Parity anomaly in even dimensions Dmitri Vassilevich UFABC VI Conference "Models in Quantum Field Theory", 30.08.2018 Dmitri Vassilevich Parity anomaly Contents Overview and history Induced Chern-Simons current on the boundary Topologically non-trivial bundles Some further developments Based on works with Maxim Kurkov, PRD 2017, JHEP 2018 Support: CNPq, FAPESP Dmitri Vassilevich Parity anomaly Consider one generation of Dirac fermions in odd d. Niemi & Semenoff, Redlich, Alwarez-Gaume et al, Deser et al, Fosco et al: – gauge invariant renormalization violates the parity symmetry. – spectral asymmetry of the Dirac operator leads to the Parity anomaly in the one-loop effective action. That reads (for d = 3, m = 0, abelian gauge field) ik Z p W odd(A) = d3x habc A @ A ; k = ± 1 4π a b c 2 which is the Chern-Simons action of the level k. Dmitri Vassilevich Parity anomaly New reincarnation: Dirac/Weyl materials in the condensed matter physics (topological insulators, Weyl semimetals, etc.). Quasiparticles in these materials obey the Dirac equation that predicts topologically protected surface modes that are in turn described by the Dirac operator in 3D. The problem: does the surface mode generate the half-integer Hall conductivity? Purpose of this work: compute the parity anomaly directly on a 4D manifold with boundary. Dmitri Vassilevich Parity anomaly The setup Consider the Dirac operator with an abelian gauge field µ D= = iγ (rµ + iAµ) on a 4D flat Euclidean manifold M with a boundary S @M = α @Mα consisting of several connected components @Mα. Bag boundary conditions Define the projectors 1 5 n Π± := 2 1 ± i"αγ γ "α = ±1. n is the inward pointing unit normal. The BC: Π− j@M = 0 Dmitri Vassilevich Parity anomaly Let λ denote the eigenvalues of D=. Define the ζ-functions as X X ζ(s; D=) = λ−s + e−iπs (−λ)−s λ>0 λ<0 and the ζ-regularized effective action s Ws = − ln det(D=)s = µ Γ(s)ζ(s; D=): Separate even and odd parts: ζ = ζeven + ζodd = 1 = = 1 −iπs = ζ(s; D)odd = 2 ζ(s; D) − ζ(s; −D) = 2 1 − e η(s; D) where X X η(s; D=) := λ−s − (−λ)−s : λ>0 λ<0 Dmitri Vassilevich Parity anomaly iπ W odd ≡ W odd = η(0; D=) s=0 2 To compute, consider the variation Aµ ! Aµ + δAµ. One can show: 2 2 δη(0; D=) = − lim Tr (δD=)t1=2e−tD= : π1=2 t!+0 If Q is any matrix-valued function, there is the following expansion for the smeared heat kernel (in 4D): 1 −tD=2 X k−4 2 Tr Qe ' t 2 ak (Q; D= ) k=0 By substituting here the known heat kernel coefficients with bag BC (Marachevsky & DV): a0 = a1 = a2 = 0 Dmitri Vassilevich Parity anomaly 1 Z p a (Q; D=2) = d3x h" nabc (δA )@ A : 3 3=2 α a b c 8π @M yielding Z p odd i 3 nabc δW = − d x h"α (δAa)@bAc 8π @M The boundary current reads i ja = − " nabc @ A odd 8π α b c This is valid for arbitrary topology of the U(1) bundle. Just the variation has to be smooth. Dmitri Vassilevich Parity anomaly Trivial U(1) bundles The variation may be integrated to give: Z p odd i 3 nabc W (A) = − d x h"α Aa@bAc : 16π @M 1 This action corresponds to the CS level jkj = 4 . This value is 1 too small, since the 3D value is jkj = 2 per mode too large, since bag boundary condtions in (let us say) half-space do not allow for surface modes at all. In fact, this value is exactly what one needs... Consider a slab between two parallel planes ar x4 = 0 and x4 = l. The KK limit l ! 0 will be considered. There are two distinct choices of the boundary conditions: Dmitri Vassilevich Parity anomaly "1 = −"2: The spectrum of the Dirac operator π2p2 λ2(k ; p) = k2 + p 2 a a l2 Z contains a massless 3D fermion (p = 0) in the KK limit. The CS 1 levels add up giving the correct 3D value k = ± 2 . "1 = "2: The spectrum of the Dirac operator π2(p + 1 )2 λ2(k ; p) = k2 + 2 p 2 a a l2 Z does not contain any 3D fermions in the KK limit. The CS actions on two sides of the slab cancel against each other. Dmitri Vassilevich Parity anomaly Non-trivial U(1) bundles (stationary configurations) 1 4 Let M = M~ × S , x 2 [0; 2π[, A4 = a = const. Then 4 D= = iγ (@4 + ia) + De 4 Let De (µ) = µψ(µ) and @4 = i! . Note that [γ ; Π±] = 0 and γ4De = −Deγ4. (i) µ 6= 0. Eigenmodes of De may be combined in pairs ( (µ); (−µ)) so that µ −(! + a) D= = −(! + a) −µ The spectrum is symmetric, does not contribute to the parity anomaly. Dmitri Vassilevich Parity anomaly (ii) µ = 0. This is an eigenspace of γ4 and De. All modes can be separated according to their γ4-chirality: 4 γ (0)± = ± (0)± D= (0)± = ∓(! + a) (0)± Let n± be the number of ± eigenmodes. 4 −tDe2 n+ − n− = Ind (De) = Ind (De) = lim Tr γ e t!0 (De is understood here as a 3D operator). The index is again given by the heat kernel expansions and reads Z 1 2 p n4bc − d x h~"α @bAc 4π @Mf (This is essentially the magnetic flux across the boundary). Dmitri Vassilevich Parity anomaly Contribution of a single − mode to the η-function: X −s X −s η−(s) = (! +a ¯) − (−! − a¯) !2N0 !∈−N = ζR (s; a¯) − ζR (s; 1 − a¯) ; ¯ where a¯ = a − bac, ζR (s; ) is the generalized Riemann (Hurwitz) zeta function. Using its’ value at s = 0, we conclude η−(0) = 2a¯ − 1, and thus Z odd i(1 − 2a¯) 2 p n4bc W = d x h~"α @bAc 8 @Mf Dmitri Vassilevich Parity anomaly Remarks: Ind(De) is always an even number. Thus, we can replace a¯ ! a without changing the partition function. W odd is invariant under all large gauge transformations. W odd is consistent with the Chern-Simons current. Naive substitution of the topologically non-trivial stationary configuration to the Chern-Simons action obtained for trivial 1 U(1) bundles gives 2 of the correct result. Dmitri Vassilevich Parity anomaly Further results Gravity theories: parity anomaly depends on extrinsic curvature of the boundary and thus cannot be reproduced in any boundary theory. Computations in massive theories: the relation between boundary conductivities and edge states seems to be even less trivial. Does the Chern-Simons term lead to Casimir repulsion between topological insulators? Not likely - the Chern-Simons coupling is too small [Fialkovsky, Khusnutdinov and D.V. PRB 2018] Dmitri Vassilevich Parity anomaly.