Fermions on Simplicial Lattices and their Dual Lattices Alan - PowerPoint PPT Presentation

Fermions on Simplicial Lattices and their Dual Lattices Alan Horowitz 2018

What Am I Talking About? Background Naive and Staggered Fermions on an A 4 lattice Naive and Staggere Fermions on an A ∗ 4 lattice Final Remarks and Sales Pitch

The isotropic lattices in every dimension The notation comes from the book by Conway and Sloane. ◮ Z n ; The hypercubic lattices. Automorphism group has 2 n n ! elements (=384 in 4-d). ◮ A n ; Also called ”simplicial.” Group order = 2 · n ! (=240 in 4-d). In 2-d, triangular lattice. FCC in 3-d. Pure gauge models were simulated on an A 4 lattice. ◮ A ∗ n ; The lattice dual to A n . In 3-d A ∗ 3 is the BCC lattice. ◮ D n ; Also known as the ”checkerboard” lattice. D 3 = A 3 is FCC. D 4 = F 4 is self-dual. Automorphism group of D 4 has 1152 elements. D 3 , D 4 , and D 5 are the densest possible lattice packings in 3, 4 and 5 dimensions. ◮ Hyperdiamond lattice is not a Bravais lattice. Union of 2 A n lattices.

Extremely Abridged History Noticed a long time ago [Celmaster and Krausz, (1983)] that fermions on non-cubic lattices are problematic: � ¯ ψ n e i · γ ( ψ n + e i − ψ n − e i ) Equations for doublers break rotational symmetry. There must be a symmetry connecting doublers to have rotational invariance and a reduction to staggered fermions. Could add Wilson term. On D 4 you have rotational symmetry broken only at O ( a 4 ). In 4-d, staggered fermions have only been satisfactorily formulated on hypercubic lattices. Drouffe and Moriarty (1983) did simulations of pure SU(2) and SU(3) gauge theories on the A 4 lattice.

A Lattice Fermion Popularity Contest Counting papers on hep-lat since 2017 using lattice fermions: ◮ 155 Wilson/clover, ◮ 86 domain wall ◮ 62 staggered ◮ 57 overlap ◮ 0 on non-cubic lattices

The A 4 lattice Coordinate vector of A d lattice: � n i = 0 ( n 1 , n 2 , . . . , n d +1 ) where Surface in Z d +1 lattice. Nearest neighbor vectors: ǫ 12 = (1 , − 1 , 0 , 0 , 0) , ǫ 13 = (1 , 0 , − 1 , 0 , 0) , . . . , ǫ 45 = (0 , 0 , 0 , 1 , − 1) and negatives of these. So 20 neighbors in 4-d, compared to 8 for hc. Take primitive lattice vectors τ µ = ǫ µ 5 : τ 1 = (1 , 0 , 0 , 0 , − 1) , . . . , τ 4 = (0 , 0 , 0 , 1 , − 1) Reciprocal lattice vectors, b µ , defined by b µ · τ ν = 2 πδ µν are b 1 = κ (4 , − 1 , − 1 , − 1 , − 1) , . . . , b 4 = κ ( − 1 , − 1 , − 1 , 4 , − 1) with κ = 2 π/ 5, generate the lattice A ∗ 4 .

Also need a set of orthonormal vectors on A 4 : √ √ e 1 = (1 , − 1 , 0 , 0 , 0) / 2, e 2 = (1 , 1 , − 2 , 0 , 0) / 6, √ √ e 3 = (1 , 1 , 1 , − 3 , 0) / 12, e 4 = (1 , 1 , 1 , 1 , − 4) / 20.

The action: √ 5 2 � � ¯ S A = ψ n γ i γ j ( ψ n + ǫ ij − ψ n − ǫ ij ) 8 i n j > i { γ i , γ j } = 2 δ µν The inverse free propagator in momentum space: 5 � D ( k ) ∝ γ i γ j sin( k · ǫ ij ) j > i which leads to the propagator � � sin 2 ( k · ǫ ij ) S ( k ) ∝ γ i γ j sin( k · ǫ ij ) / j > i j > i

The modes Poles at k = 0 and at k = b µ / 2 and sums of 2, 3 and all 4 of these, 16 in total. � 4 5 π ⇔ π 5 ( − 4 , 1 , 1 , 1 , 1) , . . . π 5 modes at | k | = 5 (1 , 1 , 1 , 1 , − 4) � 6 5 π ⇔ π 10 modes at | k | = 5 (3 , 3 , − 2 , − 2 , − 2) , . . .

Symmetries connecting modes The action is invariant under ψ n → ¯ ¯ ψ n → T ( n ) ψ n , ψ n T ( n ) where T ( n ) = ( − 1) n µ γ µ and products of these. Since all modes are equivalent need only examine the one at k ≈ 0

For k ≈ 0 4 D ( k ) ≈ − 1 � � √ γ i γ j k · ǫ ij ≡ i Γ µ k · e µ 5 j > i µ =1 Solving for Γ µ : 5 � e i Γ µ = i µ γ i A i =1 where 5 1 � γ i √ A = 5 i =1 The Γ µ comprise a set of Euclidean Dirac matrices: { Γ µ , Γ ν } = 2 δ µν Thus the action describes 16 Dirac fermions. We also have 5 1 � γ i Γ 5 = A = √ 5 i =1

Short paws

Symmetry group of the A 4 lattice Permuations of ( n 1 , n 2 , n 3 , n 4 , n 5 ), the ”symmetric” group S 5 . Negation of all the coordinates is also a symmetry. So 2 X 5! = 240 elements. S 5 is generated by single exchanges: e.g. (21345) The action is invariant provided 1 ψ n → √ ( γ 1 − γ 2 ) ψ n ′ 2 ψ n ′ 1 ψ n → ¯ ¯ √ ( γ 1 − γ 2 ) . 2

Representations of some lattice objects ǫ ij , γ i γ j , U ij = e i A ij transform as 10-d rep. of S 5 . Orthogonality of characters → 10 = 4 ⊕ 6 � 5 ǫ µ � 2 ( e i µ e j ν − e j µ e i i γ i γ j = ij Γ µ + i ν )Γ µ Γ ν ν>µ showing reduction to vector and antisymmetric tensor.

Likewise: A ij = ǫ µ � ( e i µ e j ν − e j µ e i ij B µ + ν ) Y µν ν>µ the naive continuum limit: � d 4 x ¯ ψ { Γ µ ( ∂ µ − igB µ ) + g σ µν Y µν } ψ + m ¯ ψψ Y µν is short range → four-fermion interaction with coupling of order a 2 g 2 .

The Action for the Link Variables

Absence of additive mass renormalization Additive mass renormalization is forbidden, even though there is no exact axial symmetry. The action √ 5 2 � � ¯ S A = 8 i ψ n γ i γ j U n , ij ψ n + ǫ ij + h . c . n j > i is invariant under negation of all the coordinates provided U ij → U † ψ n → − ¯ ¯ ij ; ψ n → ψ − n ; ψ − n This implies for the full propagator: S ( − p ) = − S ( p ) which forbids a mass term. Mass or Wilson terms are not invariant.

No exact chiral symmetry → fermion determinant is not real (except for free fermions). ◮ In a simulation, the pseudo-fermion action φ ( D † D + m 2 ) − 1 φ is real and ≈ det ( D + m ). ◮ Or to get to reality you can double the fermions ψ → ( ψ 1 , ψ 2 ) with a mass term m ψσ 3 ψ . ◮ Or go to a hyperdiamond lattice ( A 4 ∪ A 4 ) with ψ 1 on one A 4 with mass m and ψ 2 on the other with mass − m . The coupling → axial-vector interaction mixing 1 and 2.

Axial Vector Interaction Using µ Γ µ Γ 5 + 1 � e i γ i = − i √ Γ 5 5 µ a rotationally invariant, axial vector interaction is 5 � � ( ¯ ψ n γ i ψ n + r i + ¯ ψ n + r i γ i ψ n ) Z i ( n ) n i the same for all doublers, where r 1 = (4 , − 1 , − 1 , − 1 , − 1) , . . . , r 5 = ( − 1 , − 1 , − 1 , − 1 , 4) generate an A ∗ 4 sublattice. So axial currents live on a dual sublattice. Naive continuum limit ⇒ ¯ µ + ¯ ψ Γ µ Γ 5 ψ A 5 ψ Γ 5 ψ φ

Reduction to Staggered Fermions Naive action is diagonalized by: 4 γ ( n 1 + n 2 + n 3 + n 4 ) ψ n → γ n 1 1 γ n 2 2 γ n 3 3 γ n 4 ψ n 5 leading to the staggered fermion action � S st = χ n η i ( n ) η j ( n ) ( χ n + ǫ ij − χ n − ǫ ij ) + m ¯ ¯ χ n χ n where χ n is a single anticommuting variable and the phases are η 1 = 1 , η 2 = ( − 1) n 1 , η 3 = ( − 1) n 1 + n 2 , η 4 = ( − 1) n 1 + n 2 + n 3 , η 5 = ( − 1) n 1 + n 2 + n 3 + n 4

Can make blocks of 16 points as on hypercubic lattice. Degrees of freedom in a block couple to degrees of freedom in 20 neighboring blocks. All the symmetries of the naive fermions carry through to the staggered case. There is no additive mass renormalization.

Staggered Blocks on Triangular Lattice

Fermions on an A ∗ 4 lattice The action: 5 S = 5 � � ¯ ψ n γ i ( ψ n + f j − ψ n − f j ) 16 n j where f 1 = κ (4 , − 1 , − 1 , − 1 , − 1) , . . . , f 5 = κ ( − 1 , − 1 , − 1 , − 1 , 4) √ with κ = 1 / 20. Take the first 4 to be primitive vectors. The doubling symmetry is then ψ n → ( − 1) n µ γ µ ψ n

The propagator � � sin 2 ( k · f i ) S ( k ) ∝ γ i sin( k · f i ) / i i has a mode at k = 0, and 10 modes at √ α (1 , − 1 , 0 , 0 , 0) , . . . , α (0 , 0 , 0 , 1 , − 1); α = 2 π/ 5 and 5 modes at α (0 , 1 , 1 , − 1 , − 1) , . . . , α (1 , 1 , − 1 , − 1 , 0)

For k ≈ 0 the inverse propagator 4 2 � � √ ⇒ γ i k · f i ≡ Γ µ k · e µ 5 i µ =1 5 2 � ⇒ Γ µ = √ f i · e µ γ i 5 i =1 which obey { Γ µ , Γ ν } = 2 δ µν and as for A 4 5 1 � γ i √ Γ 5 = 5 i =1

The naive continuum limit is � d 4 x ¯ ψ { Γ µ ( ∂ µ − igB µ ) + g Γ 5 φ } ψ + m ¯ ψψ Absence of additive mass renormalization works the same. The staggered action is � S st = χ n η i ( n ) ( χ n + f i − χ n − f i ) + m ¯ ¯ χ n χ n where η 3 = ( − 1) n 1 + n 2 , η 4 = ( − 1) n 1 + n 2 + n 3 , η 2 = ( − 1) n 1 , η 1 = 1 , η 5 = ( − 1) n 1 + n 3

Axial Interactions on the A ∗ 4 lattice An axial interaction with the same charge for all the doublers is 5 � � ( ¯ ψ n γ i γ j ψ n + f i − f j + ¯ ψ n + f i − f j γ i γ j ψ n ) A ij n j > i The vectors f i − f j generate an A 4 sublattice. So, again, axial interactions live on a dual sublattice.