Investigation of the 1+1 dimensional Thirring model using the method - - PowerPoint PPT Presentation

Investigation of the 1+1 dimensional Thirring model using the method of matrix product states

Lattice 2018 East Lansing, MI 26/07/2018 C.-J. David Lin National Chiao-Tung University, Taiwan In collaboration with Mari Carmen Banuls (MPQ Munich), Krzysztof Cichy (Adam Mickiewicz Univ.), Ying-Jer Kao (National Taiwan Univ.), Yu-Ping Lin (Univ. of Colorado, Boulder), David T.-L. Tan (National Chaio-Tung Univ.)

Outline

• Preliminaries
• Lattice simulations, the MPS and DMRG
• Phase structure of the Thirring model
• Remarks and outlook

Preliminaries

Motivation

• Tensor network for lattice field theory
• Topological phase transitions
• Real-time dynamics (long-term goal)

The 1+1 dimensional Thirring model and its duality to the sine-Gordon model

STh

• ψ, ¯

ψ

• =
• d2x
• ¯

ψiγµ∂µψ − m0 ¯ ψψ − g 2 ¯ ψγµψ 2

SSG [φ] =

• d2x

1 2∂µφ(x)∂µφ(x) + α0 κ2 cos (κφ(x))

• φ→φ/κ, and κ2=t

− − − − − − − − − − − − → 1 t

• d2x

1 2∂µφ(x)∂µφ(x) + α0cos (φ(x))

• strong-weak duality g ↔ κ

Works in the zero-charge sector

Dualities and phase structure

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.

Thirring sine-Gordon ¯ µ 1 2⇡ ✏µν@ν ¯ Λ ⇡ cos

Thirring sine-Gordon XY g

4π2 t

− π

T K − π

g ⇠ π/2, Coleman’s instability point

RG flows of the Thirring model

βg ≡ µ dg dµ = −64π m2 Λ2 , βm ≡ µdm dµ = −2(g + π

2 )

g + π m − 256π3 (g + π)2Λ2 m3. Massless Thirring model is a conformal field theory

mass relevant mass irrelevant

g = π

2 , Coleman’s instability point

g

m

Lattice simulations, the MPS and DMRG

Operator formalism and the Hamiltonian

• Operator formaliam of the Thirring model Hamiltonian


• Staggering, J-W transformation ( ):

projected to a sector of total spin JW-trans of the total fermion number, Bosonise to topological index in the SG theory.

¯ HXXZ = ν(g)  1 2

N−2

X

n

• S+

n S− n+1 + S+ n+1S− n

• +a ˜

m0

N−1

X

n

(1)n ✓ Sz

n + 1

2 ◆ +∆(g)

N−1

X

n

✓ Sz

n + 1

2 ◆ ✓ Sz

n+1 + 1

2 ◆

ν(g) = 2γ π sin(γ), ˜ m0 = m0 ν(g), ∆(g) = cos (γ) , with γ = π g 2

HTh = Z dx " i ¯ ψγ1∂1ψ + m0 ¯ ψψ + g 4 ¯ ψγ0ψ 2 g 4 ✓ 1 + 2g π ◆−1 ¯ ψγ1ψ 2 #

C.R. Hagen, 1967

¯ H(penalty)

XXZ

= ¯ HXXZ + λ N−1 X

n=0

Sz

n Starget

!2

S±

j = Sx j ± iSy j

• J. Kogut and L. Susskind, 1975; A. Luther, 1976

• Matrix product operator for the Hamiltonian


• Choices of parameters

Simulation details

W [n] = B B B B B @ 12×2 − 1

2S+ − 1 2S− 2λSz ∆Sz βnSz + α12×2

S− S+ 1 Sz Sz 12×2 1 C C C C C A

βn = ∆ + (−1)n ˜ m0a − 2λ Starget , α = λ 1 4 + S2

target

N ! + ∆ 4

∆(g)

Twenty values of , ranging from -0.9 to 1.0

˜ m0a = 0.0, 0.1, 0.2, 0.3, 0.4

˜ m0a = 0.005, 0.01, 0.02, 0.03, 0.04, 0.06, 0.08, 0.13, 0.16

(run 1) (run 2) Bond dimension System size

D = 50, 100, 200, 300, 400, 500, 600 N = 400, 600, 800, 1000

• Start from random tensors at D=50, then go up in D
• DMRG converges fast at and

⇠ ˜ m0a 6= 0

6 ∆(g) > ⇠ 0.7

Convergence of DMRG

Results for the phase structure

Calabrese-Cardy scaling observed at all values of for

Calabrese-Cardy scaling and the central charge

Entanglement entropy

SN(n) = c 6 ln N π sin ⇣πn N ⌘ + k ,

d ∆(g) = 0

∼ − ˜ m0a = 0

200 400 600 800 1000

site n

0.8 1.0 1.2 1.4 1.6

SN(n) ∆(g) = −0.88, ˜ m0a = 0.0 D = 100 D = 200 D = 400 D = 600

200 400 600 800 1000

site n

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

SN(n) ∆(g) = 0.0, ˜ m0a = 0.0 D = 100 D = 200 D = 400 D = 600

Calabrese-Cardy scaling observed at for

Calabrese-Cardy scaling and the central charge

Entanglement entropy

∆(g) < ⇠ 0.7

⇠ ˜ m0a 6= 0

200 400 600 800 1000

site n

0.6 0.8 1.0 1.2 1.4 1.6

SN(n) ∆(g) = −0.88, ˜ m0a = 0.2 D = 100 D = 200 D = 400 D = 600

200 400 600 800 1000

site n

0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56

SN(n) ∆(g) = 0.0, ˜ m0a = 0.2 D = 100 D = 200 D = 400 D = 600

SN(n) = c 6 ln N π sin ⇣πn N ⌘ + k ,

Calabrese-Cardy scaling and the central charge

Central charge is unity in the critical phase

0.2 0.4 0.6 0.8 1.0

1 6 ln

N

π sin(πn N )

• 0.6

0.8 1.0 1.2 1.4 1.6

SN(n) ∆(g) = −0.88, ˜ m0a = 0.2 D = 100 D = 200 D = 400 D = 600

Entanglement entropy

SN(n) = c 6 ln N π sin ⇣πn N ⌘ + k ,

Soliton correlators

†

↵(x) ↵(y) = ⌥ i|2⇡(x y)|−1|cµ(x y)|−2g2/(2⇡)3

⇥ : exp ⇢ 2⇡i−1 Z y

x

d⇠ ˙ (⇠) ⌥ 1 2i [(y) (x)] + O(x y)2

• :

(35)

S+

mei⇡ Pn−1

j=m+1 Sz j S−

n

Soliton operators Power-law in the critical phase Exponential-law in the gapped phase

Jordan-Wigner transformation

connecting vortex and anti-vortex

Vertex operators

• S. Mandelstam, 1975

Power-law

α = ±

Soliton correlators

1 2 3 4 5

ln (r/a)

−8 −6 −4 −2

ln ¯ G ˜ m0a = 0.0, D = 600 ∆(g) =-0.86 ∆(g) =-0.78 ∆(g) =-0.74 ∆(g) =-0.72 ∆(g) =-0.68 ∆(g) =-0.62

1 2 3 4 5

ln (r/a)

−10 −8 −6 −4 −2

ln ¯ G ˜ m0a = 0.2, D = 600 ∆(g) =-0.86 ∆(g) =-0.78 ∆(g) =-0.74 ∆(g) =-0.72 ∆(g) =-0.68 ∆(g) =-0.62

Evidence for BKT phase transition

G(r) = hψ†

+(r)ψ+(0)i, ¯

G(r) = G(r)/G(0)

Chiral condensate

−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00

∆(g)

0.0 0.1 0.2 0.3 0.4

ˆ χ ˜ m0a = 0.0 ˜ m0a = 0.1 ˜ m0a = 0.2 ˜ m0a = 0.3 ˜ m0a = 0.4

ˆ χ = a

• h ¯

ψψi

• = 1

N

• X

n

(1)nSz

n

• Zero-mass results reproduced using uMPS

Extrapolated to infinite D and N

Chiral condensate

ˆ χ = a

• h ¯

ψψi

• = 1

N

• X

n

(1)nSz

n

• Curvature at small mass in the gapped phase

Extrapolated to infinite D and N chiral symmetry is not spontaneously broken

Mass gap

Heff[M] = ΠM−1 . . . Π0HΠ0 . . . ΠM−1 = H −

M−1

• k=0

Ek|Ψk⟩⟨Ψk|.

Hk

eff[M]=

Hk

eff

+ M−1

m=0 Em×

(Πm)k

eff

|Ψm⟩k

eff

−0.8 −0.6 −0.4 −0.2 0.0 0.2

∆(g)

0.0 0.2 0.4 0.6 0.8

E1 − E0 ˜ m0a = 0.0 ˜ m0a = 0.1 ˜ m0a = 0.2 ˜ m0a = 0.3 ˜ m0a = 0.4

Phase structure of the Thirring model

βg ≡ µ dg dµ = −64π m2 Λ2 , βm ≡ µdm dµ = −2(g + π

2 )

g + π m − 256π3 (g + π)2Λ2 m3. Massless Thirring model is a conformal field theory

gapped critical

g0 = 0, continuum limit

g0 = gc, Coleman’s instability point

g0

am0

Conclusion and outlook

• Evidence for BKT phase transition found using MPS

• Current work for more detailed probe of the phase structure:


• Future projects:

More simulations at small fermion mass Eigenvalue spectrum of the transfer matrix Chemical potential Real-time evolution with a quench Chiral symmetry is not spontaneously broken