Zero-temperature phase structure of the 1+1 dimensional Thirring - - PowerPoint PPT Presentation

SLIDE 1

Zero-temperature phase structure

• f the 1+1 dimensional Thirring model

from matrix product states

RIKEN R-CCS, Kobe, Japan 09/10/2019 C.-J. David Lin National Chiao-Tung University, Taiwan

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.)

arXiv:1908.04536 (submitted to Phys. Rev. D)

SLIDE 2

Outline

• Preliminaries: motivation and introduction
• Lattice formulation and the MPS
• Simulations and numerical results: 


Phase structure of the Thirring model

• Remarks and outlook (spectrum, real-time dynamics)

SLIDE 3

Preliminaries

SLIDE 4

Logic flow

Hamiltonian formalism for QFT Quantum spin model MPS & variational method for obtaining the ground state Compute correlators and excited state spectrum

SLIDE 5

Motivation

• New formulation for lattice field theory
• No sign problem
• Real-time dynamics
• Future quantum computers?

In this talk: BKT phase transition

SLIDE 6

The 1+1 dimensional Thirring model

STh[ , ¯ ] = Z d2x h ¯ iµ@µ m ¯ g 2 ¯ µ ¯ µ i

Conformality of the massless theory Duality with the sine-Gordon theory

SLIDE 7

Bosonisation and duality

• Basic ingredients from free field theories



 


• The dictionary (zero total fermion number)

n

• i=1

eiκiφ(x)

• ren.

=

• i<j

(µ |xi − xj|)κiκj/2π , where

• eiκiφ(x)

bare = (Λ/µ)−κ2

i /4π

eiκiφ(x)

ren.

And similar power law for ¯ ψψ correlators.

¯ µ $ 1 2⇡ ✏µν@ν , ¯ $ Λ ⇡ cos , 4⇡ t = 1 + g ⇡ . STh

• ψ, ¯

ψ

• =
• d2x
• ¯

ψiγµ∂µψ − m0 ¯ ψψ − g 2 ¯ ψγµψ 2 SSG [φ] = 1 t

• d2x

1 2∂µφ(x)∂µφ(x) + α0cos (φ(x))

• m0 = m (µ/Λ)g/(g+π) ,

↵0 = ↵ (µ/Λ)t/4π .

↵0 t = m0Λ ⇡ .

Coleman: Unstable vacuum at

field redifinition, anomaly

g ∼ −π/2 Works in the zero-charge sector

SLIDE 8

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

SLIDE 9

RG flows of the Thirring model

βg ⌘ µ dg dµ = 64π ⇣m Λ ⌘2 , βm ⌘ µdm dµ = m 2(g + π

2 )

g + π

• 256π3

(g + π)2 ⇣m Λ ⌘2

Perturbative expansion in mass

SLIDE 10

Lattice formulation and the MPS

SLIDE 11

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

SLIDE 12

Issue of large Hilbert space & DMRG/MPS

| i 2 |ψi =

d

X

j1,...,jn=1

cj1,...,jn|j1, . . . , jni =

d

X

j1,...,jn=1

cj1,...,jn|j1i ⌦ · · · ⌦ |jni

big: For a spin system of size n and local dimension d, the

dim(H) = O(dn).

Entanglement-based truncation

• f the Hilbert space
• S. White, 1992; M.B. Hasting, 2004; F. Verstraeten and I. Cirac, 2006; …

(Area law of the entanglement entropy)

SLIDE 13

Matrix product states in a nutshell

| i 2 |ψi =

d

X

j1,...,jn=1

cj1,...,jn|j1, . . . , jni =

d

X

j1,...,jn=1

cj1,...,jn|j1i ⌦ · · · ⌦ |jni

cj1,...,jn =

D

X

α,...,ω=1

A(1)

α;j1A(2) β,γ;i2 . . . A(n) ω;jn = A(1) j1 A(2) j2 . . . A(n) jn ,

.

• bers. While

O(dn) man many real

by O(ndD2) xponential in : so

Entanglement-based argument for choosing D (DMRG via MPS)

Bond dim

SLIDE 14

Matrix Product Operator

ˆ O = X

i

⇣ ˆ Ai ˆ Bi+1 + ˆ Bi ˆ Ai+1 ⌘

ˆ = ˆ A ⊗ ˆ B ⊗ 1 ⊗ · · · ⊗ 1 + 1 ⊗ ˆ A ⊗ ˆ B ⊗ 1 ⊗ · · · ⊗ 1 + · · · + ˆ B ⊗ ˆ A ⊗ 1 ⊗ · · · ⊗ 1 + 1 ⊗ ˆ B ⊗ ˆ A ⊗ 1 ⊗ · · · ⊗ 1 + · · ·

b -1 b σ σ´ b1 σ1 bL-1 σL σ´1 σ´L

ˆ O =

• b1,...,bL−1

M σ1,σ′

1

1,b1 M σ2,σ′

2

b1,b2 M σ3,σ′

3

b2,b3 . . . M σL−1,σ′

L−1

bL−3,bL−1 M σL,σ′

L

bL−1,1

σ σ´ σ1 σL σ´1 σ´L

bl−1 bl−1

It is simple to compute local operator matrix elements with canonical states. matrix elements

SLIDE 15

• 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

SLIDE 16

Practice of MPS for DMRG

One step in a sweep of finite-size DMRG

a -1 σ σ ´ a ´ a -1´ a L

SLIDE 17

Simulations and numerical results

SLIDE 18

• 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

SLIDE 19

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

SLIDE 20

Calabrese-Cardy scaling observed at for

Calabrese-Cardy scaling and the central charge

Entanglement entropy

∆(g) < ⇠ 0.7

⇠ ˜ m0a 6= 0

SN(n) = c 6 ln N π sin ⇣πn N ⌘ + k ,

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

0.6 0.7 0.8 0.9 1.0

SN(n) ∆(g) = −0.7, ˜ m0a = 0.2 D = 100 D = 200 D = 400 D = 600

200 400 600 800 1000

site n

0.465 0.470 0.475 0.480 0.485 0.490 0.495 0.500 0.505

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

SLIDE 21

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 ,

SLIDE 22

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

SLIDE 23

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; E. Witten, 1978

Power-law

α = ±

SLIDE 24

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

SLIDE 25

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

SLIDE 26

Probing the phase structure

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

• 0.9
• 0.8
• 0.7
• 0.6
• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

m0a ∆(g) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Cpow−exp

string

(x) = BxηAx + C

SLIDE 27

Results for the phase structure

Value of C critical undetermined massive

0.001 0.01

SLIDE 28

Conclusion and outlook

• Concluding results for phase structure

• Current and future work

KT-type transition observed using the MPS Real-time dynamics and dynamical phase transition Excited-state spectrum and the continuum limit

Exploratory spectrum results presented at Lattice 2017 Exploratory results presented at Lattice 2019

SLIDE 29

Backup slides

SLIDE 30

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

SLIDE 31

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) =

SLIDE 32

Observing DQPT

massive critical

real time t real time t

massive critical

DQPT is a “one-way” transition…

SLIDE 33

DQPT and eigenvalue crossing

D-dependence in the crossing points

SLIDE 34

“Universality” in DQPT?

SLIDE 35

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

SLIDE 36

The Jordan-Wigner transformation

• The fermion fields satisfy

• The Jordan-Wigner transformation



 
 
 expresses the the fermions fields in spins,

• cn, cm
• =
• c†

n, c† m

• = 0,
• cn, c†

m

• = δn,m .
• S±

j = Sx j ± iSy j ,

• Sa

i , Sb j

• = iδi,jabcSc

i .

cn = exp ⎛ ⎝iπ

n−1

• j=1

Sz

j

⎞ ⎠ S−

n , c† n = S+ n exp

⎛ ⎝−iπ

n−1

• j=1

Sz

j

⎞ ⎠

SLIDE 37

The singular value decomposition

Ψi,j can be regarded as elements of a DA × DB (assuming (DA ≥ DB) matrix.

Ψi,j =

DB

• α

Ui,αλα

• V †

α,j

SVD

U †U = 1, V V † = 1

Discard small singular values

Ψi,j =

D′

B<DB

• α

Ui,αλα

• V †

α,j

|Ψ⟩ =

DA

• i=1

DB

• j=1

Ψi,j|i⟩ ⊗ |j⟩

SLIDE 38

Schmidt decomposition and entanglement

|Ψ⟩ =

DA

• i=1

DB

• j=1

DB

• α

Ui,αλαV ∗

α,j|i⟩ ⊗ |j⟩ = DB

• α

λα DA

• i=1

Ui,α|i⟩

• ⊗

⎛ ⎝

DB

• j

V ∗

α,j|j⟩

⎞ ⎠ =

DB

• α

λα|α⟩A ⊗ |α⟩B

ρA = TrB|Ψ⟩⟨Ψ| =

• α

λ2

α|α⟩A A⟨α|

ρB = TrA|Ψ⟩⟨Ψ| =

• α

λ2

α|α⟩B B⟨α|

,

S = −Tr [ρA log (ρA)] = −Tr [ρB log (ρB)] = −

• α

λ2

αlogλ2 α

von Neumann entanglement entropy Reduced density matrices Truncating the Hilbert space by omitting small singular values Throwing away small-entanglement states