Anomaly Non-renormalization in Interacting Weyl Semimetals

Abstract

Weyl semimetals are 3D condensed matter systems characterized by a degenerate Fermi surface, consisting of a pair of ‘Weyl nodes’. Correspondingly, in the infrared limit, these systems behave effectively as Weyl fermions in $$3+1$$ 3 + 1 dimensions. We consider a class of interacting 3D lattice models for Weyl semimetals and prove that the quadratic response of the quasi-particle flow between the Weyl nodes is universal, that is, independent of the interaction strength and form. Universality is the counterpart of the Adler–Bardeen non-renormalization property of the chiral anomaly for the infrared emergent description, which is proved here in the presence of a lattice and at a non-perturbative level. Our proof relies on constructive bounds for the Euclidean ground state correlations combined with lattice Ward Identities, and it is valid arbitrarily close to the critical point where the Weyl points merge and the relativistic description breaks down.

Energy Correlations of Non-Integrable Ising Models: The Scaling Limit in the Cylinder

“…As commented there, uniform rescalings, translations and parity transformations are the only conformal transformations mapping finite cylinders 1 , 2 to finite cylinders Method of proof, motivations and comparison with previous works As briefly mentioned above, the rigorous application of Wilsonian RG to interacting 2D Ising models at the critical point was sparked by Spencer's proposal [Spe00] of a rigorous strategy to compute the energy-energy critical exponent and by the related (unpublished) work of Pinson and Spencer [PS]. The starting point of their approach is an exact representation of the partition and generating functions in terms of a non-gaussian Grassmann integral, a sort of fermionic φ 4 2 theory, which can be studied via the constructive fermionic RG methods first developed in the mid '80 s and early '90 s [BG90,Ben+94,Fel+92,GK85,Les87] and later applied to several critical statistical mechanics models in two dimensions [BFM09,BFM10,GM04,GM05,GMT17,GMT20,Mas04] and to condensed matter systems in one [BM01,GM01], two [GM10,GMP12a,GMP12b] and higher dimensions [GMP21,Mas14]. Dimensionally, the quartic interaction of the effective φ 4 2 model which the interacting Ising model is equivalent to, is marginal in the RG jargon.…”

Anomaly cancellation condition in an effective nonperturbative electroweak theory

“…The normal and-points are n and are associated to terms in the effective potential not depending from the external fields J. The ν-endpoints are associated to the first line of (93) and have scale h v ď N `1 and there is the constraint that h v " h v 1 `1, if v 1 is the first non trivial vertex immediately preceding v. The V G -endpoints have scale h v " N `1 and are associated one of the terms in (26), and to the V A -endpoints is associated one of the terms in (59) with m " 0. The special end-endpoints have associated terms with at least an external J fields; the Z W , Z e.m. , Z 5 end-points have h v ď N `1 and there is the constraint that h v " h v 1 `1, and are associated with one of the terms in the second line of (93); the V A end-points have scale N and are associated to the terms with pn, mq " p1, 1q in (59).…”

“…QED with massive photons and massless fermions with a momentum cut-off has been constructed in [24], by integrating out the bosons and reducing to a purely fermionic theory; such a system has a natural condensed matter interpretation [25]. It has been proven in [26], [27] that the anomaly non-renormalization is true even with a finite lattice cut-off breaking chiral and Lorentz invariance, if the photons are massive; this is even true for an anisotropic lattice and velocities, depending only on the fact that the lattice preserve gauge invariance. The interaction is irrelevant in the Renormalization Group sense in that case, but this is not-essential for the anomaly non-renormalization, as shown in d " 2 QED where the interaction is marginal [28], [29].…”

Multi-channel Luttinger Liquids at the Edge of Quantum Hall Systems

“…In particular, the method allows to fully characterize the interacting phase diagram, and to prove the universality of the longitudinal conductivity on the transition curves. Finally, concerning three-dimensional systems with pointlike Fermi surface, the combination of RG with lattice and emergent Ward identities has been used to establish the non-renormalization of the lattice analogue of the chiral anomaly for Weyl semimetals [45]. In all these works, an important simplification is provided by the fact that the interaction is irrelevant in the RG sense, which allows to construct the RG fixed point of the theories in a considerably simpler way.…”