Evaluating fermion determinants through the chiral anomaly

A class of fermion operators whose determinants can be calculated exactly has recently been noted. We observe that typically such operators can be chirally rotated into the free Dirac operator; hence, their determinants are given by the chiral anomaly. Four-dimensional fermion determinants of this type are computed; the appearance of the Wess-Zumino anomaly term is noted.     

Axial anomaly and index of the overlap hypercube operator

The overlap hypercube fermion is constructed by inserting a lattice fermion with hypercubic couplings into the overlap formula. One obtains an exact Ginsparg-Wilson fermion, which is more complicated than the standard overlap fermion, but which has improved practical properties and is of current interest for use in numerical simulations. Here we deal with conceptual aspects of the overlap hypercube Dirac operator. Specifically, we evaluate the axial anomaly and the index, demonstrating that the correct classical continuum limit is recovered. Our derivation is non-perturbative and therefore valid in all topological sectors. At the non-perturbative level this result had previously only been shown for the standard overlap Dirac operator with Wilson kernel. The new techniques which we develop to accomplish this also for hypercubic kernels are of a general nature and have the potential to be extended to overlap Dirac operators with even more general kernels.Received: 27 October 2003, Published online: 25 February 2004     

Axial anomalies of Lifshitz fermions

We compute the axial anomaly of a Lifshitz fermion theory with anisotropic scaling z = 3 which is minimally coupled to geometry in 3+1 space‐time dimensions. We find that the result is identical to the relativistic case using path integral methods. An independent verification is provided by showing with spectral methods that the η‐invariant of the Dirac and Lifshitz fermion operators in three dimensions are equal. Thus, by the integrated form of the anomaly, the index of the Dirac operator still accounts for the possible breakdown of chiral symmetry in non‐relativistic theories of gravity. We apply this framework to the recently constructed gravitational instanton backgrounds of Hořava–Lifshitz theory and find that the index is non‐zero provided that the space‐time foliation admits leaves with harmonic spinors. Using Hitchin's construction of harmonic spinors on Berger spheres, we obtain explicit results for the index of the fermion operator on all such gravitational instanton backgrounds with SU(2) × U(1) isometry. In contrast to the instantons of Einstein gravity, chiral symmetry breaking becomes possible in the unimodular phase of Hořava–Lifshitz theory arising at λ = 1/3 provided that the volume of space is bounded from below by the ratio of the Ricci to Cotton tensor couplings raised to the third power. Some other aspects of the anomalies in non‐relativistic quantum field theories are also discussed.     

Fermion production despite fermion number conservation

Lattice proposals for a nonperturbative formulation of the Standard Model easily lead to a global U(1) symmetry corresponding to exactly conserved fermion number. The absence of an anomaly in the fermion current would then appear to inhibit anomalous processes, such as electroweak baryogenesis in the early universe. One way to circumvent this problem is to formulate the theory such that this U(1) symmetry is explicitly broken. However we argue that in the framework of spectral flow, fermion creation and annihilation still in fact occurs, despite the exact fermion number conservation. The crucial observation is that fermions are excitations relative to the vacuum, at the surface of the Dirac sea. The exact global U(1) symmetry prohibits a state from changing its fermion number during time evolution, however nothing prevents the fermionic ground state from doing so. We illustrate our reasoning with a model in two dimensions which has axial-vector couplings, first using a sharp momentum cutoff, then using the lattice regulator with staggered fermions. The difference in fermion number between the time evolved state and the ground state is indeed in agreement with the anomaly. Both the sharp momentum cutoff and the lattice regulator break gauge invariance. In the case of the lattice model a mass counterterm for the gauge field is sufficient to restore gauge invariance in the perturbative regime. A study of the vacuum energy shows however that the perturbative counterterm is insufficient in a nonperturbative setting and that further quartic counterterms are needed. For reference we also study a closely related model with vector couplings, the Schwinger model, and we examine the emergence of the θ-vacuum structure of both theories.     

A rigorous construction of the chiral anomaly on a spin manifold

The quantum field for Dirac fermion is rigorously formulated on a compact spin manifold by using the functional integral defined as the continuum limit of a lattice approximation with a new action. Within this framework, the chiral anomaly for a fermion interacting with gauge as well as gravitational fields is calculated with mathematical rigor.     

Grassmann phase space methods for fermions. I. Mode theory

In both quantum optics and cold atom physics, the behaviour of bosonic photons and atoms is often treated using phase space methods, where mode annihilation and creation operators are represented by c-number phase space variables, with the density operator equivalent to a distribution function of these variables. The anti-commutation rules for fermion annihilation, creation operators suggest the possibility of using anti-commuting Grassmann variables to represent these operators. However, in spite of the seminal work by Cahill and Glauber and a few applications, the use of Grassmann phase space methods in quantum–atom optics to treat fermionic systems is rather rare, though fermion coherent states using Grassmann variables are widely used in particle physics.     

Classical Behavior of the Dirac Bispinor

It is usually supposed that the Dirac and radiation equations predict that the phase of a fermion will rotate through half the angle through which the fermion is rotated, which means, via the measured dynamical and geometrical phase factors, that the fermion must have a half-integral spin. We demonstrate that this is not the case and that the identical relativistic quantum mechanics can also be derived with the phase of the fermion rotating through the same angle as does the fermion itself. Under spatial rotation and Lorentz transformation the bispinor transforms as a four-vector like the potential and Dirac current. Previous attempts to provide this form of transformational behavior have foundered because a satisfactory current could not be derived.⁽¹⁴⁾     

Discrete quantum gravity and anomalies

A new version of quantum gravity on discrete spaces (simplicial complexes) is proposed. A theory of gravitation interacting with Dirac field is considered. This theory is shown to be free of reparametrization anomaly. The problem of axial gauge anomaly and the associated problem of the doubling of fermion states on a lattice are discussed.     

Backscattering of Dirac fermions in HgTe quantum wells with a finite gap

The density-dependent mobility of n-type HgTe quantum wells with inverted band ordering has been studied both experimentally and theoretically. While semiconductor heterostructures with a parabolic dispersion exhibit an increase in mobility with carrier density, high-quality HgTe quantum wells exhibit a distinct mobility maximum. We show that this mobility anomaly is due to backscattering of Dirac fermions from random fluctuations of the band gap (Dirac mass). Our findings open new avenues for the study of Dirac fermion transport with finite and random mass, which so far has been hard to access.     

Dirac Functional Determinants in Terms of the Eta Invariant and the Noncommutative Residue

 The zeta and eta–functions associated with massless and massive Dirac operators, in a D-dimensional (D odd or even) manifold without boundary, are rigorously constructed. Several mathematical subtleties involved in this process are stressed, as the intrinsic ambiguity present in the definition of the associated fermion functional determinant in the massless case and, also, the unavoidable presence (in some situations) of a multiplicative anomaly, that can be conveniently expressed in terms of the non-commutative residue. The ambiguity is here proven to disappear in the massive case, giving rise to a phase of the Dirac determinant – that agrees with very recent calculations which appeared in the mathematical literature – and to a multiplicative anomaly – also in agreement with other calculations, in the coinciding cases (in fact our results cover much more general situations). A number of physically relevant explicit examples are worked out. After explicit, nontrivial resummation of the mass series expansions, involving zeta and eta functions, our results are finally expressed in terms of quite simple formulae. Received: 7 October 1999 / Accepted: 27 January 2003 Published online: 5 May 2003 RID="*" ID="*" On leave from: ICE/CSIC and IEEC, Edifici Nexus 201, Gran Capità 2-4, 08034 Barcelona, Spain Communicated by R.H. Dijkgraaf     

A unification of boson expansion theories: (I). Functional representations of fermion states

The method for obtaining boson expansion by representing the fermion states as holomorphic functions of many complex variables is presented. Such functional representation is explicitly constructed for each space which is the carrier space of an irreducible representation of a semisimple compact Lie group. This is achieved by proving the unity resolution in terms of holomorphically parametrized Perelomov's generalized coherent states. The functional images of fermion states are polynomials of complex variables, while those of fermion operators are differential operators of finite order with polynomial coefficients.     

Maximal symmetry and mass generation of Dirac fermions and gravitational gauge field theory in six-dimensional spacetime

The relativistic Dirac equation in four-dimensional spacetime reveals a coherent relation between the dimensions of spacetime and the degrees of freedom of fermionic spinors. A massless Dirac fermion generates new symmetries corresponding to chirality spin and charge spin as well as conformal scaling transformations. With the introduction of intrinsic W-parity, a massless Dirac fermion can be treated as a Majorana-type or Weyl-type spinor in a six-dimensional spacetime that reflects the intrinsic quantum numbers of chirality spin. A generalized Dirac equation is obtained in the six-dimensional spacetime with a maximal symmetry. Based on the framework of gravitational quantum field theory proposed in Ref. [1] with the postulate of gauge invariance and coordinate independence, we arrive at a maximally symmetric gravitational gauge field theory for the massless Dirac fermion in six-dimensional spacetime. Such a theory is governed by the local spin gauge symmetry SP(1,5) and the global Poincar′e symmetry P(1,5)= SO(1,5) P~(1,5) as well as the charge spin gauge symmetry SU(2). The theory leads to the prediction of doubly electrically charged bosons. A scalar field and conformal scaling gauge field are introduced to maintain both global and local conformal scaling symmetries. A generalized gravitational Dirac equation for the massless Dirac fermion is derived in the six-dimensional spacetime. The equations of motion for gauge fields are obtained with conserved currents in the presence of gravitational effects. The dynamics of the gauge-type gravifield as a Goldstone-like boson is shown to be governed by a conserved energy-momentum tensor, and its symmetric part provides a generalized Einstein equation of gravity. An alternative geometrical symmetry breaking mechanism for the mass generation of Dirac fermions is demonstrated.     

The Phase of the Scattering Operator from the Geometry of Certain Infinite-Dimensional Groups

We revisit the computation of the phase of the Dirac fermion scattering operator in external gauge fields. The computation is through a parallel transport along the path of time evolution operators. The novelty of the present paper compared with the earlier geometric approach by Langmann and Mickelsson (J Math Phys 37(8):3933–3953, 1996) is that we can avoid the somewhat arbitrary choice in the regularization of the time evolution for intermediate times using a natural choice of the connection form on the space of appropriate unitary operators.     

On the boson mapping of fermion collective pairs

We study the problem of the mapping of fermion collective pairs onto particle-particle bosons and of different fermion operators (hamiltonian, one- and two-particle transfer operators) onto corresponding boson ones and we test the consequences of the truncation to lowest orders of these boson operators. We find that, although the lowest-order terms in the expansion of the operators in boson space lead to matrix elements between boson states which display the qualitative behaviour of the corresponding ones between fermion states, higher-order terms are required to get a quantitative agreement when a large number of particles are involved, as a direct consequence of the increased role of the Pauli principle.     

Dirac equation in metric-affine gravitation theories and superpotentials

We consider a metric-affine gravitational framework in which the dynamical fields are the spin structures, the general linear connections, and the Dirac fermion fields. Using a spin structure and a linear connection on the world manifold, we construct a principal connection on the spinor bundle. By applying general ideas concerning the conservation laws in the Lagrangian approach to field theory, we examine the corresponding conserved currents. The main result is that the currents associated with infinitesimal vertical (internal) transformations of the covariance group are shown to vanish identically. It follows that to every vector field on the world manifold there corresponds a well-defined current, the stress-energymomentum of the fields. It turns out that the fermion fields do not contribute at all to the superpotential terms. Actually the expression we get for the superpotential generalizes the well-known expression obtained by Komar.     

A mathematical analysis of the axial anomaly

Letters in Mathematical Physics - As is well known to physicists, the axial anomaly of the massless free fermion in Euclidean signature is given by the index of the corresponding Dirac operator. We...     

The invariant factor of the chiral determinant

The coupling of spin 0 and spin 1 external fields to Dirac fermions defines a theory which displays gauge chiral symmetry. Quantum-mechanically, functional integration of the fermions yields the determinant of the Dirac operator, known as the chiral determinant. Its modulus is chiral invariant but not so its phase, which carries the chiral anomaly through the Wess–Zumino–Witten term. Here we find the remarkable result that, upon removal from the chiral determinant of this known anomalous part, the remaining chiral-invariant factor is just the square root of the determinant of a local covariant operator of the Klein–Gordon type. This procedure bypasses the integrability obstruction allowing one to write down a functional that correctly reproduces both the modulus and the phase of the chiral determinant. The technique is illustrated by computing the effective action in two dimensions at leading order (LO) in the derivative expansion. The results previously obtained by indirect methods are indeed reproduced.     

Interaction of phonons and dirac fermions on the surface of Bi2Se3: a strong Kohn anomaly

We report the first measurements of phonon dispersion curves on the (001) surface of the strong three-dimensional topological insulator Bi2Se3. The surface phonon measurements were carried out with the aid of coherent helium beam surface scattering techniques. The results reveal a prominent signature of the exotic metallic Dirac fermion quasiparticles, including a strong Kohn anomaly. The signature is manifest in a low energy isotropic convex dispersive surface phonon branch with a frequency maximum of 1.8 THz and having a V-shaped minimum at approximately 2kF that defines the Kohn anomaly. Theoretical analysis attributes this dispersive profile to the renormalization of the surface phonon excitations by the surface Dirac fermions. The contribution of the Dirac fermions to this renormalization is derived in terms of a Coulomb-type perturbation model.