Identify UMTCs by central charge with rank lower than 5

src/aliases.jl

@@ -14,6 +14,9 @@ const RepD5 = PMFC{4, 1, 0, 3, 0, 0}
 const RepD6 = PMFC{6, 1, 0, 2, 0, 0}
 const RepD7 = PMFC{5, 1, 0, 4, 0, 0}
 
+# Rep[Q₈]
+# const RepQ8 = PMFC{5, 1, 0, 1, ?, ?}
+
 # Vec[Dₙ]
 const VecD3 = UFC{6, 1, 2, 1, 0}
 
@@ -34,5 +37,135 @@ Base.getindex(::CenterTable, ::Type{VecS3}) = ZVecS3
 @objectnames ZVecS3 = ZVecS3 A B C F G H D E
 
 # Varia
+
+# --- SU2 ---
+@objectnames SU2_1 = PMFC{2, 1, 0, 1, 1, 0} _0 _1
+@objectnames SU2_2 = PMFC{3, 1, 0, 1, 0, 3} _0 _2 _1
+@objectnames SU2_3 = PMFC{4, 1, 0, 2, 1, 3} _0 _3 _1 _2
+@objectnames SU2_4 = PMFC{5, 1, 0, 3, 0, 0} _0 _4 _3 _1 _2
+@objectnames SU2_5 = PMFC{6, 1, 0, 6, 0, 1} _0 _5 _1 _4 _2 _3
+
+# --- PSU2 ---
+@objectnames PSU2_3 = PMFC{2, 1, 0, 2, 0, 0} _0 _2
+@objectnames PSU2_4 = PMFC{3, 1, 0, 2, 0, 1} _0 _4 _2 # Non-modular. {0, 4} form a Rep[Z₂] braided subcategory. Isomorphic to Rep[D₃] as UFC, but has different braiding.
+@objectnames PSU2_5 = PMFC{3, 1, 0, 3, 0, 0} _0 _4 _2
+@objectnames PSU2_6 = PMFC{4, 1, 0, 4, 0, 0} _0 _6 _4 _2 # Non-modular. Hard to distinguish 2 and 4. They seems to be equivalent. {0, 6} form a sVec braided subcategory.
+@objectnames PSU2_7 = PMFC{4, 1, 0, 6, 0, 0} _0 _6 _2 _4
+@objectnames PSU2_8 = PMFC{5, 1, 0, 7, 0, 0} _0 _8 _2 _6 _4 # Non-modular. Hard to distinguish 2 and 6. They seems to be equivalent. {0, 8} form a Rep[Z₂] braided subcategory.
+@objectnames PSU2_9 = PMFC{5, 1, 0, 10, 0, 1} _0 _8 _2 _6 _4
+@objectnames PSU2_10 = PMFC{6, 1, 0, 16, 0, 0} _0 _10 _8 _2 _6 _4 # Non-modular
+@objectnames PSU2_11 = PMFC{6, 1, 0, 18, 0, 1} _0 _10 _2 _8 _4 _6
+
+# --- Kitaev 16-fold way ---
+@objectnames Kitaev16_0 = PMFC{4, 1, 0, 1, 0, 2} I e m ψ
+@objectnames Kitaev16_1 = PMFC{3, 1, 0, 1, 1, 3} I ψ σ
+@objectnames Kitaev16_2 = PMFC{4, 1, 2, 1, 2, 0} _0 _2 _1 _3
+@objectnames Kitaev16_3 = PMFC{3, 1, 0, 1, 0, 3} I ψ σ
+@objectnames Kitaev16_4 = PMFC{4, 1, 0, 1, 1, 1} II ϵI ϵϵ Iϵ
+@objectnames Kitaev16_5 = PMFC{3, 1, 0, 1, 0, 0} I ψ σ
+@objectnames Kitaev16_6 = PMFC{4, 1, 2, 1, 2, 1} _0 _2 _1 _3
+@objectnames Kitaev16_7 = PMFC{3, 1, 0, 1, 1, 0} I ψ σ
+@objectnames Kitaev16_8 = PMFC{4, 1, 0, 1, 0, :?} I e m ψ
+@objectnames Kitaev16_9 = PMFC{3, 1, 0, 1, 1, 2} I ψ σ
+@objectnames Kitaev16_10 = PMFC{4, 1, 2, 1, 2, 2} _0 _2 _1 _3
+@objectnames Kitaev16_11 = PMFC{3, 1, 0, 1, 0, 2} I ψ σ
+@objectnames Kitaev16_12 = PMFC{4, 1, 0, 1, 1, :?} II ϵI ϵϵ Iϵ
+@objectnames Kitaev16_13 = PMFC{3, 1, 0, 1, 0, 1} I ψ σ
+@objectnames Kitaev16_14 = PMFC{4, 1, 2, 1, 2, 3} _0 _2 _1 _3
+@objectnames Kitaev16_15 = PMFC{3, 1, 0, 1, 1, 1} I ψ σ
+
+# All rank-2 PMFCs are identified!
 @objectnames Fib = PMFC{2, 1, 0, 2, 0, 0} I τ
+@objectnames sVec = PMFC{2, 1, 0, 1, 0, 1} I ψ # Non-modular
+@objectnames Semion = PMFC{2, 1, 0, 1, 1, 0} I ϵ
+@objectnames U1_1 = PMFC{2, 1, 0, 1, 1, 0} 0 1
+
+
+
+# All rank-3 PMFCs are identified!
+
 @objectnames Ising = PMFC{3, 1, 0, 1, 1, 3} I ψ σ
+@objectnames Ising1 = PMFC{3, 1, 0, 1, 1, 3} I ψ σ # c = 1//2 Ising type UMTC
+@objectnames Ising3 = PMFC{3, 1, 0, 1, 0, 3} I ψ σ # c = 3//2 Ising type UMTC
+@objectnames Ising5 = PMFC{3, 1, 0, 1, 0, 0} I ψ σ # c = 5//2 Ising type UMTC
+@objectnames Ising7 = PMFC{3, 1, 0, 1, 1, 0} I ψ σ # c = 7//2 Ising type UMTC
+
+
+@objectnames Z3MTC = PMFC{3, 1, 2, 1, 0, 1} 0 1 2
+
+@objectnames Z2sVec = PMFC{4, 1, 0, 1, 0, 1} 0I 1I 0ψ 1ψ # Non-modular
+
+# @objectnames ThreeFermion = PMFC{4, 1, 0, 1, 0, 3} I f1 f2 f3 # Not found. Should be here.
+@objectnames ZSemion = PMFC{4, 1, 0, 1, 1, 0} II ϵI ϵϵ Iϵ
+@objectnames SemionSemion = PMFC{4, 1, 0, 1, 1, 1} II ϵI ϵϵ Iϵ
+@objectnames sVecSemion = PMFC{4, 1, 0, 1, 1, 2} II Iϵ ψI ψϵ # Non-modular
+# @objectnames Z2Semion = PMFC{4, 1, 0, 1, 1, ?} # Non-modular and not found. Should be here.
+@objectnames Z2Fib = PMFC{4, 1, 0, 2, 0, 2} 0I 1I 1τ 0τ # Non-modular.
+@objectnames sVecFib = PMFC{4, 1, 0, 2, 0, 3} II ψI Iτ ψτ # Non-modular.
+@objectnames SemionFib = PMFC{4, 1, 0, 2, 1, 1} II ϵI ϵτ Iτ # Others can be obtained by taking braided-reverse to each layers respectively.
+# How to understand PMFC{4, 1, 0, 3, 0, 1} and PMFC{4, 1, 0, 3, 0, 2}? They are non-modular, isomorphic to Rep[D₅] as UFC, with the Muger center Rep[Z₂]. What is its minimal modular extension?
+
+@objectnames ZFib = PMFC{4, 1, 0, 5, 0, 1} II τI Iτ ττ
+# @objectnames FibFib = PMFC{4, 1, 0, 5, 0, 2} II τI Iτ ττ # Not found. Should be here.
+
+@objectnames sRepZ4 = PMFC{4, 1, 2, 1, 0, 2} 0 2 1 3 # Non-modular. {0, 2} form a Rep[Z₂] braided subcategory.
+# Some other Z4PMFCs are not identified: do not know how to understand. Some are not found.
+@objectnames U1_2 = PMFC{4, 1, 2, 1, 2, 0} 0 2 1 3
+
+
+
+# @objectnames JK4 = PMFC{5, 1, 0, 3, 1, 1} 0 4 3 1 2 # Not found. Should be here.
+
+_umtcs_lowrank_centralcharge = [
+    Dict(
+        1 // 1 => PMFC{2, 1, 0, 1, 1, 0}, # Semion, U1_1, SU2_1
+        -1 // 1 => PMFC{2, 1, 0, 1, 1, 1}, # TimeReversed{Semion}
+        -14 // 5 => PMFC{2, 1, 0, 2, 0, 0}, # TimeReversed{Fib}
+        14 // 5 => PMFC{2, 1, 0, 2, 0, 1}, # Fib, PSU2_3
+    ),
+    Dict(
+        2 // 1 => PMFC{3, 1, 2, 1, 0, 1}, # Z3MTC
+        -2 // 1 => PMFC{3, 1, 2, 1, 0, 2}, # TimeReversed{Z3MTC}
+        5 // 2 => PMFC{3, 1, 0, 1, 0, 0}, # The following are 8-fold ways of Ising-type UMTCs
+        -3 // 2 => PMFC{3, 1, 0, 1, 0, 1}, # TimeReversed{SU2_2}
+        -5 // 2 => PMFC{3, 1, 0, 1, 0, 2},
+        3 // 2 => PMFC{3, 1, 0, 1, 0, 3}, # SU2_2
+        7 // 2 => PMFC{3, 1, 0, 1, 1, 0},
+        -1 // 2 => PMFC{3, 1, 0, 1, 1, 1}, # TimeReversed{Ising}
+        -7 // 2 => PMFC{3, 1, 0, 1, 1, 2},
+        1 // 2 => PMFC{3, 1, 0, 1, 1, 3}, # Ising
+        8 // 7 => PMFC{3, 1, 0, 3, 0, 0}, # PSU2_5
+        -8 // 7 => PMFC{3, 1, 0, 3, 0, 1}, # TimeReversed{PSU2_5}
+    ),
+    Dict(
+        (0 // 1, 1) => PMFC{4, 1, 0, 1, 0, 2}, # Toric code ≅ Z(Rep[Z₂])
+        # 4//1 => PMFC{4, 1, 0, 1, 0, 3}, # ThreeFermion Not found. Should be here.
+        (0 // 1, 2) => PMFC{4, 1, 0, 1, 1, 0}, # Z(Semion) ≅ Semion ⊠ TimeReversed{Semion}
+        2 // 1 => PMFC{4, 1, 0, 1, 1, 1}, # Semion ⊠ Semion
+        # -2//1 => PMFC{4, 1, 0, 1, 1, ?}, # TimeReversed{Semion} ⊠ TimeReversed{Semion} Not found. Should be here.
+        1 // 1 => PMFC{4, 1, 2, 1, 2, 0}, # U1_2
+        3 // 1 => PMFC{4, 1, 2, 1, 2, 1},
+        -3 // 1 => PMFC{4, 1, 2, 1, 2, 2},
+        -1 // 1 => PMFC{4, 1, 2, 1, 2, 3}, # TimeReversed{U1_2}
+        -9 // 5 => PMFC{4, 1, 0, 2, 1, 0}, # Semion ⊠ TimeReversed{Fib}
+        19 // 5 => PMFC{4, 1, 0, 2, 1, 1}, # Semion ⊠ Fib
+        -19 // 5 => PMFC{4, 1, 0, 2, 1, 2}, # TimeReversed{Semion} ⊠ TimeReversed{Fib}
+        9 // 5 => PMFC{4, 1, 0, 2, 1, 3}, # TimeReversed{Semion} ⊠ Fib
+        12 // 5 => PMFC{4, 1, 0, 5, 0, 0}, # TimeReversed{Fib} ⊠ TimeReversed{Fib}
+        (0 // 1, 3) => PMFC{4, 1, 0, 5, 0, 1}, # TimeReversed{Fib} ⊠ Fib
+        # -12//5 => PMFC{4, 1, 0, 5, 0, 2}, # Fib ⊠ Fib Not found. Should be here.
+        10 // 3 => PMFC{4, 1, 0, 6, 0, 0}, # PSU2_7
+        -10 // 3 => PMFC{4, 1, 0, 6, 0, 1}, # TimeReversed{PSU2_7}
+    ),
+    Dict(
+        4 // 1 => PMFC{5, 1, 4, 1, 0, 1},
+        0 // 1 => PMFC{5, 1, 4, 1, 0, 2},
+        (2 // 1, 1) => PMFC{5, 1, 0, 3, 0, 0}, # SU2_4
+        # Cannot find TimeReversed{SU2_4}, but it should be here with PMFC{5, 1, 0, 3, 0, 1}
+        (2 // 1, 2) => PMFC{5, 1, 0, 3, 1, 0}, # TimeReversed{JK4} http://dx.doi.org/10.1103/PhysRevA.92.012301
+        # Cannot find JK4, but it should be here somewhere with PMFC{5, 1, 0, 3, 1, 1}
+        -16 // 11 => PMFC{5, 1, 0, 10, 0, 0}, # TimeReversed{PSU2_9}
+        16 // 11 => PMFC{5, 1, 0, 10, 0, 1}, # PSU2_9
+        # Cannot find c = ± 18//7 UMTCs. They are not SimpleFusion. Should be PMFC{5, 2, ?, ?, ?, ?}
+    ),
+]

test/aliases.jl

@@ -0,0 +1,5 @@
+@testset "Test all aliases" begin
+    for (alias, name) in aliases
+        @test alias == name
+    end
+end