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

2019/07/29 CAQMP2019 @ ISSP , Kashiwa Critical behavior of the two-dimensional dodecahedron model Icosahedron model: HU, Okunishi, Krcmar, Gendiar, Yunoki and Nishino, Phys. Rev. E 96 , 062112 (2017). Dodecahedron model: HU, Okunishi, Yunoki and Nishino, in preparation. HIROSHI UEDA (RIKEN R-CCS)

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: [Mermin and Wagner, PRL, 1966] spontaneous breaking of a continuous symmetry does not occur in 2D  Discretization  Spontaneous symmetry breaking ex) 𝑟 -state clock  XY [ O(2) ] 2 1 𝑟 � 2 : Ising 𝑟 � 3 : Three-state Potts 0 𝑟 � 4 : Ising � 2 𝑟 � 1 𝑟 � 5 : BKT

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 : 4 6 8 12 20 Ising � 3 Class of P .T.: 4-state Potts 2nd Order 2nd Order massless? [Wu,1982] ［Surungan&Okabe, 2012］ [Patrascioiu, [Patrascioiu, et al ., 2001] et al ., 1991] MC MC MC ↓ [Surungan& ↓ Weak 1st Okabe, 2012] MC 2nd Order [Roman, et al ., 2016] CTMRG ［Surungan&Okabe, 2012］ MC

Motivation and conclusion N or m T C     𝑑 ��.� 3.0 ��.� [Patrascioiu, 320&FSS ��.� 0.555 1 1.7 ��.� � � � et al ., 2001] [Surungan& 256&FSS 0.555 1 1.30 1 � 0.199 1 � � Okabe, 2012] 500&FmS 0.5550 1 Our work 1.62 2 � � 0.12 1 1.90 2 12 PRE (2017) N or m T C     𝑑 [Patrascioiu, 200&FSS 0.36,0.47 � � � � � et al., 1991] [Surungan& 64&FSS 0.438 1 2.0�1� � 0.149�1� � � Okabe, 2012] 800&FmS Our work 20 0.433 2 2.5 3 � � � 2.3�2� Unpublished numerical results (2019) We use an empirical relation c𝜆/6 � 12/𝑑 � 1 with c𝜆/6 � 0.304�7� . [Pollmann, 2009]

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 ) 2 3 𝑀 4  Partition function

Classical analogue of Entanglement Entropy • Quantum 1D Hamiltonian • Classical 2D Transfer matrix �𝝉� �𝝉� 𝑰 �𝝉′� �𝝉′� • Eigenvector • Ground state 𝑰 � 𝛀 � 𝐹 � 𝛀 Corner transfer matrix : 𝑀 � ∞ , 𝑀 � 4

Classical analogue of Entanglement Entropy • Reduced density matrix︓ 𝜍 � Ω Ω 𝑾 ∗ 𝑾 𝛀 ∗ �𝝉 � � � � 𝑽 ∗ �𝝉 � � 𝑽 �𝝉 � � 𝛀 � � �𝝉 � 𝑽 ∗ 𝑽 Λ 𝑽 ∗ 𝑾 � 𝑾 ∗ 𝑾 𝑾 ∗ 𝑽 Λ Ω Ω 𝑽 ∗ 𝑽 ∗ � � Λ � Ω � 𝑽 𝑽

Classical analogue of Entanglement Entropy CTM: 𝑴 � ∞ • Entanglement Entropy � � 4 log Ω � 𝑇 � � � ∑ Ω � 𝑀 ≫ 𝜊�𝑛, 𝑈� � � � 2 log 𝜇 � 𝑇 � � � ∑ 𝜇 � � • CTM of CTMRG 【Nishino,Okunishi(1996)】 ︓ 𝑀 � 𝑀 Same � 𝑀 ≫ 𝜊�𝑛, 𝑈� 𝑛 : # of renormalized states ※ finite 𝑛 ⇒ finite 𝜊 𝑛, 𝑈 2 3 𝑀 4

Magnetization Rotational symmetry along Icosahedral the axis passing through symmetry vertexes 1 and 12. 𝑞�𝑡� m=500

Finite- scaling near criticality HU et al., PRE (2017)  Finite size scaling [ Fisher and Barber, 1972, 1983 ] Nishino, Okunishi and Kikuchi, PLA, 1996 Tagliacozzo, Oliveira, Iblisdir, and Latorre, PRB, 2008 + Finite- 𝑛 scaling at criticality Pollmann, Mukerjee, Turner, and Moore, PRL, 2009 Pirvu, Vidal, Verstraete, and Tagliacozzo, PRB, 2012  Scaling assumption 1 𝑐 : Intrinsic length scale of the system

Finite- scaling near criticality HU et al., PRE (2017)  Definition of Correlation length 𝜂 � and 𝜂 � : 1st and 2nd eigenvalues of the transfer matrix  Scaling assumption 2  𝑐 ∼ 𝜊�𝑛, 𝑢� & Scaling assumption 1

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 Shoulder-like structure 𝑈 � � 0 . 5550 disappears at 𝑛 → ∞ 𝜉 � 1 . 617 𝜆 � 0 . 898 Single order-disorder phase transition occurs.

Von Neumann entropy  Definition: Analogue of the entanglement entropy near criticality [ ℓ ≫ 𝜊�𝑛, 𝑢� ] Vidal, Latorre, Rico, and Kitaev, PRL, 2003 Calabrese and Cardy, J. Stat. Mech., 2004  Finite- 𝑛 scaling function 𝑏 : non-universal constant 𝑑 : central charge

Finite- scaling for  One parameter 𝑑 � 1.894  Empirical relation [ Pollmann, Mukerjee, Turner, and Moore, PRL, 2009 ] 𝑈 � � 0 . 5550 � 𝜆 � 𝜉 � 1 . 617 � ��/��� 𝜆 � 0 . 898 This work: � ��/��� � 𝜆 � 0.003 �

Summary for Icosahedron model  Single continuous phase transition  Ordered phase: five-fold rotational symmetry  Estimated central charge︓beyond the minimal model of CFT 6 � 𝜆 𝑈 𝐃 𝜉 𝜆 𝛾 𝑑 𝑑 12/𝑑 � 1 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 Unpublished numerical results 𝜉 � 3.12 𝜆 � 0.860  Single continuous phase transition  Large-m calculations are required  Necessity of Parallelized CTMRG

Pseudo code of Symmetric CTMRG ����� � 0) Set 𝐗 � 𝑥 �,� ����� � , 𝑥 ��� ��� ,� � ���� � ��� � 𝓍 ��� � � � EigenExa • 1×1 2D Block-Cyclic ����� , 𝑞 �,� � �� ��� � 𝓆 �� � � Distribution Set 𝐐 � 𝑞 �,� ������ BLACS • Reshape by MPI_ALLTOALLV • where 𝓆 �� � � � 𝓆 � � �� Set 𝛁 � 𝜀 �,� � 𝜕 � , 𝜕 � � 0 [ 𝑛 ≫ 𝑒 ] FUNC_CTMRG[ 𝐐, 𝐐 � , 𝐗 ] 1) 𝐐 � � 𝑞 �,� � ≔ 𝜕 � 𝑞 �,� 1) 𝛀 ≔ 𝐐 � 𝐐 � 2) 𝛀 � 𝜔 �,� � ≔ FUNC_CTMRG[ 𝐐, 𝐐 � , 𝐗 ] ����� � 2) 𝛀 � � 𝜔 �,� � , ����� � 3) Diagonalization: 𝛀 � 𝐕𝛁𝐕 �  Update 𝐕 � 𝑣 �,� � and 𝛁 � 𝜔 ��� � � �� ,��� � � �� ≔ 𝜔 ��� ��� ,� � �� � � �� 4) 𝐕 � � 𝑣 �,� � � , 𝑣 �,� � �� ��� 3) 𝚾 ≔ 𝛀 � 𝐗 ≔ 𝑣 ��� ��� ,� � 5) 𝛀 ≔ FUNC_CTMRG( 𝐐, 𝐕 � , 𝐗� 4) 𝜔 ��� ��� ,� � �� � � �� ≔ 𝜚 ��� � � �� ,��� � � �� 6) 𝐐 ≔ 𝐕 � 𝛀 �,� 5) return 𝛀 7) go to 1)

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

Benchmarks of parallel CTMRG System: Icosahedron model ( 𝑂 � 12𝑛 ) [sec.] 10,000 itr. For 𝑂 � 10000 24 hours For 𝑂 � 16000 # of nodes required for the dodecahedron model with 𝑛 � 500~800 : 𝑜 � 7 � ~24 �

Finite- scaling for Unpublished numerical results 𝑛 ∗ � 𝑛 � � 𝑛 � /2 Extrapolation : liner fitting as a function of 1/𝑛 ∗

Summary of Dodecahedron model  Single continuous phase transition  Estimated central charge︓ beyond the minimal model 𝑈 𝐃 𝜆/𝜉 𝑑𝜆/6 𝜉 𝑑 0.433 2 0.322�9� 0.304 7 2.5 3 2.3�2� Unpublished numerical results Using the relation  Future issue: Calculation of 𝜊 and 𝑁 c𝜆/6 � 12/𝑑 � 1 Employment of a solver for partial eigenvalue problems