Tensor network approach for chiral symmetry restoration of 1 ......June 22, 2015 @ DESY & HU Lattice seminar H. Saito （NIC, DESY Zeuthen） Tensor network approach for chiral symmetry

June 22, 2015 @ DESY & HU Lattice seminar H. Saito 　（NIC, DESY Zeuthen）

Tensor network approach for chiral symmetry restoration of

1-flavor Schwinger model at finite temperature

Hana Saito (NIC, DESY Zeuthen)

with M. C. Bañuls, K. Cichy, J. I. Cirac and K. JansenH. Saito et al. PoS Lattice, 302 (2014), arXiv:1412.0596

M. C. Bañuls et al, arXiv:1505.00279


QCD Phase diagram• Quark: one of elementary particles • Confinement at low T (QCD Phase diagram)⇒ non-perturbative aspect

• Lattice QCD simulation ✴ At finite temperature and zero chemical potential : Successful!!

✴ At finite chemical potential μ : failing

• But, a lot of interests in dense QCD • critical point at finite μ, • unknown phases at large μ

2

T

Confinement

Ex. Early universe after Big Bang

Fukushima, Hatsuda, Rept. Prog. Phys. 74, 014001 (2011), arXiv:1005.4814[hep-ph]

sQGP

uSCdSCCFL

2SC

Critical Point

Quarkyonic Matter

Quark-Gluon Plasma

Hadronic Phase

Color Superconductors?Te

mpe

ratu

re T

Baryon Chemical Potential mB

Inhomogeneous ScB

Liquid-Gas

Nuclear SuperfluidCFL-K , Crystalline CSCMeson supercurrentGluonic phase, Mixed phase

0


Sign problem• Complex action at finite μ spoils Markov Chain Monte Carlo built on positive probability measure

• To avoid the problem in convention lattice QCD • Reweighting: • Taylor expansion of μ:

• For real time dynamics of high energy physics, alternative approaches required

3

to change the distribution of Monte Carlo ensemble → overlap problem

coefficients calculated with μ = 0 ensemble → a problem of convergence

Complex quark determinant

To establish a promising numerical tool ⇒ our strategy: to employ Hamiltonian approach


Outline • Motivation • 1-flavor Schwinger model • Hamiltonian of Schwinger model

• Tensor Network • Chiral condensate at T = 0 • Chiral symmetry restoration at finite T • Thermal calculation in TN and outcomes • Summary

4


1-flavor Schwinger model• 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 1-flavor Schwinger model

5

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)


Hamiltonian of Schwinger model in continuum• Lagrangian of Schwinger model:

• Hamiltonian via Legendre transformation:

• Gauss’s law: 6

, : 2-component fermion field

∵ i) , ii) temporal gauge

: covariant derivative, : gamma matrix: Field strength

For details, e.g. T. Byrnes, et al Phys. Rev. D66 (2002), 013002


Hamiltonian of Schwinger model on lattice• For fermion, Kogut-Susskind type discretization:

• For gauge field, discretization: • Jordan-Wigner transformation:

• Gauss law on Lattice: • Hamiltonian:

7

1-component fermion field:2-component fermion field:

n n+1

for keeping anti-commutative relation

in continuum

hopping term gauge partmass termzero back ground field

For details, T. Byrnes, et al Phys. Rev. D66 (2002), 013002

s0 s1 s2 s3


Tensor network (TN)• Hamiltonian approach with exact diagonalization available for only small size: Ex. In 1D, with chain length N ~ O(10), not enough to take thermodynamic limit

• Tensor Network: An efficient approximation of quantum many-body state from quantum information

• Matrix product state (MPS): tensor network for 1d

8

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

physical indexvirtual indices

site


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

• Supposing D = 2 for tensors

• By computing trace of products

9

ik = ↑ , ↓ for k = 1, 2


• An efficient way to express sub-space of Hilbert space

• From quantum information aspect, much smaller value D than dN/2 is enough to derive ground state

• Hilbert space growing exponentially as increasing system size ⇔ With TN, one can investigate sub-space growing polynomially

• systematically improved

Bond dimension

10

F. Verstraete et al. PRL 93, 227204

dN ⇔ NdD2 d : d.o.f of physical index at each site, N : chain length

By using MPS with D,

⇒ If D ~ dN/2, no advantage of TN

size of the whole Hilbert space

Ex.) 1d spin system

polynomial

Hilbert space

exponential Ex.) In our studies, D ~ 100 is enough (<< dN/2~1015 )


Graphical representation of TN• Ex.1)

11

i k

j

ij

klm

j

i kl

m

l

m

j

i

(i) A general tensor Tijklm

(ii) A tensor with three indices Mijk (iii)Product of two tensors Mijk Aklm = Bijlm

• Ex.2) MPS state:

i1 i2 i3 i4

MPS state

:physical indices

bond indices from TN approximation

ex. N = 4open boundary condition


• 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

Variational search

12

with fixing the other elements

: a trial MPS state

Hamiltoniantrial MPS


• More concretely,updating through the whole chain, until convergence

• Techniques to solve it efficiently, i) canonical form derived by SVD: ii)MPO for Hamiltonian

Variational search

13

=

=

for given in, kn, kn+1

=

MPS state:: a function in terms of with fixing the others


• Hamiltonian of Schwinger model

• Basis of Hamiltonian dN ⇒ Hamiltonian dN x dN matrix

• Another way to express H mapping into operator space

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

Matrix Product Operator (MPO)

14

hopping term gauge partmass term

:hopping

where ,,

, , , ,

for left boundary

for right boundaryfor bulk n = 2,3

Useful to compute the trial energy


Our previous study• 1-flavor Schwinger model at zero T • With variational method, computing:

✴ mass spectrum ✴ (subtracted) chiral condensate:

• Continuum extrapolation:

15

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. Bañuls et al JHEP 1311, 158, LAT2013, 332

(H.) Y. Hosotani arXiv:9703153

Logarithmic correction from analytic form of free theory

Fit function:

x = 1/g2a2 ⇒ continuum limit


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 ✴ On higher dimension TN ✴ Real time evolution ✴ With quantum link model ✴ Tensor Renormalization Group

16

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

B. Buyens, et al. PRL, 113, 091601

Y. Shimizu, Y. Kuramashi, PRD90, 014508 (2014), PRD90, 074503 (2014)

E. Rico, et al. PRL112, 201601 (2014)

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

T. Pichler, et al. arXiv:1505.04440

S. Kühn, et al. arXiv:1505.04441

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

• Analytic formula

18

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, Helv. Phys. Acta, 65, 652 (1992)

where

Euler constant


Thermal state calculation• Expectation value at finite T :

• How to calculate the ρ(β) ✴ to approximate ρ(β) ✴ ρ (β/2) to ensure positivity: ✴ imaginary time evolution:

19

thermal density operator where

high T → low T

Ex. ) For fixed δ, larger Nstep corresponds to lower T


Thermal state calculation✴ Each factor ,

• Approximation (global optimization) (1st step)(2nd step)

20

dN x dN elements tensors M[k] in size of d2 x D2

obtained by variational search

and so on

In β = 0, ρ (β) is identity.

d :physical indices

tensors M’[k] in size of d2 x D2


Bond dimension and Step size dependence

21

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

difference


The other errors

22

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

continuum extrapolation with fixed

approaching

(fixed physical length)


• Four simulation parameters

• Four extrapolations • To avoid too large finite size effect: • Open boundary condition

Simulation setup1.From MPS approx., bond dimension D 2. From T evol., step size δ 3. chain length N 4. inverse coupling x

23

D → ∞

δ → 0

N → ∞


Extrapolation ① & ②

24

① D → ∞ with fixed δ, N, x ② δ → 0 with fixed N, xfrom a mathematical proof, convergence in D

linear convergence in δ theoretically predicted

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


Extrapolation ③

25

③ N → ∞ with fixed x

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

at gβ = 0.4

0 0.005 0.01 0.015 0.02 0.0250.1

0.12

0.14

0.16

0.18

0.2

1/N

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

at gβ = 4.0

In high/low T, the slope changing between them.


0 0.2 0.4 0.6

0.14

0.16

0.18

0.2

1/√

x

Extrapolation ④

26

gβ = 4.0

④ continuum extrapolation

(solid blue)

(dashed red)


Chiral condensatein continuum limit

27

After eliminating those systematic errors …

0 2 4 60

0.05

0.1

0.15

gβ


Cut-off effect• continuum extrapolation in high T

28

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• Massless Schwinger Hamiltonian:

• basis states in strong coupling limit

• Exponential of Hamiltonian:

• Gauss’s law for the :

29

s0 s1 s2 s3

l0 l1 l2

where

with l-1 = 0,

exponentially suppressed


Another approach• chiral condensate at high T

30

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

• Dependence of each parameters: bond dimension, step size, system size, inverse of coupling

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

31

I. Sachs and A. Wipf, arXiv:1005.1822

Backup slides


• Hamiltonian of Schwinger model

• Basis of Hamiltonian dN ⇒ Hamiltonian dN x dN matrix

• Matrix Product Operator (MPO) form of Hamiltonian: a mapping into operator space (locally expressed form of H )

Matrix Product Operator 1

33

hopping term gauge partmass term

possible to decompose indices into physical (i, j) and virtual one (l, r)l r

i

j wherep


Matrix Product Operator 2• Ex. For only hopping term with N = 4, open boundary

• Automata • useful to compute the efficiently

34

:hopping

where ,,

, , , ,

for left boundary

for right boundaryfor bulk n = 2,3

For the detail, G. M. Crosswhite and D. Bacon, PRA78, 012356

Documents

Tensor network approach for chiral symmetry restoration of 1 ......June 22, 2015 @ DESY & HU Lattice seminar H. Saito （NIC, DESY Zeuthen） Tensor network approach for chiral symmetry