Lattice spinor gravity

We propose a regularized lattice model for quantum gravity purely formulated in terms of fermions. The lattice action exhibits local Lorentz symmetry, and the continuum limit is invariant under general coordinate transformations. The metric arises as a composite field. Our lattice model involves no signature for space and time, describing simultaneously a Minkowski or euclidean theory. It is invariant both under Lorentz transformations and euclidean rotations. The difference between space and time arises from expectation values of composite fields. Our formulation includes local gauge symmetries beyond the generalized Lorentz symmetry. The lattice construction can be employed for formulating models with local gauge symmetries purely in terms of fermions.     

Calculation of the color axial anomaly on the basis of the gauge-invariant spatial point-splitting regularization

The nonsinglet axial anomaly is calculated by employing Schwinger’s point-splitting regularization of the interaction between fermions and a non-Abelian gauge field. This method makes it possible to obtain a covariant expression for the anomaly directly from the effective action for the gauge field. Previously, the anomaly under study was calculated by many other methods. However, all calculations based on the point-splitting regularization (from the pioneering study of Bardeen in 1969) involve a number of intermediate steps and subtractions of specially chosen polynomials in the field.     

Optimised Dirac operators on the lattice: construction,properties and applications

We review a number of topics related to block variable renormalisation group transformations of quantum fields on the lattice, and to the emerging perfect lattice actions. We first illustrate this procedure by considering scalar fields. Then we proceed to lattice fermions, where we discuss perfect actions for free fields, for the Gross‐Neveu model and for a supersymmetric spin model. We also consider the extension to perfect lattice perturbation theory, in particular regarding the axial anomaly and the quark gluon vertex function. Next we deal with properties and applications of truncated perfect fermions, and their chiral correction by means of the overlap formula. This yields a formulation of lattice fermions, which combines exact chiral symmetry with an optimisation of further essential properties. We summarise simulation results for these so‐called overlap‐hypercube fermions in the two‐flavour Schwinger model and in quenched QCD. In the latter framework we establish a link to Chiral Perturbation Theory, both, in the p‐regime and in the ϵ‐regime. In particular we present an evaluation of the leading Low Energy Constants of the chiral Lagrangian – the chiral condensate and the pion decay constant – from QCD simulations with extremely light quarks.     

The Schwinger model and its axial anomaly

The Schwinger model (quantum electrodynamics with massless fermions in one spatial dimension) is solved, supposing that space is a circle. This clarifies aspects of the usual version of the model, where space is a line, without changing the physics. The Hamiltonian formalism is used. On a circle, an abelian gauge field has one physical degree of freedom, and the gauge covariant Dirac operator, which couples the fermions to this degree of freedom, exhibits spectral flow. The relationship between the spectral flow and the axial anomaly is explained. Some variants of the Schwinger model are also discussed.     

Axial-vector anomaly for Dirac-Kähler fermions on the lattice

The axial-vector current of Dirac-Kähler fermions on the lattice is studied. We consider a U(1) gauge theory in two dimensions as well as an SU(N) gauge theory in four dimensions. Using a short-distance expansion of the fermion propagator in an external gauge field, we show that the correct anomaly is reproduced in the continuum limit.     

Vectorial versus axial Goldstone bosons

The Yukawa interactions of fermions with Goldstone bosons are given in closed form for an arbitrary renormalizable field theory to all orders of perturbation theory or for a general effective Lagrangian. Although the diagonal couplings are always pseudoscalar there is an important difference between spontaneously broken vector and axial-vector global symmetries. Compared to the axial case, the diagonal couplings of “vectorial” Goldstone bosons to charged fermions are suppressed by mixing angles or appear only via radiative corrections involving gauge fields. This general result may be relevant for the problem of flavour symmetry breaking in composite models.     

Monte Carlo simulation of quantum statistical lattice models

In this article we review recent developments in computational methods for quantum statistical lattice problems. We begin by giving the necessary mathematical basis, the generalized Trotter formula, and discuss the computational tools, exact summations and Monte Carlo simulation, that will be used to examine explicit examples. To illustrate the general strategy, the method is applied to an analytically solvable, non-trivial, model: the one-dimensional Ising model in a transverse field. Next it is shown how to generalized Trotter formula most naturally leads to different path-integral representations of the partition function by considering one-dimensional fermion lattice models. We show how to analyze the different representations and discuss Monte Carlo simulation results for one-dimensional fermions. Then Monte Carlo work on one- and two-dimensional spin-$\text{1}\text{2}$ models based upon the Trotter formula approach is reviewed and the more dedicated Handscomb Monte Carlo method is discussed. We consider electron-phonon models and discuss Monte Carlo simulation data on the Molecular Crystal Model in one, two and three dimensions and related one-dimensional polaron models. Exact numerical results are presented for free fermions and free bosons in the canonical ensemble. We address the main problem of Monte Carlo simulations of fermions in more than one dimension: the cancellation of large contributions. Free bosons on a lattice are compared with bosons in a box and the effects of finite size on Bose-Einstein condensation are discussed.     

Field theory of the random flux model

The random flux model (defined here as a model of lattice fermions hopping under the influence of maximally random link disorder) is analysed field theoretically. It is shown that the long range physics of the model is described by the supersymmetric version of a field theory that has been derived earlier in connection with lattice fermions subject to weak random hopping. More precisely, the field theory relevant for the behaviour of n-point correlation functions is of non-linear σ model type, where the group GL(n|n) is the global invariant manifold. It is argued that the model universally describes the long range physics of random phase fermions and provides further evidence in favour of the existence of delocalised states in the middle of the band in two dimensions. The same formalism is applied to the study of non-Abelian generalisations of the random flux model, i.e. N-component fermions whose hopping is mediated by random U(N) matrices. We discuss some physical applications of these models and argue that, for sufficiently large N, the existence of long range correlations in the band centre (equivalent to metallic behaviour in the Abelian case) can be safely deduced from the RG analysis of the model.     

One-Dimensional World as a Tool to Study Chiral Effects

JETP Letters - An analog of the chiral separation effect and the chiral magnetic effect has been studied for fermions in one spatial dimension. The relation of these effects to axial anomaly in the...     

Mixtures of correlated bosons and fermions: Dynamical mean‐field theory for normal and condensed phases

We derive a dynamical mean‐field theory for mixtures of interacting bosons and fermions on a lattice (BF‐DMFT). The BF‐DMFT is a comprehensive, thermodynamically consistent framework for the theoretical investigation of Bose‐Fermi mixtures and is applicable for arbitrary values of the coupling parameters and temperatures. It becomes exact in the limit of high spatial dimensions d or coordination number Z of the lattice. In particular, the BF‐DMFT treats normal and condensed bosons on equal footing and thus includes the effects caused by their dynamic coupling. Using the BF‐DMFT we investigate two different interaction models of correlated lattice bosons and fermions, one where all particles are spinless (model I) and one where fermions carry a spin one‐half (model II). In model I the local, repulsive interaction between bosons and fermions can give rise to an attractive effective interaction between the bosons. In model II it can also lead to an attraction between the fermions.     

Wilson fermions and the topological charge on the lattice

We investigate the realization on the lattice of the relation between the chiral properties of the fermions and the topological properties of the gauge field. A lattice definition of the topological charge density via the U(1) Adler-Bardeen anomaly is analysed with the help of given configurations of nontrivial topology and for 2-dimensional quenched QED.     

Axial anomaly in lattice abelian gauge theory in arbitrary dimensions

The axial anomaly of lattice abelian gauge theory on a hyper-cubic regular lattice in arbitrary even dimensions is investigated by applying the method of exterior differential calculus. The topological invariance, gauge invariance and locality of the axial anomaly determine the explicit form of the topological part. The anomaly is obtained up to a multiplicative constant for finite lattice spacing and can be interpreted as the Chern character of the abelian lattice gauge theory.     

Thermodynamics of the System of Massive Dirac Fermions in a Uniform Magnetic Field

We construct the grand partition function of the system of massive Dirac fermions in a uniform magnetic field from Landau levels,through which all thermodynamic quantities can be obtained.Making use of the Abel-Plana formula,these thermodynamic quantities can be expanded as power series with respect to the dimensionless variable b=2eB/T~2.The zero-field magnetic susceptibility is expanded at zero mass,and the leading order term is logarithmic.We also calculate scalar,vector current,axial vector current and energy-momentum tensor of the system through ensemble average approach.Mass correction to chiral separation effect is discussed.For massless chiral fermions,our results recover the chiral magnetic effect for right-and left-handed fermions,as well as chiral separation effect.     

Chiral symmetry on the lattice with Wilson fermions

The chiral properties of the continuum limit of lattice QCD with Wilson fermions are studied. We show that a partially conserved axial current can be defined, satisfying the usual current algebra requirements.A proper definition of the chiral symmetry order parameter, $\text{〈0|}\text{ψ}\text{ψ|0〉}$, is given, and the chiral properties of composite operators are investigated. The implications of our analysis to the lattice determination of non-leptonic weak amplitudes are also discussed.     

Induced fermion current in the σ model in (2 + 1) dimensions

The topological current in the (2 + 1)-dimensional O(3) nonlinear σ model is shown to be identical to the fermionic current induced by a Yukawa-type interaction with the classical field of the σ model. Alternatively, the system can be described in terms of two species of fermions interacting with an abelian gauge field, and hence related to the recently discovered three-dimensional anomaly. The role of fermion zone modes is 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.     

Anomalous production of chiral charge from black holes

The Reissner Nordström solution to Einstein equations coupled to a Yang-Mills field is unstable against the emission of light-charged fermions. We study the flux chirality in this decay process and illustrate the consistency of the result with the prediction of the axial anomaly.     

Renormalized Ward-Takahashi relations and topological susceptibility with staggered fermions

Local (axial) vector currents and (pseudo) scalar fields are constructed, renormalized and used to derive Ward-Takahashi relations for staggered fermions in lattice QCD. With the help of these a lattice derivation is given of the Witten-Veneziano formula for the flavor singlet pseudoscalar particle. We discuss how the ensuing topological susceptibility is to be evaluated and argue that it is finite.     

Heavy fermions in gauge theories: (I). Analysis of a general spinor loop

A general one-loop spinor diagram is analyzed in the coordinate space with an arbitrary number of external scalar, pseudoscalar, vector and axial vector legs. We identify chiral anomalies, and an unambiguous definition of a renormalized spinor loop amplitude in gauge theories is given by studying its symmetry properties. We then study the case when some of fermions carry very large masses compared to external momentum scales. Using a new calculational technique based on Schwinger's proper-time method, we provide the explicit forms of dominant effective local vertices induced by virtual heavy fermions in general spontaneously broken gauge theories. In the sequel to the present paper, these results will be applied to various interesting field theory models.