Matrix Product State 法の Schwinger 模型への応用...2015年9月5日, 有限温度有限温度密度系の物理と格子QCDシミュレーション齋藤華（CCS. 筑波大）

2015年9月5日, 有限温度有限温度密度系の物理と格子ＱＣＤシミュレーション齋藤　華（CCS. 筑波大）

Matrix Product State 法のSchwinger 模型への応用

齋藤　華(CCS, 筑波大)

with M. C. Bañuls, K. Cichy, J. I. Cirac and K. Jansen

H. Saito et al. PoS LATTICE2014, 302, 2014, arXiv:1412.0596M. C. Bańuls et al, arXiv:1505.00279


QCD Phase diagram• QCD Phase diagram : non-perturbative aspect • Lattice QCD simulation • A lot of interests in dense QCD

• critical point • unknown phases

• Lattice QCD at finite chemical potential μ : sign problem

• “Is there another method?”

2

T

Confinement

QGP

??

μ


Tensor network (TN)• An efficient approximation of quantum many-body state from quantum information

• Hamiltonian formalism

• Matrix product state (MPS) : TN for 1d

3

ik: physical indices at site k, m, n (=1,…, D) : indices from this approximation, D : bond dimension

physical indexvirtual indices

site

cf. Lagrangian formalism Y. Shimizu, Y. Kuramashi PRD90, 074503, PRD90, 014508


An example of MPS• 1/2-spin 2 particle system:

• Supposing D = 2,

• By computing trace of products

4

ik = ↑ or ↓ for k = 1, 2


Variational search• For ground (and some excited) state search • Ground state derived by searching minimum of trial energy computed by trial MPS state:

• the minimum searched with variational approach

5with fixing the other elements

: a trial MPS state


Bond dimension D• “the relevant tiny corner” of Hilbert space

• Approximation improved systematically • From quantum information, D << dN/2

6

polynomial

Hilbert space

exponential

MPS with finite D : size of the full Hilbert spaceEx.) 1d spin system

NdD2dN ⇔

The full Hilbert space by D = (dN-1/N)1/2 ~ dN/2


Convergence in D

7

0 0.005 0.01 0.015 0.02 0.025 0.03−1

0

1

2

3

4

5

6x 10−6

1/D

Σ−Σ

Dm

ax

✦ Pure stateN = 300 ⇨ dN/2 = 2150 ~ O (1045)

✦ Mixed state

1.5087e-02

1.5088e-02

1.5089e-02

1.5090e-02

1.5091e-02

0 0.01 0.02 0.03

Σ

1/D

(N = 40)

M. C. Bañuls et al, LAT2013, 332 (2013)

non-zero T (gβ = 0.4)

T = 0 (Dmax = 140)


Schwinger model for Nf = 1• 1+1 dimensional QED model

✴ not QCD, but similar to QCD : confinement, chiral symmetry breaking (via anomaly for Nf =1)

✴ exactly solvable in massless case • Hamiltonian of Schwinger model

8

hopping term

gauge part

mass term

J. Schwinger Phys.Rev. 128 (1962)

T. Banks, L. Susskind and J. Kogut, PRD13, 4 (1973)

N. L. Pak and P. Senjanovic, Phys.Let.B71, 2 (1977), K. Johnson Phys.Let. 5, 4(1963)

Gauss law

⇒ a good test case

where inverse coupling x=1/a2g2, dimensionless mass μ=2m/ag2 and

in spin language(staggered discrieization)


Additional tools: Matrix Product Operator (MPO)

Ex.) MPO of one hopping term with N = 4, open boundary

9

:hopping

where

,

,

, ,

,

,

for left boundary

for right boundary

for bulk n = 2,3


Our previous study• Schwinger model with MPS method • With variational method, computing:

✴ spectrum ✴ (subtracted) chiral condensate:

• Continuum limit: with inverse coupling x = 1/g2a2

10

0 0.05 0.1 0.15 0.2 0.250.16

0.165

0.17

0.175

0.18

0.185

0.19

0.195

1/√

x

m/g = 0

exact 0.1599290.159930(8)

0 0.05 0.1 0.15 0.2 0.250.09

0.095

0.1

0.105

0.11

0.115

0.12

1/√

x

m/g = 0.125

0.092023(4)(H.) 0.0929

in spin language

M. C. Bańuls et al JHEP 1311, 158, LAT2013, 332

(H.) Y. Hosotani arXiv:9703153

Logarithmic correction from analytic form of free theory

Fit function:


Lattice gauge theory (LGT) with TN approach

• Earlier Study: critical behavior of Schwinger model with Density Matrix Renormalization Group

• Nowadays: various branches ✴ Our previous studies ✴ LGT with TN on higher dimension ✴ Real time evolution ✴ Quantum link model ✴ Tensor Renormalization Group

11

T. Byrnes, et al. PRD.66.013002 (2002)

B. Buyens, et al. arXiv:1312.6654

Y. Shimizu, Y. Kuramashi arXiv:1403.0642 (With Lagrangian)

P. Silvi, et al arXiv:1404.7439E. Rico, et al. PRL112, 201601 (2014)

M. C. Bańuls et al JHEP 1311, 158, LAT2013, 332 (2013)

This study


Chiral symmetry restoration of Schwinger model for Nf = 1• Chiral symmetry breaking at T = 0 (via anomaly) ⇔ At high T, the symmetry restoration

• Order parameter : chiral condensate

• Analytic formula

13

where 0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1 1.2

Y

`

zero T limit

smooth curve

I. Sachs and A. Wipf, arXiv:1005.1822

Euler constant


Thermal calculation• Expectation value at finite T • How to calculate the ρ(β)

• Approximation to ρ(β) (MPO ansatz, global opt.)

14

dN x dN elements as variational search, updating tensors M[k] in size of d2 x D2

In β = 0, ρ (β) is identity.

1st step

2nd step

high T → low TEx. ) For fixed δ, larger Nstep (= β / δ ) corresponds to lower Twhere


Simulation

1.From MPS approx., bond dimension D 2. From T evol., step size δ 3. chain length N 4. inverse coupling x

• MPO approximation for ρ(β)

• Open Boundary condition • Four simulation parameters

• To avoid large finite size effect:

15

D → ∞

δ → 0

N → ∞


Four systematic errors

16

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5

Y

g`

N= 20, x=1N= 50, x=6.25N= 80, x=16N=100, x=25N=120, x=36N=160, x=64Sachs and Wipf

From chain length N, inverse coupling x continuum limit with fixed physical length

0

0.05

0.1

0.15

0.2

0 1 2 3 4 5

Y

g`

Sachs and WipfD=20, b=0.001 D=20, b=0.0001D=40, b=0.001 D=40, b=0.0001

From bond dimension D, step size δ


Extrapolations ① ② ③

17

① D → ∞ with fixed δ, N, x

② δ → 0 with fixed N, x

③ N → ∞ with fixed x

at gβ = 0.4

0 5 10 15 200

1

2

3

4x 10ï7

(1/D) × 103

0 10 200246x 10ï5

δ = 5 · 10−4

δ = 7 · 10−4

δ = 10−3

0 0.5 1 1.5x 10ï3

0.014

0.0145

0.015

δ

N =40N =50N =60

0 0.005 0.01 0.015 0.02 0.0250.01

0.012

0.014

0.016

1/N

x =5x =11x =20x =32x =45


0 0.2 0.4 0.6

0.14

0.16

0.18

0.2

1/√

x

Extrapolation ④

18

gβ = 4.0

④ continuum extrapolation

(solid blue)

(dashed red)

at each T


Chiral condensatein continuum limit

19

After eliminating those systematic errors …

0 2 4 60

0.05

0.1

0.15

gβ


Cut-off effect• continuum extrapolation in high T

20

0 0.2 0.4 0.60

0.005

0.01

1/√

x

gβ = 0.4 (solid)

(dashed)

large x required→ large N for finite size effect → large norm of H → extremely small δ for approx. exp (-δHz) ~ 1 - δHz

→ the huge number of steps in Suzuki-Trotter exp.


Another approach• chiral condensate at high T

21

0 0.5 1 1.5 20

0.02

0.04

0.06

gβ

• large electric flux is exponentially suppressed in thermal state:

• additional extrapolation needed


Summary• Computing chiral condensate at finite T in Hamiltonian formalism with tensor network methods

• Evaluating dependence of parameters: bond dimension, step size, system size, inverse of coupling

• By taking continuum limit, we obtained results consistent with an analytic formula.

• Future plans …

22

I. Sachs and A. Wipf, arXiv:1005.1822

Documents

Matrix Product State 法の Schwinger 模型への応用...2015年9月5日, 有限温度有限温度密度系の物理と格子QCDシミュレーション 齋藤 華（CCS. 筑波大）

Matrix Product State 法の Schwinger 模型への応用...2015年9月5日, 有限温度有限温度密度系の物理と格子QCDシミュレーション齋藤華（CCS. 筑波大）