Phase structure analysis of CP(N-1) model

using Tensor renormalization group

Hikaru Kawauchi with

Shinji Takeda Kanazawa University

July 25, 2016

Lattice 2016 Southampton, July 24-30, 2016

Outline

• Introduction

• Tensor Renormalization Group (TRG)

• CP(N-1) model and Numerical results

• Summary

Introduction• The two dimensional CP(N-1) model including the θ term is a toy model of QCD.

• The phase structure is left vague because of the sign problem.

• One possible way to avoid the sign problem is to simply abandon Monte Carlo simulation.

• We apply the Tensor Renormalization Group method to analyze this phase structure.

Phase structure of CP(N-1) model?

confining phase

deconfinement phase?

0

1

θ/π

β 1

G. Schierholz, Nucl. Phys. B, Proc. Suppl. 37A, 203 (1994).

Schierholz suggested a possibility that θ becomes 0 in the continuum limit.

Other researches indicated that the deconfinement phase boundary may appear due to the statistical errors. J. C. Plefka and S. Samuel, Phys. Rev. D 56, 44 (1997).

M. Imachi, S, Kanou, and H. Yoneyama, Prog. Theor. Phys. 102, 653 (1999).

Strong coupling analysis of the phase structure of the CP(1) model

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

θ/π

β

confining phase

deconfinement phase

Deconfinement phase appears, but strong coupling analysis is not appropriate for large β region.

J. C. Plefka and S. Samuel, Phys. Rev. D 55, 3966 (1997).

confining phase

Monte Carlo analysis of the phase structure of CP(1) model

V. Azcoiti, G. Di Carlo, and A. Galante, Phys. Rev. Lett. 98, 257203 (2007).

Haldane conjecture: 2d O(3) model at θ=π is gapless.

The order of phase transition changes around β=0.5.0

1

θ/π

β0.5

1st 2ndHow about CP(1) model?F. D. M. Haldane, Phys. Lett. A 93, 464 (1983).

F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983).

Tensor Renormalization Group (TRG)Michael Levin and Cody P. Nave, Phys. Rev. Lett. 99, 120601 (2007).

1. Tensor network representation

2. Reduce the number of tensors

Truncate the bond dimensions according to the hierarchy of the singular values of the tensors.

CP(N-1) model in two dimensionsTensor Network Representation

za⇤za = 1, a = 1, · · · , NUij = exp{iAij}

zj

Uij

z⇤i

CP(N-1) model in two dimensionsTensor Network Representation

Expand the weight with new integers

za⇤za = 1, a = 1, · · · , NUij = exp{iAij}

CP(N-1) model in two dimensionsTensor Network Representation

Integrate out old d.o.f

Tensor

Expand the weight with new integers

za⇤za = 1, a = 1, · · · , NUij = exp{iAij}

Tensor network representation of CP(N-1) model

((lt;mt), {b}, tp)

((lu;mu), {c}, up)

((lv;mv), {d}, vp)

T

((ls;ms), {a}, sp)

truncate the bond dimensions to Dβ×Dθ

In(x): modified Bessel functions of the first kind

ls, ms: non-negative integers {a}={a1, …, al+m} (an=1, …, N) sp: integer

H. K. and S. Takeda, Phys. Rev. D 93, 114503 (2016).

Tstuv ⌘T((ls;ms),{a},sp)((lt;mt),{b},tp)((lu;mu),{c},up)((lv;mv),{d},vp)

/p

IN�1+ls+ms(2N�)⇥2sin ✓+2⇡sp

2

✓ + 2⇡sp

Free energy density of CP(1) modelTRG method vs. Strong coupling analysis

β=0.2 L=220

J. C. Plefka and S. Samuel, Phys. Rev. D 55, 3966 (1997).

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2

F

θ/π

Strong coupling:β=0.2TRG:Dβ=5,Dθ=3,β=0.2,L=1048576

V dependence of susceptibility χ

0

20

40

60

80

100

120

140

3.12 3.13 3.14 3.15 3.16 3.17 3.18

χ

θ

TRG:Dβ=5, Dθ=3, β=0.2, L=4TRG:Dβ=5, Dθ=3, β=0.2, L=8

TRG:Dβ=5, Dθ=3, β=0.2, L=16TRG:Dβ=5, Dθ=3, β=0.2, L=32

V dependence of the peak of susceptibility χ

0

1

2

3

4

5

6

1 1.5 2 2.5 3 3.5

logχ m

ax

logL

TRG:Dβ=5,Dθ=3,β=0.2TRG:Dβ=5,Dθ=5,β=0.2

TRG:Dβ=17,Dθ=3,β=0.2

β = 0.2

�max

/ L2.003±0.031

when D� = 17, D✓ = 3

Truncation dependence of

The truncation dependence does not converge in the region β ≧ 0.4.

�max

/ L�⌫

1st

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

γ/ν

β

Dβ=5,Dθ=3Dβ=5,Dθ=5

Dβ=17,Dθ=3

Phase structure analysis of CP(1) model using TRG

0

1

θ/π

β0.3

1st

0.5

2nd?

Tensor Network Renormalization (TNR)TRG does not work well especially in critical region.

G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015).

?

Summary• We analyze the phase structure of CP(1) model including the θ term using TRG.

• We reconfirm that the order of the phase transition at θ=π is 1st order until β=0.3.

• Truncation dependence of the susceptibility arises for β ≧ 0.4.

• Next, we will apply TNR method to the critical region.