Critical behavior of the two-dimensional dodecahedron model - - PowerPoint PPT Presentation

Critical behavior of the two-dimensional dodecahedron model

HIROSHI UEDA (RIKEN R-CCS)

2019/07/29 CAQMP2019 @ ISSP , Kashiwa Icosahedron model: HU, Okunishi, Krcmar, Gendiar, Yunoki and Nishino, Phys. Rev. E 96, 062112 (2017). Dodecahedron model: HU, Okunishi, Yunoki and Nishino, in preparation.

Collaborators

 K. Okunishi (Niigata Univ.)  R. Krcmar (Slovak Academy of Sciences)  A. Gendiar (Slovak Academy of Sciences)  S. Yunoki (RIKEN)  T

. Nishino (Kobe Univ.)

Background

 Mermin–Wagner theorem:

spontaneous breaking of a continuous symmetry does not occur in 2D

 Discretization  Spontaneous symmetry breaking

ex) 𝑟-state clock  XY [ O(2) ] 𝑟 2: Ising 𝑟 3: Three-state Potts 𝑟 4: Ising 2 𝑟 5: BKT

[Mermin and Wagner, PRL, 1966]

1 2 𝑟 1

Regular polyhedron model

Tetrahedron

Regular polyhedron model

Octahedron

Regular polyhedron model

Cube

Regular polyhedron model

Icosahedron

Regular polyhedron model

Dodecahedron

Discretization and variety of phase transition

# of Vertexes : Class of P .T.: 4 4-state Potts

[Wu,1982]

6 2nd Order

［Surungan&Okabe, 2012］ ↓

Weak 1st

[Roman,et al., 2016]

8 Ising 3 12 2nd Order

[Patrascioiu, et al., 2001] [Surungan& Okabe, 2012]

20 massless?

[Patrascioiu, et al., 1991] ↓

2nd Order

［Surungan&Okabe, 2012］

MC MC MC MC MC CTMRG

Motivation and conclusion

20 12

N or m TC     𝑑

320&FSS 0.555 1 1.7.

. 3.0. .

• 256&FSS

0.555 1 1.30 1

• 0.199 1
• 500&FmS 0.5550 1

1.62 2

• 0.12 1

1.90 2

[Surungan& Okabe, 2012] [Patrascioiu, et al., 2001]

N or m TC     𝑑

200&FSS 0.36,0.47

• 64&FSS

0.438 1 2.01

• 0.1491
• 800&FmS

0.433 2 2.5 3

• 2.32

[Surungan& Okabe, 2012] Our work PRE (2017) Our work (2019) We use an empirical relation c𝜆/6 12/𝑑 1 with c𝜆/6 0.3047.

[Pollmann, 2009]

[Patrascioiu, et al., 1991] Unpublished numerical results

Regular Icosahedron

 Icosahedral symmetry

• Centers of edges (two-fold)
• Centers of faces (three-fold)
• Two opposite vertexes (five-fold)

 Q. Which symmetry is broken

in ordered phases?

Icosahedron Model

 Vertex representation

Icosahedron Model

 Vertex representation

Method: CTMRG

 Corner transfer matrix renormalization group [Nishino, Okunishi, JPSJ, 1996]

• CTM: partition function of the quadrant (edge spins are specified )

𝑀 2 3 4

 Partition function

Classical analogue of Entanglement Entropy

𝛀

• Ground state
• Eigenvector
• Quantum 1D Hamiltonian
• Classical 2D Transfer matrix

𝑰

𝝉 𝝉′

𝑰

𝐹

𝛀

𝝉 𝝉′

• Corner transfer matrix : 𝑀 ∞, 𝑀 4

Classical analogue of Entanglement Entropy

• Reduced density matrix︓𝜍

𝝉 𝝉

𝛀 𝛀∗

• 𝑽

𝑽∗ Λ

• 𝑽

𝑽∗ 𝑾∗ 𝑾 Λ Λ

𝝉 𝝉

• 𝑾∗

𝑾 𝑽 𝑽∗ 𝑾 𝑾∗ 𝑽∗ 𝑽 Ω Ω Ω Ω

• 𝑽

𝑽∗ Ω

Classical analogue of Entanglement Entropy

Same

• Entanglement Entropy

𝑇 ∑ 𝜇

2 log 𝜇

• 𝑇 ∑ Ω

4 log Ω

• CTM of CTMRG 【Nishino,Okunishi(1996)】︓𝑀 𝑀

𝑀 2 3 4

𝑀 ≫ 𝜊𝑛, 𝑈 𝑀 ≫ 𝜊𝑛, 𝑈

𝑛: # of renormalized states ※finite 𝑛 ⇒ finite 𝜊 𝑛, 𝑈

CTM:𝑴 ∞

Magnetization

m=500

𝑞𝑡

Icosahedral symmetry

Rotational symmetry along the axis passing through vertexes 1 and 12.

Finite- scaling near criticality

 Finite size scaling [Fisher and Barber, 1972, 1983]

+ Finite-𝑛 scaling at criticality

 Scaling assumption 1

Nishino, Okunishi and Kikuchi, PLA, 1996 Tagliacozzo, Oliveira, Iblisdir, and Latorre, PRB, 2008 Pollmann, Mukerjee, Turner, and Moore, PRL, 2009 Pirvu, Vidal, Verstraete, and Tagliacozzo, PRB, 2012

𝑐: Intrinsic length scale of the system

HU et al., PRE (2017)

Finite- scaling near criticality

 Definition of Correlation length  Scaling assumption 2  𝑐 ∼ 𝜊𝑛, 𝑢 & Scaling assumption 1

𝜂 and 𝜂: 1st and 2nd eigenvalues of the transfer matrix HU et al., PRE (2017)

Finite- scaling for

 Bayesian scaling

[Harada, PRE, 2011]

 Three parameters

𝑈

0.5550

𝜉 1.617 𝜆 0.898

Finite- scaling for

 One parameter

𝛾 0.129 𝑈

0.5550

𝜉 1.617 𝜆 0.898

Shoulder-like structure disappears at 𝑛 → ∞ Single order-disorder phase transition occurs.

Von Neumann entropy

 Definition:

Analogue of the entanglement entropy near criticality [ ℓ ≫ 𝜊𝑛, 𝑢 ]

 Finite-𝑛 scaling function

Vidal, Latorre, Rico, and Kitaev, PRL, 2003 Calabrese and Cardy, J. Stat. Mech., 2004

𝑏: non-universal constant 𝑑: central charge

Finite- scaling for

 One parameter

𝑑 1.894

 Empirical relation

𝜆

• /

This work:

• / 𝜆 0.003

𝑈

0.5550

𝜉 1.617 𝜆 0.898

[ Pollmann, Mukerjee, Turner, and Moore, PRL, 2009 ]

Summary for Icosahedron model

 Single continuous phase transition  Ordered phase: five-fold rotational symmetry  Estimated central charge︓beyond the minimal

model of CFT 𝑈𝐃 𝜉 𝜆 𝛾 𝑑

6 𝑑 12/𝑑 1 𝜆 0.5550(1) 1.62(2) 0.89(2) 0.12(1) 1.90(2) ~0.009

• Phys. Rev. E 96, 062112 (2017)

Dodecahedral model

𝑈 0.441 𝜉 3.12 𝜆 0.860

 Single continuous phase transition  Large-m calculations are required  Necessity of Parallelized CTMRG

Unpublished numerical results

Pseudo code of Symmetric CTMRG

0) Set 𝐗 𝑥,

• , 𝑥 , 𝓍

Set 𝐐 𝑞,

, 𝑞, 𝓆

where 𝓆 𝓆 Set 𝛁 𝜀,𝜕 , 𝜕 0 [ 𝑛 ≫ 𝑒 ] 1) 𝐐 𝑞,

• ≔ 𝜕𝑞,

2) 𝛀 𝜔, ≔FUNC_CTMRG[𝐐, 𝐐, 𝐗] 3) Diagonalization: 𝛀 𝐕𝛁𝐕  Update 𝐕 𝑣, and 𝛁 4) 𝐕 𝑣,

• , 𝑣,
• ≔ 𝑣 ,

5) 𝛀 ≔FUNC_CTMRG(𝐐, 𝐕, 𝐗 6) 𝐐 ≔ 𝐕𝛀 , 7) go to 1) FUNC_CTMRG[𝐐, 𝐐, 𝐗] 1) 𝛀 ≔ 𝐐𝐐 2) 𝛀 𝜔,

• ,

𝜔 ,

• ≔ 𝜔 ,

3) 𝚾 ≔ 𝛀𝐗 4) 𝜔 , ≔ 𝜚 , 5) return 𝛀

• EigenExa
• BLACS
• Reshape by MPI_ALLTOALLV

1×1 2D Block-Cyclic Distribution

EigenExa

*http://www.r-ccs.riken.jp/labs/lpnctrt/assets/img/EPASA2014_dense_poster_ImamuraT_only.pdf

Benchmarks of parallel CTMRG

[sec.] System: Icosahedron model (𝑂 12𝑛) 10,000 itr. 24 hours

For 𝑂 10000 For 𝑂 16000

# of nodes required for the dodecahedron model with 𝑛 500~800: 𝑜 7~24

Finite- scaling for

𝑛∗ 𝑛 𝑛 /2 Extrapolation : liner fitting as a function of 1/𝑛∗ Unpublished numerical results

Summary of Dodecahedron model

 Single continuous phase transition  Estimated central charge︓

beyond the minimal model 𝑈𝐃 𝜆/𝜉 𝑑𝜆/6 𝜉 𝑑 0.433 2 0.3229 0.304 7 2.5 3 2.32

Using the relation c𝜆/6 12/𝑑 1

 Future issue: Calculation of 𝜊 and 𝑁

Employment of a solver for partial eigenvalue problems

Unpublished numerical results