-zero and gets contribution from the sectors, Q = 0 and Q=1.

χ = 1 ZV Z [dA dψ dψ] e−S n Z d4xψ(x)ψ(x)

• n Z

d4yψ(y)ψ(y)

• χ =

1 ZV Z [dAQ=0] e−SG (AQ=0) Det(D(AQ=0)) n X

λ

1 λ2

• + …

Spontaneous chiral symmetry breaking requires χ ∼ Σ2V Chiral condensate vanishes due to additional chiral symmetries! hψψi = 0 This implies that the smallest non-zero eigenvalues in the Q=0 must scale as λ ∼

1 ΣV

Nf = 2, Fermion Bag Analogy

Nf = 2, Fermion Bag Analogy

Action (Nf = 2 model)

S = 1 2 X

x,α,i=1,2

ηx,α ⇣ ψx,iψx+α,i − ψx+α,iψx,i ⌘ − U X

x

ψx,1ψx,1ψx,2ψx,2

Ayyar, SC, PRD91 (2015) 6, 065035, PRD93 (2016) 8, 081701, arXiv:1606.06312

Nf = 2, Fermion Bag Analogy

Z = Z [dψ] e− 1

2 ψT Mψ Y

x

• 1 + U ψx,1ψx,2ψx,3ψx,4
• Z =

X

B

ek log(U) ⇣ Det(W (B)) ⌘2

Partition function Action (Nf = 2 model)

S = 1 2 X

x,α,i=1,2

ηx,α ⇣ ψx,iψx+α,i − ψx+α,iψx,i ⌘ − U X

x

ψx,1ψx,1ψx,2ψx,2

Ayyar, SC, PRD91 (2015) 6, 065035, PRD93 (2016) 8, 081701, arXiv:1606.06312

Nf = 2, Fermion Bag Analogy

Z = Z [dψ] e− 1

2 ψT Mψ Y

x

• 1 + U ψx,1ψx,2ψx,3ψx,4
• Z =

X

B

ek log(U) ⇣ Det(W (B)) ⌘2

Partition function

ψxψx

Action (Nf = 2 model)

S = 1 2 X

x,α,i=1,2

ηx,α ⇣ ψx,iψx+α,i − ψx+α,iψx,i ⌘ − U X

x

ψx,1ψx,1ψx,2ψx,2

Ayyar, SC, PRD91 (2015) 6, 065035, PRD93 (2016) 8, 081701, arXiv:1606.06312

Nf = 2, Fermion Bag Analogy

Z = Z [dψ] e− 1

2 ψT Mψ Y

x

• 1 + U ψx,1ψx,2ψx,3ψx,4
• Z =

X

B

ek log(U) ⇣ Det(W (B)) ⌘2

Partition function Chiral condensate vanishes

hψψi = 0

ψxψx

Action (Nf = 2 model)

S = 1 2 X

x,α,i=1,2

ηx,α ⇣ ψx,iψx+α,i − ψx+α,iψx,i ⌘ − U X

x

ψx,1ψx,1ψx,2ψx,2

Ayyar, SC, PRD91 (2015) 6, 065035, PRD93 (2016) 8, 081701, arXiv:1606.06312

hψx,aψx,b ψy,aψy,bi = Z [dψdψ] e−S ψx,aψx,b ψy,aψy,b

Fermion bilinear correlation function If B = B1 + B2 + … then, it vanishes unless x and y are in the same bag!

hψx,aψx,b ψy,aψy,bi = Z [dψdψ] e−S ψx,aψx,b ψy,aψy,b

Fermion bilinear correlation function If B = B1 + B2 + … then, it vanishes unless x and y are in the same bag! Non-zero!

hψx,aψx,b ψy,aψy,bi = Z [dψdψ] e−S ψx,aψx,b ψy,aψy,b

Fermion bilinear correlation function If B = B1 + B2 + … then, it vanishes unless x and y are in the same bag! Non-zero! Zero!

hψx,aψx,b ψy,aψy,bi = Z [dψdψ] e−S ψx,aψx,b ψy,aψy,b

Fermion bilinear correlation function If B = B1 + B2 + … then, it vanishes unless x and y are in the same bag! At sufficiently large coupling U all two point correlation functions will exponentially decay! Non-zero! Zero!

Analogy with QCD?

As in QCD Chiral Susceptibility can get contributions from Q=0,1 sectors.

Analogy with QCD?

As in QCD Chiral Susceptibility can get contributions from Q=0,1 sectors. As in QCD spontaneous chiral symmetry breaking requires

χ ∼ Σ2V

Analogy with QCD?

As in QCD Chiral Susceptibility can get contributions from Q=0,1 sectors. As in QCD spontaneous chiral symmetry breaking requires

χ ∼ Σ2V

This implies that the smallest non-zero eigenvalues in the Q=0 must sector scale as λ ∼ 1 ΣV

Analogy with QCD?

As in QCD Chiral Susceptibility can get contributions from Q=0,1 sectors. A necessary condition is that fermion bags span the entire volume for SSB! As in QCD spontaneous chiral symmetry breaking requires

χ ∼ Σ2V

This implies that the smallest non-zero eigenvalues in the Q=0 must sector scale as λ ∼ 1 ΣV

Analogy with QCD?

As in QCD Chiral Susceptibility can get contributions from Q=0,1 sectors. A necessary condition is that fermion bags span the entire volume for SSB! As in QCD spontaneous chiral symmetry breaking requires

χ ∼ Σ2V

This implies that the smallest non-zero eigenvalues in the Q=0 must sector scale as λ ∼ 1 ΣV But at large U fermion bags are small. Smallest eigenvalues λ 6⇠ 1 ΣV

Analogy with QCD?

As in QCD Chiral Susceptibility can get contributions from Q=0,1 sectors. A necessary condition is that fermion bags span the entire volume for SSB! As in QCD spontaneous chiral symmetry breaking requires

χ ∼ Σ2V

This implies that the smallest non-zero eigenvalues in the Q=0 must sector scale as λ ∼ 1 ΣV But at large U fermion bags are small. Smallest eigenvalues λ 6⇠ 1 ΣV At large U, unlike QCD no SSB, but fermions are still massive!

Analogy with QCD?

Results in the Nf=2 model

Results in the Nf=2 model

In 3D there is evidence for a single exotic transition

U = ∞ U = 0 Scenario B Uc Symmetric Massive Symmetric Massless

PRD91 (2015) 6, 065035, PRD 93 (2016), 081701

Results in the Nf=2 model

In 4D we find evidence for a narrow intermediate spontaneously broken phase

Symmetric Massive Massless Fermions U = 0 U = ∞ Broken Massive

arXiv:1606.06312

In 3D there is evidence for a single exotic transition

U = ∞ U = 0 Scenario B Uc Symmetric Massive Symmetric Massless

PRD91 (2015) 6, 065035, PRD 93 (2016), 081701

Results in the Nf=2 model

In 4D we find evidence for a narrow intermediate spontaneously broken phase

Symmetric Massive Massless Fermions U = 0 U = ∞ Broken Massive

arXiv:1606.06312

In 3D there is evidence for a single exotic transition

U = ∞ U = 0 Scenario B Uc Symmetric Massive Symmetric Massless

PRD91 (2015) 6, 065035, PRD 93 (2016), 081701

In 2D we find evidence for a single asymptotically free (massive) phase

Symmetric Massive U = 0 U = ∞

work in progress

Origin of Fermion Masses

Origin of Fermion Masses

Fermion bilinear condensates form due to anomalous or explicit chiral symmetry breaking through topological space-time defects. Examples: Nf=1 QCD, Nf =1 massive free staggered fermions

Bilinear Condensate Mechanism: A

Origin of Fermion Masses

Fermion bilinear condensates form due to anomalous or explicit chiral symmetry breaking through topological space-time defects. Examples: Nf=1 QCD, Nf =1 massive free staggered fermions

Bilinear Condensate Mechanism: A

Fermion bilinear condensates form due to spontaneous breaking of symmetries that prevent their formation. Nf = 2 QCD is an example of this phenomena.

Bilinear Condensate Mechanism: B

Origin of Fermion Masses

Fermion bilinear condensates form due to anomalous or explicit chiral symmetry breaking through topological space-time defects. Examples: Nf=1 QCD, Nf =1 massive free staggered fermions

Bilinear Condensate Mechanism: A

Fermion bilinear condensates form due to spontaneous breaking of symmetries that prevent their formation. Nf = 2 QCD is an example of this phenomena.

Bilinear Condensate Mechanism: B Four-fermion Condensate Mechanism: C

Four-fermion condensate can form while preserving symmetries that ensure fermion bilinear condensates vanish. But, fermions can still become massive. Nf = 2 interacting fermion model is one example.

The concepts of topology and index theorems that arise in the context of QCD, have analogies in simple fermion lattice field theories with staggered fermions, when formulated in the fermion bag approach

Summary

This connection gives a more complete perspective

• n fermion mass generation mechanisms,

including a mechanism where fermions acquire a mass through four-fermion condensates instead of fermion bilinear condensates. Can non-Abelian gauge theories also demonstrate this alternate mechanism of fermion mass generation?