Two body interactions induces the axion-phason field in Weyl semimetals
aa r X i v : . [ c ond - m a t . o t h e r] O c t Two body interactions induces the axion-phason field in Weylsemimetals
D. Schmeltzer
Physics Department, City College of the City Universityof New York, New York, New York 10031, USA
Abstract
Following our results that the two-body interaction can induce a space and time dependent topo-logical axion term e π ~ θ ( z, t )( ~E · ~B ), we show that by applying this theory to a Weyl semimetal withtwo nodes in a magnetic field a one-dimensional sliding charge density wave (CDW) in agreementwith the recent experimental finding of J. Gooth et al. [Nature, https://doi.org/10.1038/s41586-019-1630-4 (2019)]. Wee show that the theory is equivalent to a time and space dependent of thetopological angle which represents the phason. . INTRODUCTION Few years ago we showed that in topological insulator there is no need to use dimensionalprojection from from 4 + 1 dimension to 3 + 1 dimension [1, 2] to derive the second Chernnumber, instead we can use a two body CDW interaction which has two solution. A gap isopen in the electronic spectrum when the 2 k F electronic momentum is commensurate withthe two body interaction resulting a topological term with a constant θ H = R d xθ ( e π ~ )( ~E · ~B ). For θ = ± π the system is time reversal invariant . The two body system allows anadditional solution. The second solution corresponds to an incommensurate case with anoder parameter ∆ = | ∆ | e iα ( ~r,t ) where ∆ = 0 and | ∆ | 6 = 0. For this case the fluctuation of thephason α ( ~r, t ) generates the axion term δH = R d xθ ( ~x, t )( e π ~ )( ~E · ~B ). A C.D.W. theory [5]has been apply by [4] et al. Other explanation based on the Edelstein effect has been given[3].We aim to present a microscopic theory for the case that the electric field and the magneticfield are parallel. For this case the low energy model is equivalent to a one dimensional WeylHamiltonian which can be Bosonized. Including the two body interaction we recover theaxion-phason model for the Weyl case. The two W eyl bands are linearized around eachWeyl node. 2 Q is the distance between the Weyl nodes in the momentum space. TheHamiltonian with the two nodes is given by: (for more than two nodes we consider that twonodes are aligned with the magnetic field) H R = Z d x Ψ † R ( ~x ) h ~ v~σ ⊥ · ( ~k ⊥ − e ~A ⊥ ) + ~ vσ ( k − Q ) i Ψ R ( ~x ) H L = Z d x Ψ † L ( ~x ) h − ~ v~σ ⊥ · ( ~k ⊥ − e~ A ⊥ ) − ~ vσ ( k + Q i Ψ L ( ~x ) (1)Following [8] the two component spinor can be written as a Dirac spinor :.¯Ψ( ~x, t ) = Ψ † ( ~x, t ) γ ~γ = ~σ × iτ , γ = I × τ , γ = − I × τ (2)The Dirac Weyl Hamiltonian takes the form: H W eyl = Z d x ¯Ψ( ~x, t ) h ~ v~γ ⊥ · ( ~k ⊥ − e ~A ⊥ ) + ~ vγ ( k − γ Q ) − γ µ i Ψ( ~x, t ) (3)2he momentum Q plays the role of the axial field when Q is by space and time depen-dent field Q ( x, t ) [8] The basic property of the Weyl semimetals is the chiral anomalywhich with the non-conserving the axial current .The axial anomaly is obtained from lin-ear diverging triangle diagram [10] which consist from two Electro-Magnetic fields and thethird axial field Q [8, 9] .In the presence of attractive two body interactions − U eff. n R n L we replace the interaction with a Hubbard Stratonovici field ∆ Real ( ~x, t ) ¯Ψ( ~x, t )Ψ( ~x, t ) + i ∆ Imag. ( ~x, t ) ¯Ψ( ~x, t ) γ Ψ( ~x, t ) The Hamiltonian becomes: H W eyl = Z d x ¯Ψ( ~x, t ) h ~ v~γ ⊥ · ( i∂ x ⊥ − e ~A ⊥ ) + ~ vγ ( − i∂ z − γ Q ) + i ∆ Imag. ( ~x, tγ + M ( ~x, t ) − γ µ i Ψ( ~x, t ) M ( ~x, t ) = ∆ Real ( ~x, t ); ∆ Imag. ( ~x, t ) (4)We compute the triangle diagram in the presence of the mass term M ( ~x, t ) and chiral field i ∆ Imag. ( ~x, t ) γ . The contribution to the effective action from fields ∆ Imag. ( ~x, t ) ~A ⊥ becomes: S (3) eff. = Z d x Z dt Z d x Z dt Z d x Z dt T r h G ( ~x , t ; ~x , t ) γ ∆ Imag. ( ~x , t ) G ( ~x , t ; ~x , t ) ~γ perp · ~A ⊥ ( ~x , t ) ~γ perp · ~A ⊥ ( ~x , t ) i (5)Performing the computation using the Green’s function G ( ~x, t ; ~x ′ , t ′ ) we obtain a lin-ear diverging result [10]. We regularize the effective action ,replacing ∆ Imag. ( ~x, t ) → +∆ Imag. ( ~x, t ) + ∂ z β ( z, t ). We obtain: S (3)) eff. = Z dt Z d x h e h β ( z, t )( ~E · ~B ) i (6)The action depends on the angle between the electric field and the magnetic field which isa function of the interaction -phason.The phason gives the sliding CDW phase is the originto the axion field. In a recent paper [4] it was shown that when E is paralel to B, from thepositive magneto conductivity we can deduce the presence of the sliding phase phason-axionphase. In this paper we will not consider the pinning phase by the impurities,for this reasonour theory will be applicable only for E > E treshold − pining II-The Weyl representation in the presence of interactions
W eyl two nodes are in the ”Z” direction .For this case we can write a one dimensional model based on the
W eyl dispersion for theLandau level n = 0 and σ = − k + Q ) (right moover) and − ( k − Q )(left moover) [7]. We mention that this descriptionwill hold better for ( T aSe ) I which is a quasi-one-dimensional material. Projecting theHamiltonian to the n = 0 Landau level we obtain the one dimensional model : P n =0 H W eyl P n =0 = H d,W eyl = Z dz ~ v h ˜Ψ † R,σ = ↓ ( − i∂ z + Q ) ˜Ψ R,σ = ↓ − ˜Ψ † L,σ = ↓ ( − i∂ z − Q ) ˜Ψ L,σ = ↓ i (7)We include the two body attractive interaction projected to the n = 0 Landau level − U eff. n R,sigma = ↓ n L,sigma = ↓ Decoupling the interaction term we obtain, P n =0 H ( int. ) P n =0 = H (1 d,int. ) = Z dz h ∆( z ) ˜Ψ † R,σ = ↓ ˜Ψ L,σ = ↓ + ˜Ψ † L,σ = ↓ ˜Ψ R,σ = ↓ + | ∆( z ) | U eff. i (8)We introduce the one dimensional fields:˜Ψ R,σ = ↓ ( z ) = e i ( k F + Q ) z C R ( z ); ˜Ψ L,σ = ↓ ( z ) = e − i ( k F + Q ) z C L ( z ) (9)We use the Bosonization rule: C R ( z ) = r πa e i √ πθ R ( z,t ) C L ( z ) = r πa e − i √ πθ L ( z,t ) θ R ( z, t ) + θ L ( z, t ) = 2 θ ( z, t ); θ R ( z, t ) − θ L ( z, t ) = 2 ϕ ( z, t ) = 2 ϕ ( z, t ) (10)The charge density wave order parameter is given by, < Ψ † L,σ = ↓ ( z )Ψ R,σ = ↓ ( z ) > = ∆( z ) = | ∆ | e iα ( z,t ) (11)4he Weyl Hamiltonian with the interaction term is: H ( W eyl + int. ) = Z dz ~ h v ∂ z ϕ ( z, t )) + v ∂ z θ ( z, t )) + | ∆ | π cos[ √ πθ ( z, t ) + α ( z, t ) + (2 Q + 2 k F ) z ] i (12)The Fermi momentum K F is shifted by the nodal momentum Q . The C.D.W. order param-eter is given by ∆ = | ∆ | e iα ( z,t ) .In the comensurate case α ( z, t ) = qz with q ≈ − (2 Q + 2 k F )and ∆ = ∆ ∗ .In the incomensurate case | ∆ | 6 = 0 and α ( z, t ) = + ˆ α ( z, t ) + qz Under this condi-tions we expand H ( W eyl + int. ) around the minimum which is given by √ πθ ( z, t )+ ˆ α ( z, t ) ≈ π .This gives the phason action H ( W eyl + int. ) = Z dz h K c v c ∂ z ( ∂ z ϕ ( z, t )) + v c K c ( ∂ z ˆ α ( z, t )) ....... i , K c = √ π > K c > III- The sliding phase responce to an Electric field
The sliding CDW phase proposed by [5] has been observed by [4].The action in the presence of the electic field E ( z, t ) = ∂ z a ( z, t ) − ∂ t a ( z, t ) >E treshold − pining [6] takes the form. S ( W eyl + int. ) = 18 π Z dz Z dt h K c (cid:16) v c ( ∂ z ˆ α ( z, t )) − v c ( ∂ t ˆ α ( z, t )) (cid:17)i S ( E ) = e π Z dz Z dt h − a ∂ z ˆ α ( z, t ) + a ∂ t ˆ α ( z, t ) i (14)We find the phason driven by the electric fiield E ( z ) − ∂ z (cid:16) v c K c ( ∂ z ˆ α ( z ) (cid:17) = eE ( z ) (15)The conductivity due to the phason will be , σ = e h K c III-The Chiral transformation Q . This canbe seen from the following chiral transformation:Ψ( ~x, z ) ′ = e iγ β ( z ) Ψ( ~x, z ) γ Ψ † ( z ) ′ = γ Ψ † ( ~x, z ) e iγ β ( z ) γ = I × τ (16)Equation (3) will be modified to Q → Q + ∂ z β ( z ). In 3 + 1 dimension the trianglediagram is superficial linear divergent .This divergence can be eliminated by a linear shift ofsubstraction of the linear divergent divergent triangle diagram [10]. Performing the chiraltransformation replaces Q to Q + ∂ z β ( z ).This change of variables gives a finite change forthe action 3 + 1 dimensions . δS = Z d x Z dz Z dt h ( e h ) β ( z )( ~E · ~B ) i (17)The phase β ( z )i s fixed by the phason term. The one dimensional interaction is modified bythe chiral transformation, | ∆ | π cos[ √ πθ ( z, t ) + α ( z, t ) + 2 β ( z ) + (2 Q + 2 k F ) z ]This equation fixes the chiral phase β ( z ) − α ( z, t ) = β ( z, t )This shows that the phason plays the role of the axion. IV-Conclusions
The axion-phason action has been obtained for the Weyl semimetal without relying onthe phenomenological theory for charge density wave [5] or the asymptotic theory given by R d x R dz R dt e h β ( z, t )( ~E · ~B ) [1] D.Schmeltzer, Electrodynamics in 3 + 1 dimensions induced by interaction in topologicalinsulators,arXiv :112.5470[2] D.Schmeltzer,Avadh Saxena,Physics Letter A(377)(2013)1631-1636[3] G.Masarelli,B.Wu,and A. Paramekanti cond-matt 1904.04280v1
4] Gooth J. et al Axionic charge -density wave in the Weyl semimetal (
T aSe ) I. Naturehttps://doi.org/10.1038/s41586-0191630-4(2019)[5] Lee,P.A. Rice,T.M.and Anderson P.W. Solid State Commun,14, 703-709(1974).[6] D.L.Maslov and M.Stone Phys.Rev. B. 52 R5539(1995)[7] M.Ninoomiya, Physics Letters [8] K.Lansteiner Phys.Rev.B (1979)1195[10] V.P.Nair ”Quantum Field Theory -a modern-perspective” pages 283-287,Springer 2005.(1979)1195[10] V.P.Nair ”Quantum Field Theory -a modern-perspective” pages 283-287,Springer 2005.