Remarks on the Chern-Simons photon term in the QED description of graphene

Abstract

We revisit the Coleman-Hill theorem in the context of reduced planar QED. Using the global U(1) Ward identity for this non-local but still gauge invariant theory, we can confirm that the topological piece of the photon self-energy at zero momentum does not receive further quantum corrections apart from the potential one-loop contribution, even when considering the Lorentz non-invariant case due to the Fermi velocity v F < c. This is of relevance to probe possible time parity odd dynamics in a planar sheet of graphene which has an effective description in terms of (2 + 1)-dimensional planar reduced QED.Keywords: Reduced QED, dynamical Chern-Simons term. * david.dudal@kuleuven.be † ana.mizher@kuleuven.be ‡ pablo.pais@kuleuven.be I. CONTEXT AND MOTIVATIONQuantum Electrodynamics in (2 + 1) dimensions (QED 3 ) has been widely used as a toy model for Quantum Chromodynamics (QCD). This is due to the fact that although being Abelian, QED 3 exhibits similar features as non-Abelian gauge theories, making it possible, for instance, to map and investigate chiral symmetry breaking and confinement into it [1][2][3][4][5]. The similarity is reinforced by the fact that a non-Abelian gauge theory at high temperature suffers a dimensional reduction and, if coupled to N f fermion families, the non-Abelian interactions are suppressed by a factor of N −1 f , so that in the large N f limit the theory can be considered approximately Abelian.Recently, the emergence of the so-called Dirac and Weyl planar materials [6], converted QED 3 into a playground in which a potential link between high energy physics (including quantum fields in curved spacetimes) and condensed matter can emerge [7][8][9][10][11][12][13]. Those are materials in which, due to the specific structure of their underlying lattice, the charge carriers present a relativistic-like behavior, being correctly described by a Dirac-like equation in some regimes. Particularly, the physical realization of graphene and other materials in two space dimensions, that are proved to contain a priori massless Dirac spinors, naturally yields the fermionic part of QED 3 [14, 15] through the continuum limit of the tight-binding theory, usually applied to describe their conduction electrons, which in turn implies a direct connection to QCD, as discussed above.Nevertheless, even though in these systems the fermions are constrained to remain in-plane and therefore are correctly described by a theory in (2 + 1) dimensions, the gauge fields responsible for the interaction between these electrons are not subject to the same constraint. One of the most remarkable consequences of this fact is that the interaction between electrons remains the familiar ∼ 1/r potential rather than the logarithmic one that would take place if the gauge fields were also restricted to the plane. Therefore, it is convenient and necessary to modify QED 3 in order to merge the desired features of the two sectors of the theory, starting with a general (3 + 1) theory and dimensionally reducing it to a non-local effective (2 ...

New point-like sources and a conducting surface in Maxwell–Chern–Simons electrodynamics

“…Theoretical aspects of Maxwell-Chern-Simons electrodynamics have been investigated in Casimir effect [4][5][6][7][8][9], quantum dissipation of harmonic systems [10], quantum electrodynamics (QED 3 ) [11][12][13][14][15], dynamical mass generation [16,17], condensed matter physics (see, for instance, Ref. [18] and references therein), description of graphene properties [19][20][21][22][23][24], noncommutativity [25][26][27][28], strings theory [29], dynamics of vortices [30,31], and with a planar boundary [32][33][34][35][36], to mention just a few. In fact, there is a vast literature concerning this model.…”

Half-integer anomalous currents in 2D materials from a QFT viewpoint

“…This is the Coleman-Hill theorem 36 generalized to RQED, resulting in a “topological” Chern-Simons mass term in the effective action. Moreover, that relevant contribution is also independent of the Fermi velocity 35 . At one-loop the (transverse and gauge invariant) polarization tensor is only depending on fermion propagators, which are just the same as in usual planar QED.…”

“…Assuming first a standard Dirac mass m for a single two-component spinor and a chemical potential , it can be shown that the low-energy limit of the induced -odd part in the RQED photon self-energy is one-loop exact 35 . This is the Coleman-Hill theorem 36 generalized to RQED, resulting in a “topological” Chern-Simons mass term in the effective action.…”

Torsion in quantum field theory through time-loops on Dirac materials

“…In such approach, the gauge field propagates in a threedimensional space and one direction is integrated out to obtain an effective interaction with the electrons constrained to move in a two-dimensional plane [38,39]. This approach could shed some light on the appearance of a photon Chern-Simons term [40,41].…”