Matrix Product States and 1+1 dimensional Thirring model

Matrix Product States and

1+1 dimensional Thirring model

QCD and QFT workshop 10/11/2017, NTU

C.-J. David Lin

National Chiao-Tung University, Taiwan

arXiv:1710.09993 (LATTICE 2017)

Collaborators

• Mari-Carmen Banuls (MPI Munich)

• Krzysztof Cichy (Mickiewicz U., Poznan)

• Ying-Jer Kao (National Taiwan U.)

• Yu-Ping Lin (U. of Colorado, Boulder)

• David T.-L. Tan (Nat’l Chiao-Tung U.)

Outline

• Motivation.

• Thirring model as a quantum spin chains.

• Methods for spin chains:


1. Density matrix renormalisation group.

• Methods for spin chains:


2. Matrix product states.

• Exploratory results for Kosterlitz-Thouless phase transition.

• Remarks and outlook.

Motivation

Interests in tensor-network methods

• Hamiltonian formalism, real-time dynamics.

• No sign problem.

• Future quantum simulations?

• New for lattice practirioners.

• Investigating topological aspects of QFT.


A forward-looking method for computational QFT.

The 1+1 dimensional Thirring model and its representations as scalar models

2

I. INTRODUCTION

S

^Th

!ψ, ¯ ψ " =

#

d

²

x $ ¯ ψiγ

^µ

∂

_µ

ψ − m

⁰

ψψ − ¯ g 2

% ψγ ¯

_µ

ψ &

²

'

(1)

Acknowledgments

The authosr thank people for very useful discussions. This work is supported by grants.

2

I. INTRODUCTION

S

^Th

!ψ, ¯ ψ " =

#

d

²

x $ ¯ ψiγ

^µ

∂

_µ

ψ − m

⁰

ψψ − ¯ g 2

% ψγ ¯

_µ

ψ &

²

'

(1)

S

^SG

[φ] =

#

d

²

x ( 1

2 ∂

_µ

φ (x)∂

^µ

φ (x) + α

⁰

κ

²

cos (κφ(x)) )

φ→φ/κ, and κ²=t

− −−−−−−−−−−− → 1 t

#

d

²

x ( 1

2 ∂

_µ

φ (x)∂

^µ

φ (x) + α

⁰

cos (φ(x)) )

(2)

H

^XY

T = − K T

*

⟨i,j⟩

cos (θ

_i

− θ

_j

) (3)

Acknowledgments

The authosr thank people for very useful discussions. This work is supported by grants.

strong-weak duality

high-low temperature duality

4 v _σ

_l

_a

_l−1

_a

_l

M _σ

_l

_a

_l−1

_,a

_l

G(|a − b|) = ⟨e i(θ(a)−θ(b)) ⟩ (16)

G(r) = A r ^{−T /2πK} . (17)

G(r) = A ^′ e ^−r/ξ . (18)

T _c ∼ Kπ/2. (19)

g T

g _c ∼ −π/2. (20)

g ↔ κ

Acknowledgments

The authosr thank people for very useful discussions. This work is supported by grants.

2

I. INTRODUCTION

S

_Th

⇥

, ¯ ⇤

= Z

d

²

x h

i ¯

^µ

@

_µ

m

₀

¯ g

2 ¯

_µ ²

i

(1)

S

_SG

[ ] = Z

d

²

x

 1

2 @

_µ

(x)@

^µ

(x) + ↵

₀



²

cos ( (x))

! /, and ²=t

! 1 t

Z

d

²

x

 1

2 @

_µ

(x)@

^µ

(x) ↵

₀

cos ( (x)) (2)

S

_SG

[ ] = 1 t

Z

d

²

x

 1

2 @

_µ

(x)@

^µ

(x) + ↵

₀

cos ( (x)) (3)

H

_XY

T = K

T

X

hi,ji

cos (✓

_i

✓

_j

) (4)

*

_n

Y

i=1

e

^{ii (x)}

+

ren.

= Y

i<j

(µ |x

ⁱ

x

_j

|)

^{ij/2⇡}

, where h

e

^{ii (x)}

i

bare

= (⇤/µ)

^2i/4⇡

h

e

^{ii (x)}

i

ren.

(5)

And similar power law for ¯ correlators.

The K-T phase transition at T ⇠ K⇡/2 in the XY model.

The phase boundary at t ⇠ 8⇡ in the sine-Gordon theory.

The cosine term becomes relevant or irrelevant.

H

_Th

= Z

dx { i (

d^⇤

@

_{1 u}

+

_u^⇤

@

_{1 d}

) + m

₀

(

_u^⇤ _{u d}^⇤ _d

) + 2g (

_u^⇤ _{u d}^⇤ _d

) } (6)

H

_Th^(latt)

= i 2a

N

X

2 n=0

⇣ c

^†_n

c

_n+1

c

^†_n+1

c

_n

⌘

+ m

₀

N

X

1 n=0

( 1)

ⁿ

c

^†_n

c

_n

+ 2g a

N

2 1

X

n=0

c

^†_2n

c

_2n

c

^†_2n+1

c

_2n+1

(7)

0

=

₃

,

₁

= i

₂

u

!

^p1_a

c

_2n

,

_d

!

^p1_a

c

_2n+1

c

_n

, c

_m

= c

^†_n

, c

^†_m

= 0, c

_n

, c

^†_m

=

_n,m

. c

_n

= exp ⇣

i⇡ P

n 1

j=1

S

_j^z

⌘

S

_n

, c

^†_n

= S

_n⁺

exp ⇣

i⇡ P

n 1

j=1

S

_j^z

⌘ ,

Euclidean

Topological phase transition, critical phase,…

Summary of the dualities

2

I. INTRODUCTION

S

_Th

!ψ, ¯ ψ " =

#

d

²

x $ ¯ ψiγ

^µ

∂

_µ

ψ − m

₀

ψψ − ¯ g 2

% ψγ ¯

_µ

ψ &

2

'

(1)

S

_SG

[φ] =

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

κ

²

cos (κφ(x)) )

φ→φ/κ, and κ²=t

−−−−−−−−−−−−→ 1 t

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

cos (φ(x)) )

(2)

S

_SG

[φ] = 1 t

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

cos (φ(x)) )

(3)

H

_XY

T = −K T

*

⟨i,j⟩

cos (θ

_i

− θ

_j

) (4)

+

_n

,

i=1

e

^iκiφ(x)

-

ren.

= ,

i<j

(µ |x

_i

− x

_j

|)

^κiκj/2π

, where $

e

^iκiφ(x)

'

bare

= (Λ/µ)

^−κ2i/4π

$

e

^iκiφ(x)

'

ren.

(5)

And similar power law for ¯ ψψ correlators.

The K-T phase transition at T ∼ Kπ/2 in the XY model.

The phase boundary at t ∼ 8π separates the phases where the cosine term becomes relevant or irrelevant.

Acknowledgments

The authosr thank people for very useful discussions. This work is supported by grants.

2

I. INTRODUCTION

S

_Th

!ψ, ¯ ψ" =

#

d

²

x $ ¯ ψiγ

^µ

∂

_µ

ψ − m

₀

ψψ − ¯ g 2

% ψγ ¯

_µ

ψ &

2

'

(1)

S

_SG

[φ] =

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

κ

²

cos (κφ(x)) )

φ→φ/κ, and κ²=t

−−−−−−−−−−−−→ 1 t

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

cos (φ(x)) )

(2)

S

_SG

[φ] = 1 t

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

cos (φ(x)) )

(3)

H

_XY

T = −K T

*

⟨i,j⟩

cos (θ

_i

− θ

_j

) (4)

+

_n

,

i=1

e

^iκiφ(x)

-

ren.

= ,

i<j

(µ |x

_i

− x

_j

|)

^κiκj/2π

, where $

e

^iκiφ(x)

'

bare

= (Λ/µ)

^−κ2i/4π

$

e

^iκiφ(x)

'

ren.

(5)

And similar power law for ¯ ψψ correlators.

The K-T phase transition at T ∼ Kπ/2 in the XY model.

The phase boundary at t ∼ 8π in the sine-Gordon theory.

The cosine term becomes relevant or irrelevant.

Acknowledgments

The authosr thank people for very useful discussions. This work is supported by grants.

2

I. INTRODUCTION

S

_Th

!ψ, ¯ ψ" =

#

d

²

x $ ¯ ψiγ

^µ

∂

_µ

ψ − m

₀

ψψ − ¯ g 2

% ψγ ¯

_µ

ψ &

2

'

(1)

S

_SG

[φ] =

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

κ

²

cos (κφ(x)) )

φ→φ/κ, and κ²=t

−−−−−−−−−−−−→ 1 t

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

cos (φ(x)) )

(2)

S

_SG

[φ] = 1 t

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

cos (φ(x)) )

(3)

H

_XY

T = −K T

*

⟨i,j⟩

cos (θ

_i

− θ

_j

) (4)

+

_n

,

i=1

e

^iκiφ(x)

-

ren.

= ,

i<j

(µ |x

_i

− x

_j

|)

^κiκj/2π

, where $

e

^iκiφ(x)

'

bare

= (Λ/µ)

^−κ2i/4π

$

e

^iκiφ(x)

'

ren.

(5)

And similar power law for ¯ ψψ correlators.

The K-T phase transition at T ∼ Kπ/2 in the XY model.

The phase boundary at t ∼ 8π in the sine-Gordon theory.

The cosine term becomes relevant or irrelevant.

Acknowledgments

The authosr thank people for very useful discussions. This work is supported by grants.

Quantities Thirring sine-Gordon XY

vector current ¯

_µ

1

2⇡ ✏

_µ⌫

@

_⌫

chiral condensate ¯ ⇤

⇡ cos

Table 2: Correspondence between the massive Thirring model, sine-Gordon model and the classical XY model.

4 Tensor Network methods

4.1 Singular Value Decomposition (SVD) 4.2 Matrix Product States (MPS)

4.3 Matrix Product Operators (MPO)

4.4 Density Matrix Renormalization Group (DMRG)

5 Preliminaries on the lattice calculation

In this section we will carefully investigate the lattice version of the Hamiltonian, and the other physical quantities. We will first examine the discretization, and then write down those quantities in the spin language by the Jordan-Wigner transformation.

5.1 Staggered fermions in the Hamiltonian formalism

Let’s first consider the staggered fermion in the Hamiltonian formalism. The di↵erence between the Hamiltonian and action is that the spin diagonalization must be done in the di↵erent way. This is basically because of the additional

0

appeared in the Hamiltonian formalism, while in action

₀

was absorbed as the part of ¯ .

Let’s look into an example of the free Dirac fermion in two dimensions S[ , ¯ ] =

Z

d

²

x ⇥ ¯ i

^µ

@

_µ

m

₀

¯ ⇤

. (27)

We first compute the conjugate momentum

⇧ = L

(@

₀

) = ¯ i

⁰

= i

⁺

. (28) Therefore, the Hamiltonian reads to

H = Z

dx (⇧@

₀

L )

= Z

dx ⇥

i

^{+ 0 1}

@

₁

+ m

₀ ^{+ 0}

⇤ .

(29)

7

4

Thirring sine-Gordon XY

g

^4π

2

t

− π

T

K

− π

v _σ

_l

_a

_l−1

_a

_l

M _σ

_l

_a

_l−1

_,a

_l

G(|a − b|) = ⟨e i(θ(a)−θ(b)) ⟩ (16)

G(r) = A r ^{−T /2πK} . (17)

G(r) = A ^′ e ^−r/ξ . (18)

T _c ∼ Kπ/2. (19)

g T

g _c ∼ −π/2. (20)

g ↔ κ

E _pair ∼ log (|r ₁ − r ₂ |/a)

S _r = ! cosθ _r sinθ _r

"

(21)

Acknowledgments

The authosr thank people for very useful discussions. This work is supported by grants.

Thirring model as a spin chain

Staggering the Thirring model

• The continuum Hamiltonian



 


• Staggered regularisation a’la Kogut and Susskind

2

I. INTRODUCTION

S

_Th

!ψ, ¯ ψ" =

#

d

²

x $ ¯ ψiγ

^µ

∂

_µ

ψ − m

₀

ψψ − ¯ g 2

% ψγ ¯

_µ

ψ &

2

'

(1)

S

_SG

[φ] =

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

κ

²

cos (κφ(x)) )

φ→φ/κ, and κ²=t

−−−−−−−−−−−−→ 1 t

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

cos (φ(x)) )

(2)

S

_SG

[φ] = 1 t

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

cos (φ(x)) )

(3)

H

_XY

T = −K T

*

⟨i,j⟩

cos (θ

_i

− θ

_j

) (4)

+

_n

,

i=1

e

^iκiφ(x)

-

ren.

= ,

i<j

(µ |x

_i

− x

_j

|)

^κiκj/2π

, where $

e

^iκiφ(x)

'

bare

= (Λ/µ)

^−κ2i/4π

$

e

^iκiφ(x)

'

ren.

(5)

And similar power law for ¯ ψψ correlators.

The K-T phase transition at T ∼ Kπ/2 in the XY model.

The phase boundary at t ∼ 8π in the sine-Gordon theory.

The cosine term becomes relevant or irrelevant.

H

_Th

=

#

dx {−i (ψ

_d^∗

∂

₁

ψ

_u

+ ψ

_u^∗

∂

₁

ψ

_d

) + m

₀

(ψ

_u^∗

ψ

_u

− ψ

_d^∗

ψ

_d

) + 2g (ψ

_u^∗

ψ

_u

ψ

_d^∗

ψ

_d

)} (6)

H

_Th^(latt)

= − i 2a

N −2

*

n=0

. c

^†_n

c

_n+1

− c

^†_n+1

c

_n

/

+ m

₀

N −1

*

n=0

(−1)

ⁿ

c

^†_n

c

_n

+ 2g a

N 2 −1

*

n=0

c

^†_2n

c

_2n

c

^†_2n+1

c

_2n+1

(7)

γ

₀

= σ

₃

, γ

₁

= iσ

₂

ψ

_u

→

^√1_a

c

_2n

, ψ

_d

→

^√1_a

c

_2n+1

Acknowledgments

The authosr thank people for very useful discussions. This work is supported by grants.

2

I. INTRODUCTION

S

_Th

!ψ, ¯ ψ" =

#

d

²

x $ ¯ ψiγ

^µ

∂

_µ

ψ − m

₀

ψψ − ¯ g 2

% ψγ ¯

_µ

ψ &

2

'

(1)

S

_SG

[φ] =

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

κ

²

cos (κφ(x)) )

φ→φ/κ, and κ²=t

−−−−−−−−−−−−→ 1 t

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

cos (φ(x)) )

(2)

S

_SG

[φ] = 1 t

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

cos (φ(x)) )

(3)

H

XY

T = −K T

*

⟨i,j⟩

cos (θ

_i

− θ

_j

) (4)

+

_n

,

i=1

e

^iκiφ(x)

-

ren.

= ,

i<j

(µ |x

_i

− x

_j

|)

^κiκj/2π

, where $

e

^iκiφ(x)

'

bare

= (Λ/µ)

^{−κ2i /4π}

$

e

^iκiφ(x)

'

ren.

(5)

And similar power law for ¯ ψψ correlators.

The K-T phase transition at T ∼ Kπ/2 in the XY model.

The phase boundary at t ∼ 8π in the sine-Gordon theory.

The cosine term becomes relevant or irrelevant.

H

_Th

=

#

dx {−i (ψ

_d^∗

∂

₁

ψ

_u

+ ψ

_u^∗

∂

₁

ψ

_d

) + m

₀

(ψ

_u^∗

ψ

_u

− ψ

_d^∗

ψ

_d

) + 2g (ψ

_u^∗

ψ

_u

ψ

_d^∗

ψ

_d

)} (6)

H

_Th^(latt)

= − i 2a

N −2

*

n=0

. c

^†_n

c

_n+1

− c

^†_n+1

c

_n

/

+ m

₀

N −1

*

n=0

(−1)

ⁿ

c

^†_n

c

_n

+ 2g a

N 2 −1

*

n=0

c

^†_2n

c

_2n

c

^†_2n+1

c

_2n+1

(7)

γ

₀

= σ

₃

, γ

₁

= iσ

₂

ψ

_u

→

^√1_a

c

_2n

, ψ

_d

→

^√1_a

c

_2n+1

Acknowledgments

The authosr thank people for very useful discussions. This work is supported by grants.

2

I. INTRODUCTION

S

_Th

!ψ, ¯ ψ" =

#

d

²

x $ ¯ ψiγ

^µ

∂

_µ

ψ − m

₀

ψψ − ¯ g 2

% ψγ ¯

_µ

ψ &

2

'

(1)

S

_SG

[φ] =

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

κ

²

cos (κφ(x)) )

φ→φ/κ, and κ²=t

−−−−−−−−−−−−→ 1 t

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

cos (φ(x)) )

(2)

S

_SG

[φ] = 1 t

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

cos (φ(x)) )

(3)

H

_XY

T = −K T

*

⟨i,j⟩

cos (θ

_i

− θ

_j

) (4)

+

_n

,

i=1

e

^iκiφ(x)

-

ren.

= ,

i<j

(µ |x

_i

− x

_j

|)

^κiκj/2π

, where $

e

^iκiφ(x)

'

bare

= (Λ/µ)

^−κ2i/4π

$

e

^iκiφ(x)

'

ren.

(5)

And similar power law for ¯ ψψ correlators.

The K-T phase transition at T ∼ Kπ/2 in the XY model.

The phase boundary at t ∼ 8π in the sine-Gordon theory.

The cosine term becomes relevant or irrelevant.

H

_Th

=

#

dx {−i (ψ

_d^∗

∂

₁

ψ

_u

+ ψ

_u^∗

∂

₁

ψ

_d

) + m

₀

(ψ

_u^∗

ψ

_u

− ψ

_d^∗

ψ

_d

) + 2g (ψ

_u^∗

ψ

_u

ψ

_d^∗

ψ

_d

)} (6)

H

_Th^(latt)

= − i 2a

N −2

*

n=0

. c

^†_n

c

_n+1

− c

^†_n+1

c

_n

/

+ m

₀

N −1

*

n=0

(−1)

ⁿ

c

^†_n

c

_n

+ 2g a

N 2 −1

*

n=0

c

^†_2n

c

_2n

c

^†_2n+1

c

_2n+1

(7)

Acknowledgments

The authosr thank people for very useful discussions. This work is supported by grants.

No doubler

2

I. INTRODUCTION

S

_Th

⇥

, ¯ ⇤

= Z

d

²

x h

i ¯

^µ

@

_µ

m

₀

¯ g

2 ¯

_µ ²

i

(1)

S

_SG

[ ] = Z

d

²

x

 1

2 @

_µ

(x)@

^µ

(x) + ↵

₀



²

cos ( (x))

! /, and ²=t

! 1 t

Z

d

²

x

 1

2 @

_µ

(x)@

^µ

(x) ↵

₀

cos ( (x)) (2)

S

_SG

[ ] = 1 t

Z

d

²

x

 1

2 @

_µ

(x)@

^µ

(x) + ↵

₀

cos ( (x)) (3)

H

_XY

T = K

T

X

hi,ji

cos (✓

_i

✓

_j

) (4)

*

_n

Y

i=1

e

^{ii (x)}

+

ren.

= Y

i<j

(µ |x

ⁱ

x

_j

|)

^{ij/2⇡}

, where h

e

^{ii (x)}

i

bare

= (⇤/µ)

^2i/4⇡

h

e

^{ii (x)}

i

ren.

(5)

And similar power law for ¯ correlators.

The K-T phase transition at T ⇠ K⇡/2 in the XY model.

The phase boundary at t ⇠ 8⇡ in the sine-Gordon theory.

The cosine term becomes relevant or irrelevant.

=

✓

1 2

◆

(6)

H

_Th

= Z

dx { i (

2^⇤

@

_{x 1}

+

₁^⇤

@

_{x 2}

) + m

₀

(

₁^⇤ _{1 2}^⇤ ₂

) + 2g (

₁^⇤ _{1 2}^⇤ ₂

) } (7)

H

_Th^(latt)

= i 2a

N 2

X

n=0

⇣ c

^†_n

c

_n+1

c

^†_n+1

c

_n

⌘

+ m

₀

N 1

X

n=0

( 1)

ⁿ

c

^†_n

c

_n

+ 2g a

N

2 1

X

n=0

c

^†_2n

c

_2n

c

^†_2n+1

c

_2n+1

(8)

0

=

₃

,

₁

= i

₂

u

!

^p1_a

c

_2n

,

_d

!

^p1_a

c

_2n+1

c

_n

, c

_m

= c

^†_n

, c

^†_m

= 0, c

_n

, c

^†_m

=

_n,m

.

,

2

I. INTRODUCTION

S

_Th

⇥

, ¯ ⇤

= Z

d

²

x h

i ¯

^µ

@

_µ

m

₀

¯ g

2 ¯

_µ ²

i

(1)

S

_SG

[ ] = Z

d

²

x

 1

2 @

_µ

(x)@

^µ

(x) + ↵

₀



²

cos ( (x))

! /, and ²=t

! 1 t

Z

d

²

x

 1

2 @

_µ

(x)@

^µ

(x) ↵

₀

cos ( (x)) (2)

S

_SG

[ ] = 1 t

Z

d

²

x

 1

2 @

_µ

(x)@

^µ

(x) + ↵

₀

cos ( (x)) (3)

H

_XY

T = K

T

X

hi,ji

cos (✓

_i

✓

_j

) (4)

*

_n

Y

i=1

e

^{ii (x)}

+

ren.

= Y

i<j

(µ |x

ⁱ

x

_j

|)

^{ij/2⇡}

, where h

e

^{ii (x)}

i

bare

= (⇤/µ)

^2i/4⇡

h

e

^{ii (x)}

i

ren.

(5)

And similar power law for ¯ correlators.

The K-T phase transition at T ⇠ K⇡/2 in the XY model.

The phase boundary at t ⇠ 8⇡ in the sine-Gordon theory.

The cosine term becomes relevant or irrelevant.

=

✓

1 2

◆

(6)

H

_Th

= Z

dx { i (

2^⇤

@

_{x 1}

+

₁^⇤

@

_{x 2}

) + m

₀

(

₁^⇤ _{1 2}^⇤ ₂

) + 2g (

₁^⇤ _{1 2}^⇤ ₂

) } (7)

H

_Th^(latt)

= i 2a

N 2

X

n=0

⇣ c

^†_n

c

_n+1

c

^†_n+1

c

_n

⌘

+ m

₀

N 1

X

n=0

( 1)

ⁿ

c

^†_n

c

_n

+ 2g a

N

2 1

X

n=0

c

^†_2n

c

_2n

c

^†_2n+1

c

_2n+1

(8)

0

=

₃

,

₁

= i

₂

u

!

^p1_a

c

_2n

,

_d

!

^p1_a

c

_2n+1

c

_n

, c

_m

= c

^†_n

, c

^†_m

= 0, c

_n

, c

^†_m

=

_n,m

.

The Jordan-Wigner transformation

• The fermion fields satisfy


• The Jordan-Wigner transformation



 
 


expresses the the fermions fields in spins,

2

I. INTRODUCTION

S

_Th

!ψ, ¯ ψ" =

#

d

²

x $ ¯ ψiγ

^µ

∂

_µ

ψ − m

₀

ψψ − ¯ g 2

% ψγ ¯

_µ

ψ &

2

'

(1)

S

_SG

[φ] =

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

κ

²

cos (κφ(x)) )

φ→φ/κ, and κ²=t

−−−−−−−−−−−−→ 1 t

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

cos (φ(x)) )

(2)

S

_SG

[φ] = 1 t

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

cos (φ(x)) )

(3)

H

_XY

T = −K T

*

⟨i,j⟩

cos (θ

_i

− θ

_j

) (4)

+

_n

,

i=1

e

^iκiφ(x)

-

ren.

= ,

i<j

(µ |x

_i

− x

_j

|)

^κiκj/2π

, where $

e

^iκiφ(x)

'

bare

= (Λ/µ)

^−κ2i/4π

$

e

^iκiφ(x)

'

ren.

(5)

And similar power law for ¯ ψψ correlators.

The K-T phase transition at T ∼ Kπ/2 in the XY model.

The phase boundary at t ∼ 8π in the sine-Gordon theory.

The cosine term becomes relevant or irrelevant.

H

_Th

=

#

dx {−i (ψ

_d^∗

∂

₁

ψ

_u

+ ψ

_u^∗

∂

₁

ψ

_d

) + m

₀

(ψ

_u^∗

ψ

_u

− ψ

_d^∗

ψ

_d

) + 2g (ψ

_u^∗

ψ

_u

ψ

_d^∗

ψ

_d

)} (6)

H

_Th^(latt)

= − i 2a

N −2

*

n=0

. c

^†_n

c

_n+1

− c

^†_n+1

c

_n

/

+ m

₀

N −1

*

n=0

(−1)

ⁿ

c

^†_n

c

_n

+ 2g a

N 2 −1

*

n=0

c

^†_2n

c

_2n

c

^†_2n+1

c

_2n+1

(7)

γ

₀

= σ

₃

, γ

₁

= iσ

₂

ψ

_u

→

^√1_a

c

_2n

, ψ

_d

→

^√1_a

c

_2n+1

0c

_n

, c

_m

1 = 0c

^†_n

, c

^†_m

1 = 0, 0c

_n

, c

^†_m

1 = δ

_n,m

. c

_n

= exp .

iπ 2

_n−1

j=1

S

_j^z

/

S

_n⁻

, c

^†_n

= S

_n⁺

exp .

−iπ 2

_n−1

j=1

S

_j^z

/ ,

Acknowledgments

The authosr thank people for very useful discussions. This work is supported by grants.

2

I. INTRODUCTION

S

_Th

!ψ, ¯ ψ " =

#

d

²

x $ ¯ ψiγ

^µ

∂

_µ

ψ − m

₀

ψψ − ¯ g 2

% ψγ ¯

_µ

ψ &

2

'

(1)

S

_SG

[φ] =

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

κ

²

cos (κφ(x)) )

φ→φ/κ, and κ²=t

−−−−−−−−−−−−→ 1 t

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

cos (φ(x)) )

(2)

S

_SG

[φ] = 1 t

#

d

²

x ( 1

2 ∂

_µ

φ(x)∂

^µ

φ(x) + α

₀

cos (φ(x)) )

(3)

H

_XY

T = −K T

*

⟨i,j⟩

cos (θ

_i

− θ

_j

) (4)

+

_n

,

i=1

e

^iκiφ(x)

-

ren.

= ,

i<j

(µ |x

_i

− x

_j

|)

^κiκj/2π

, where $

e

^iκiφ(x)

'

bare

= (Λ/µ)

^−κ2i/4π

$

e

^iκiφ(x)

'

ren.

(5)

And similar power law for ¯ ψψ correlators.

The K-T phase transition at T ∼ Kπ/2 in the XY model.

The phase boundary at t ∼ 8π in the sine-Gordon theory.

The cosine term becomes relevant or irrelevant.

H

_Th

=

#

dx {−i (ψ

_d^∗

∂

₁

ψ

_u

+ ψ

_u^∗

∂

₁

ψ

_d

) + m

₀

(ψ

_u^∗

ψ

_u

− ψ

_d^∗

ψ

_d

) + 2g (ψ

_u^∗

ψ

_u

ψ

_d^∗

ψ

_d

)} (6)

H

_Th^(latt)

= − i 2a

N −2

*

n=0

. c

^†_n

c

_n+1

− c

^†_n+1

c

_n

/

+ m

₀

N −1

*

n=0

(−1)

ⁿ

c

^†_n

c

_n

+ 2g a

N

2 −1

*

n=0

c

^†_2n

c

_2n

c

^†_2n+1

c

_2n+1

(7)

γ

₀

= σ

₃

, γ

₁

= iσ

₂

ψ

_u

→

^√1_a

c

_2n

, ψ

_d

→

^√1_a

c

_2n+1

0c

_n

, c

_m

1 = 0c

^†_n

, c

^†_m

1 = 0, 0c

_n

, c

^†_m

1 = δ

_n,m

. c

_n

= exp .

iπ 2

_n−1

j=1

S

_j^z

/

S

_n⁻

, c

^†_n

= S

_n⁺

exp .

−iπ 2

_n−1

j=1

S

_j^z

/ , S

_j^±

= S

_j^x

± iS

_j^y

, !S

_i^a

, S

_j^b

" = iδ

_i,j

ϵ

^abc

S

_i^c

.

3

c

_n

= exp

⎛

⎝ iπ

n−1

#

j=1

S

_j^z

⎞

⎠ S

_n⁻

, c

^†_n

= S

_n⁺

exp

⎛

⎝ −iπ

n−1

#

j=1

S

_j^z

⎞

⎠ (8)

S

_j^±

= S

_j^x

± iS

_j^y

, &S

_i^a

, S

_j^b

' = iδ

_i,j

ϵ

^abc

S

_i^c

.

|Ψ⟩ =

DA

#

i=1 DB

#

j=1

Ψ

_i,j

|i⟩ ⊗ |j⟩ (9)

Ψ

_i,j

can be regarded as elements of a D

_A

× D

_B

(assuming (D

_A

≥ D

_B

) matrix.

Ψ

_i,j

=

DB

#

α

U

_i,α

λ

_α

(V

^†

)

α,j

(10)

Ψ

_i,j

=

D_B^′ <DB

#

α

U

_i,α

λ

_α

(V

^†

)

α,j

(11)

U

^†

U = 1, V V

^†

= 1

|Ψ⟩ =

DA

#

i=1

DB

#

j=1 DB

#

α

U

_i,α

λ

_α

V

_α,j^∗

|i⟩ ⊗ |j⟩ =

DB

#

α

λ

_α

*

_D

A

#

i=1

U

_i,α

|i⟩

+

⊗

⎛

⎝

DB

#

j

V

_α,j^∗

|j⟩

⎞

⎠ =

DB

#

α

λ

_α

|α⟩

_A

⊗ |α⟩

_B

. (12)

ρ

_A

= Tr

_B

|Ψ⟩⟨Ψ| = #

α

λ

²_α

|α⟩

_{A A}

⟨α| (13)

ρ

_B

= Tr

_A

|Ψ⟩⟨Ψ| = #

α

λ

²_α

|α⟩

_{B B}

⟨α| (14)

S = −Tr [ρ

_A

log (ρ

_A

)] = −Tr [ρ

_B

log (ρ

_B

)] = − #

α

λ

²_α

logλ

²_α

(15)

U

_σ1,a¹

= A

^σ_a1¹

U

_(a1σ2),a2

= A

^σ_a²_1,a2

Γ−Λ

O = ˆ #

b1,...,b_L−1

M

_1,b^σ1,σ1′

1

M

_b^σ2,σ2′

1,b2

M

_b^σ3,σ3′

2,b3

. . . M

_b^{σL−1,σL−1′}

L−3,b_L−1

M

_b^σL,σL′

L−1,1

(16)

Thirring model as a quantum spin chain

• JW transformation on the Thirring model gives



 
 


• The “penalty term”

5.3.3 Spin Hamiltonian

The lattice Hamiltonian (44) can be further transformed into the spin Hsmilto- nian though the Jordan-Wigner transformation.

H

_spin

= 1 2a

X

n

S

_n⁺

S

_n+1

+ S

_n+1⁺

S

_n

+ m

₀

X

n

( 1)

ⁿ

✓

S

_n^z

+ 1 2

◆

+ 2g a

X

n

✓

S

_2n^z

+ 1 2

◆ ✓

S

_2n+1^z

+ 1 2

◆ .

(45)

Note that the only coupled terms in the four-fermion term are site 2n and site 2n + 1, rather than all the nearest-neighbour coupling as in the Heisenberg XXZ model.

5.3.4 The penalty term

In order to target the excited states with specific total spin z, we can add a penalty term to the spin Hamiltonian (45). That is,

H

_spin^(penalty)

= H

_spin

+

N 1

X

n=0

S

_n^z

S

_target

!

²

. (46)

11 5.3.3 Spin Hamiltonian

The lattice Hamiltonian (44) can be further transformed into the spin Hsmilto- nian though the Jordan-Wigner transformation.

H

_spin

= 1 2a

X

n

S

_n⁺

S

_n+1

+ S

_n+1⁺

S

_n

+ m

₀

X

n

( 1)

ⁿ

✓

S

_n^z

+ 1 2

◆

+ 2g a

X

n

✓

S

_2n^z

+ 1 2

◆ ✓

S

_2n+1^z

+ 1 2

◆ .

(45)

Note that the only coupled terms in the four-fermion term are site 2n and site 2n + 1, rather than all the nearest-neighbour coupling as in the Heisenberg XXZ model.

5.3.4 The penalty term

In order to target the excited states with specific total spin z, we can add a penalty term to the spin Hamiltonian (45). That is,

H

_spin^(penalty)

= H

_spin

+

N

X

1 n=0

S

_n^z

S

_target

!

²

. (46)

11

projected to a sector of total spin

JW-trans of the total fermion number,

Bosonise to topological index in the SG theory.

XXZ model with two external fields

Density matrix RG

The large Hilbert space

2 The Density Matrix Renormalization Group 35

Fig. 2.1 Pictorial represen- tation of the Hamiltonian building recursion explained in the text. At each step, the block size is increased by adding a spin at a time

few dozen states with largest weights, and get rid of the rest. However, this long tail of states with small weights are responsible for most of the interesting physics: the quantum fluctuations, and the difference in weight from one state to another in this tail cannot be necessarily ignored, since they are all of the same order of magnitude.

However, one may notice a simple fact: this is a basis dependent problem! What if, by some smart choice of basis, we find a representation in which the distribution of weights is such that all the weight on the tail is ‘shifted to the left’ on the plot, as shown on the right panel of Fig.2.2. Then, if we truncate the basis, we would not need to worry about the loss of ‘information’. Of course, this is a nice and simple concept that might work in practice, if we knew how to pick the optimal representation. And as it turns out, this is not in principle an easy task. As we shall learn, what we need is a method for quantifying ‘information’.

2.2.3 A Simple Geometrical Analogy

Let us consider a vector in two dimensional space v = (x, y), as shown in Fig.2.3.

We need two basis vectors ˆe¹ and ˆe² to expand it as v = x ˆe¹ + y ˆe². A simple 2D rotation by an angle φ would be represented by an orthogonal matrix

Size of the Hilbert space increases exponentially when the chain grows.

Challenging to diagonalise the Hamiltonian and look for the ground state.