[PPT] - Phase structure and real-time dynamics of the massive Thirring model PowerPoint Presentation

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Phase structure and real-time dynamics of the massive Thirring model in 1+1 dimensions using tensor-network methods

TNSAA 2019 Taipei 06/12/2019 C.-J. David Lin National Chiao-Tung University, Taiwan with Mari Carmen Banuls (MPQ Munich), Krzysztof Cichy (Adam Mickiewicz Univ.), Hao-Ti Hung (National Taiwan Univ.), Ying-Jer Kao (National Taiwan Univ.), Yu-Ping Lin (Univ. of Colorado, Boulder), David T.-L. Tan (National Chaio-Tung Univ.)

Phys. Rev. D 100 (2019) 094504

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LGT in the early days

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Beginning of MC simulations for LGT

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The BMW collaboration, science 322 (2008)

Success for simple quantities

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G. A. Cowan (LHCb collaboration), arXiv:1708.08628.

Success for less simple quantities

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Motivation for HEP

Things that are challenging for Euclidean MC simulations

Further examples: light-cone physics, inelastic scattering,…

…….

See talks by Kuhn and Nakamura

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Motivation for HEP

Topology freezing

Bazavov et al., Phys. Rev. D 98 (2018) 074512

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Feasibility (toy-model) studies for HEP

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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

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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

Picture from: K. Huang and J. Polonyi, 1991

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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

m a s s i r r e l e v a n t

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Beyond the SM, composite Higgs?

12

Higgs boson ~125 GeV Searched up here ~2 TeV

?

The Higgs boson is light

Fermion favours ~1000 TeV ?

Need large anomalous dim to suppress FCNC

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The “conformal windows”

13

Figure credit: F. Sannino

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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

S±

j = Sx j ± iSy j

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

¯ Hsim = ¯ HXXZ ν(g) + λ N−1 X

n=0

Sz

n Starget

!2

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Matrix product operator for the Hamiltonian (bulk)

             

Simulation parameters

Simulation details for the phase structure

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 Bond dimension System size

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

Fourteen values of , ranging from 0 to 0.4

˜ m0a

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Convergence

different convergence properties observed

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Scaling observed at for , and for all values of at

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 ,

∼ − ˜ m0a = 0

d ∆(g) = 0

In the critical phase, c = 1

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Density-density correlators

Evidence for a critical phase

fitted values of A

0.8 0.85 0.9 0.95 1 1.05 1.1

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

0.1 ∆(g) ma=0.005 ma=0.02 ma=0.08 ma=0.3

4
3.5
3
2.5
2
1.5
1
0.8
0.6
0.4
0.2

0.2 0.4 0.6 0.8 1 ∆(g) Czz

pow fit α

Czz

pow-exp fit η

the parameter α and η fo rs: N = 1000, ˜ m0a = 0.02.

Cpow

zz (x) = βxα and

try fitting to

Czz(x) = h ¯ ψψ(x0 + x) ¯ ψψ(x0)iconn

JW trans

!

1 Nx X

n

Sz(n)Sz(n + x) 1 N0 X

n

Sz(n) X

n

Sz(n + 1)

Cpow−exp

zz

(x) = BxηAx

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0.1 0.2 0.3 0.4 0.5 0.6 0.7

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

0.1 ∆(g) ma=0.005 ma=0.02 ma=0.08 ma=0.3

2.00
1.80
1.60
1.40
1.20
1.00
0.80
0.60
0.40
0.20

0.00

1
0.8
0.6
0.4
0.2

0.2 0.4 ∆(g) Cstring

pow fit α

Cstring

pow-exp fit η

Soliton (string) correlators

and

try fitting to

Cstring(x) = hψ†(x0 + x)ψ(x0)i

JW trans

!

1 Nx X

n

S+(n)Sz(n + 1) · · · Sz(n + x 1)S−(n + x)

Similar behaviour in A. Evidence for a critical phase

Cpow

string(x) = βxα + C

the parameter α and η fo rs: N = 1000, ˜ m0a = 0.02.

fitted values of C

the string order

Cpow−exp

string

(x) = BxηAx + C

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Chiral condensate

ˆ χ = a

h ¯

ψψi

= 1

N

X

n

(1)nSz

n

Chiral condensate is not an order parameter

Extrapolated to infinite D and N

Evidence for criticality from other quantities M a s s i v e p h a s e Massive phase

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Phase structure of the Thirring model

quench and real-time dynamic

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Uniform MPS and real-time evolution

doi:10.6342/NTU201802766

A

A A A

⋯ ⋯

A .

,
doi:10.6342/NTU201802766
⋯

⋯

,
A

A l l

= ,

doi:10.6342/NTU201802766
⋯

⋯

,
A

A

=

r r

, .

doi:10.6342/NTU201802766

𝐼 𝐵𝑀 𝐵𝑀 𝐵𝑆 𝐵𝑆 𝐵 𝑀 𝐵 𝑀 𝐵 𝑆 𝐵 𝑆

⋯ ⋯ ⋯ ⋯ ⋯ ⋯

𝐼 𝐵𝑀 𝐵𝑀 𝐵𝑆 𝐵𝑆 𝐵 𝑀 𝐵 𝑀 𝐵 𝑆 𝐵 𝑆

⋯ ⋯ ⋯ ⋯ ⋯ ⋯

𝐼 𝐵𝑀 𝐵𝑀 𝐵𝑆 𝐵𝑆 𝐵 𝑀 𝐵 𝑀 𝐵 𝑆 𝐵 𝑆

⋯ ⋯ ⋯ ⋯ ⋯ ⋯

𝐵𝐷 𝐼 𝐵𝑀 𝐵𝑀 𝐵𝑆 𝐵𝑆 𝐵 𝑀 𝐵 𝑀 𝐵 𝑆 𝐵 𝑆

⋯ ⋯ ⋯ ⋯ ⋯ ⋯

𝐷

= 𝑀 𝐵𝑀 𝐵𝑀 𝐵 𝑀 𝐵 𝑀

⋯ ⋯ ⋯

𝑃 𝑃 𝑆 = 𝐵𝑆 𝐵𝑆 𝐵 𝑆 𝐵 𝑆

⋯ ⋯ ⋯

𝑃 𝑃 , .

𝛽

𝛾 1 2 3 4 5 6 1 2 3 4 5 6 𝑃 𝛽 𝛾 = .

doi:10.6342/NTU201802766

𝐼 𝐵𝑀 𝐵𝑀 𝐵𝑆 𝐵𝑆 𝐵 𝑀 𝐵 𝑀 𝐵 𝑆 𝐵 𝑆

⋯ ⋯ ⋯ ⋯ ⋯ ⋯

𝐼 𝐵𝑀 𝐵𝑀 𝐵𝑆 𝐵𝑆 𝐵 𝑀 𝐵 𝑀 𝐵 𝑆 𝐵 𝑆

⋯ ⋯ ⋯ ⋯ ⋯ ⋯

𝐼 𝐵𝑀 𝐵𝑀 𝐵𝑆 𝐵𝑆 𝐵 𝑀 𝐵 𝑀 𝐵 𝑆 𝐵 𝑆

⋯ ⋯ ⋯ ⋯ ⋯ ⋯

𝐵𝐷 𝐼 𝐵𝑀 𝐵𝑀 𝐵𝑆 𝐵𝑆 𝐵 𝑀 𝐵 𝑀 𝐵 𝑆 𝐵 𝑆

⋯ ⋯ ⋯ ⋯ ⋯ ⋯

𝐷

= 𝑀 𝐵𝑀 𝐵𝑀 𝐵 𝑀 𝐵 𝑀

⋯ ⋯ ⋯

𝑃 𝑃 𝑆 = 𝐵𝑆 𝐵𝑆 𝐵 𝑆 𝐵 𝑆

⋯ ⋯ ⋯

𝑃 𝑃 , .

𝛽

𝛾 1 2 3 4 5 6 1 2 3 4 5 6 𝑃 𝛽 𝛾 = .

Translational invariance in MPS

Finding the infinite BC for amplitudes &

(largest eigenvalue normalised to be 1)

H.N. Phien, G. Vidal and I.P. McCulloch, Phys. Rev. B86, 2012

V. Zauner-Stauber et al, Phys. Rev. B97, 2018

Similar (more complicated) procedure in the variation search for the ground state & Real-time evolution via time-dependent variational principle

J. Haegeman et al, Phys. Rev. Lett.107, 2011

doi:10.6342/NTU201802766 𝑊

𝑀

𝐵 𝐵 𝑊 𝑀 𝑚−1

2

𝑠−1

2

𝐵

⋯ ⋯

𝑡𝑜 𝑡𝑜

𝑚−1

2

𝑠−1

2

𝑄|𝜔(𝐵) =

i d

dt|Ψ(A(t))⟩ = P|Ψ(A)⟩ ˆ H|Ψ(A(t))⟩

𝑊

𝑀

𝐵 𝐵 𝑊 𝑀 𝑚−1

2

𝑠−1

2

𝐵

⋯ ⋯

𝑡𝑜

𝑚−1

2

𝑠−1

2

𝑄|𝜔(𝐵) 𝐼 |𝜔 𝐵 𝑃 𝑃 𝑃 𝑃

⋯ ⋯

=

𝑗 𝑒

𝑒𝑢 |𝜔(𝐵) =

𝑡𝑜−1 𝑡𝑜 𝑡𝑜+1

𝐵

⋯ ⋯

𝐵 (𝑢)
Key: projection to MPS in

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Dynamical quantum phase transition

“Quenching” : Sudden change of coupling strength in time evolution

H(g1)|01i = E(1)

0 |01i

|ψ(t)i = e−iH(g2)t|01i

and Questions: Any singular behaviour? Related to equilibrium PT? L(t) = h01|e−iH(g2)t|01i

g(t) = lim

N→∞

1 N lnL(t)

The Loschmidt echo and the return rate & c.f., the partition function and the free energy In uMPS computed from the largest eigenvalue of the “transfer matrix"

¯ A01

A(t)

{ }

i j Ti,j(t) =

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Observing DQPT

massive critical

real time t real time t

massive critical

DQPT is a “one-way” transition…

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DQPT and eigenvalue crossing

D-dependence in the crossing points

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Bond-dim dependence in DQPT?

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Conclusion and outlook

Concluding results for phase structure 
Exploratory results for real-time dynamics