1. Introduction
This paper concerns relativistic fermions interacting strongly in three spacetime dimensions, in the context of a field theory known as the Thirring model with Lagrangian density
Here, the fields are reducible spinors, so the Dirac matrices are . In the Euclidean metric, they obey and . The index runs over N distinct fermion species. The contact interaction between conserved fermion currents results in a repulsive force between fermions but attraction between fermions and antifermions. A question of interest is then whether in the massless limit a bilinear condensate forms as a result of strong interactions, leading to the dynamical generation of fermion mass.
It is natural to analyse bilinear condensation in terms of symmetry breaking. In three dimensions, there are two elements of the reducible Dirac algebra
and
, which anticommute with the kinetic term of (
1). Accordingly,
(
1) is invariant under the following field rotations:
Together, these rotations generate a U(2
N) global symmetry, which can be broken either explicitly by
or spontaneously by
to U(
U(
N), when the rotations (
3) no longer leave the ground state invariant. Goldstone’s theorem implies spontaneous symmetry breaking yields
massless bosons in the theory’s spectrum.
It is suspected that symmetry breaking occurs for a sufficiently large interaction strength and a sufficiently small N; it is even possible that the resulting quantum critical point observed at might be a UV-stable fixed point of the renormalisation group, implying that a continuum limit at this point is possible. The fixed-point theory is expected to display universal features of the strongly interacting dynamics characterised by the pattern of symmetry breaking. However, there are no small parameters to enable a systematic investigation of this phenomenon by analytic means. Determination of the critical exponents and even the critical flavour number below which symmetry breaking can occur are essentially non-perturbative problems.
A natural approach employs numerical simulations of lattice field theory (a recent review can be found in [
1]). The most recent work uses a lattice fermion formulation which seeks to respect the U(
) symmetry such as the SLAC derivative [
2,
3], or domain wall fermions [
4,
5]. However, there is also a substantial body of earlier simulations [
6] employing the more primitive staggered formulation, in which fermion fields are represented by single-component Grassmann objects
located on the sites of a cubic lattice. As well as U(
N) flavour rotations, staggered fermions also enjoy a second U(
N) global symmetry protecting them from acquiring mass, of the form
where
is an alternating sign in effect partitioning the sites
x into distinct odd and even sublattices. This time, therefore, bilinear condensation drives a symmetry breaking U
U
U(
N), resulting in just
Goldstones. For a strongly interacting system, therefore, we expect distinct fixed-point behaviour and, indeed, simulations of the staggered model [
7] support a critical
that is significantly larger than that found for the U(
)-symmetric variants [
3,
4,
5]. Moreover, simulation studies of the minimal staggered model with
. (it is very common in the literature to designate
N staggered flavours in 3
d as describing
“continuum flavors” [
8]). Ref. [
9] find critical indices indistinguishable from those of the Gross–Neveu model having the same global symmetries [
10], even though in the latter case symmetry breaking can be described analytically using a
expansion.
Despite these apparent shortcomings, the staggered Thirring model does exhibit interesting behaviour; in particular, the critical exponents characterising the fixed-point are sensitive to the value of
[
7]. Could there exist a continuum-based description of the corresponding fixed-point theories? One question which needs addressing is the significance of
N—in a weak-coupling long-wavelength limit it is natural to interpret staggered femions in terms of
autonomous flavours [
8], or in modern parlance, each staggered flavour describes two continuum “tastes”. However, even in early staggered Thirring studies [
6], the factorisation of interaction currents into distinct and mutually independent taste sectors was not manifest, and there is no reason a priori to require this in a strongly coupled setting. In what follows, we will refer to the difficulty in separating taste and spin components as “spin/taste entanglement”. A related question is how to engineer the U
U(
N) symmetry in the continuum where we have no lattice partition to help recover (
4).
This paper will answer such questions using a framework introduced into lattice field theory by Becher and Joos in 1982 [
11], who found that a version of the Dirac equation rooted in concepts of differential geometry originally noted by Kähler [
12] in 1962 is in fact the formal continuum limit of staggered lattice fermions. As set out in the next few sections, in the Kähler-Dirac approach fermions are not spinor fields but rather are complexes of
p-forms, where
, with
d the dimension of spacetime. This is a natural way to prepare for transcribing continuum fields to a lattice [
13]; indeed, it even proves possible to formulate Kähler-Dirac fermions on simplicial lattices [
14], thereby extending the staggered approach to the random geometries explored in dynamical triangulation models of quantum gravity. Each
p-form has
components. In the four dimensional case analysed in [
11] a fermion field has thus
components, which are recast as four independent tastes of four-component spinor fields. For the case
to be developed in what follows, the corresponding field has eight components recast as two tastes of reducible four-component spinor. The algebraic details very closely mirror the assignment of spin/taste degrees of freedom to staggered lattice fermions in three spacetime dimensions originally set out by Burden and Burkitt [
8].
The remainder is organised as follows.
Section 2 is a brief but hopefully self-contained introduction to the differential geometry machinery required. Readers who are already expert will find our notations and conventions set out; those less familiar might also benefit from the helpful Appendix of [
11], or a textbook such as [
15].
Section 3 derives the equivalence between the free Kähler-Dirac equation, which with suitable notation assumes the same form in any dimension, and a continuum Dirac equation in three Euclidean dimensions describing two tastes of reducible spinor. The same framework is used in
Section 4, following the introduction of a generalised scalar product between
p forms, to identify the fermion current that will be used in building the Thirring interaction term.
Section 5 at last introduces the Thirring model action in the Kähler-Dirac language, and identifies both the U
U(
N) global symmetry and also an important parity symmetry shared in common with staggered lattice fermions. The Euclidean path integral is introduced permitting an explicit derivation of the Noether current associated with the symmetry corresponding to (
4).
In
Section 6, we begin to take the geometrical form of the theory more seriously by exploring the idea that in a suitably regularised interacting theory the renormalisation of the field components should depend on
p: the Thirring interaction term is modified in order to accommodate this possibility, and it is shown that the resulting terms when recast in a spinor basis exhibit spin/taste entanglement and are in exact correspondence with the interaction derived from the staggered Thirring model [
6] using the formalism of [
8]. This demonstrates that the proposed
p-dependent field rescaling is perfectly consistent with a properly regularised lattice model, and also that spin/taste entanglement is not a lattice artefact, but rather in fact a feature of an interacting continuum field theory. Finally, in
Section 7, the idea is taken a step further with the exploration of truncated actions resulting from retaining just field components with two consecutive values of
p. The most interesting case corresponds to keeping just
, resulting in a theory of six-component spin-one fermions, whose physical states are transverse, and for which fermion and antifermion are states of opposite polarisation.
Section 8 summarises the paper’s findings and speculates on the applicability of the exotic scenario of
Section 7 to the physics of a putative renormalisation group fixed point at strong coupling.
3. The Kähler-Dirac Equation
The starting point is the observation that
, the Laplacian operator. Hence,
is in effect the square-root of the Laplacian, and therefore linear in momentum, while still local. It is thus a candidate for incorporating in a relativistic wave equation, as first written by Kähler [
12]:
The Kähler-Dirac equation (KDE) takes the same form in any spacetime dimension. The scalar parameter
m is the fermion mass. Note that, since
d and
implement
, the equation only makes sense if
, i.e.,
admits an expansion of the form (
6), with components
having a mass dimension of 1 in three spacetime dimensions.
It is helpful to define the Clifford product between differential forms:
with particular instances
It immediately follows from (
9) and (
12) that the KDE can be rewritten
Now, the identity
is strongly reminiscent of the defining relation
for Dirac matrices in Euclidean metric, and suggests the operation
furnishes a representation of the Dirac algebra in the eight-dimensional space spanned by
. The appropriate representation of the algebra in three spacetime dimensions was identified in [
8] in a study of the staggered lattice fermion operator. It is the direct sum
of two inequivalent irreducible two-dimensional representations generated by the Pauli matrices
(
), and by
. The Pauli matrices have the property
, where * denotes a complex conjugation and
T the matrix transpose. Analysis proceeds by identifying a new basis
The key result is now
where Roman indices
. The derivation of (
20) makes repeated use of
.
In order to express the KDE in the basis (
19), we need the orthogonality relations
implying
Using (
6), we then define
where we have introduced fields
u,
d whose lower index
will turn out to be associated with spinor degrees of freedom in the non-interacting case and whose upper index
will be associated with taste. The field transformations between bases are then:
Combining the result (
20) with the KDE equation (
17) we deduce
i.e., the free Dirac equation for a two-taste four-component spinor field
, with Euclidean Dirac matrices defined
We will refer to this familar form as the free KDE in the -basis.
5. Action and Symmetries
Now, we have enough equipment to define the action and hence the Euclidean path integral. The action for free fields is
For the Thirring model, this is supplemented by a contact interaction of the form
, where the normalisation of the coupling strength, which has mass dimension-1, is somewhat conventional but has been chosen to be consistent with [
6]. In form notation, this reads
Using
, we arrive at the Thirring model action
We note in passing that four-fermi interactions in models constructed from Kähler-Dirac fermions have also been investigated in four dimensions [
16].
As a consequence of its construction from
bilinears the action (
40) has two manifest global symmetries. First:
This symmetry correponds to the conservation of fermion charge, and the corresponding Noether current is given by (
34). Second, in the limit
:
which follows because
both yield
and by inspection of the component expansion of
(
31). This is analogous to the chiral symmetry protecting fermions from additive mass renormalisation in
. The corresponding Noether current is
In order to translate to the
-basis, observe that the action of
in effect exchanges
and
in (
24). It then follows straightforwardly that
where we introduce two new hermitian
-matrices obeying
:
From here it is straightforward to extend the model by introducing
N Kähler-Dirac fermion flavours
,
. The flavour index
i is distinct from the indices
in (
41), which run over taste degrees of freedom. The two U(1) rotation symmetries (
42) and (
43) are trivially extended to U
U
, broken to U(
N) either explicitly by
, or spontaneously by dynamical generation of a non-vanishing condensate
.
Finally, consider discrete parity inversion. In odd spacetime dimensions, this is conveniently represented by inversion of all spacetime axes:
,
. The action (
40) and (
41) is invariant provided
Note that the Noether currents (
34) and (
45), along with all bilinears of the form
, are parity-odd.
The Euclidean path integral is defined by
where
are now Grassmann-valued and
is considered independent of
. We illustrate its use via a derivation of the Ward Identity for the divergence of the current
; for simplicity, we consider only the free action (
38). Consider the impact of the field transformation (
43) where
is infinitesimal but now spacetime-dependent.
Now, use (
29) together with
and the definition (
44) to write
where in the second step we have integrated the first term by parts. Since the path-integral measure
is formally invariant under the field transformation (in fact, this is only the case on a spacetime manifold with vanishing Euler characteristic [
17]), the change in variables has no impact on the path integral, and we conclude
Since (
51) holds for any
, we conclude the expectation value of the three-form in square brackets is identically zero, which is the Ward Identity. In the
-basis, it has the familiar form
6. Impact of Quantum Corrections
Our treatment up to this point has been either classical or formal. In any application to a genuine interacting quantum field theory, it is inevitable that the theory will need to be regularised somehow in order to control the calculation of quantum corrections. As a concrete example, we have already discussed the close parallels between the KDE continuum formalism and staggered lattice fermions, and will assume without further discussion that the proof of [
11] that the KDE is the formal continuum limit of staggered fermions continues to apply in three dimensions.
Regularisation is essentially some kind of truncation of the degrees of freedom present in the classical field theory and inevitably violates some of the symmetries of the classical theory. In many cases, this leads to the requirement of renormalisation of both the fields and the coupling parameters in the theory, which depends on some physical scale. Consider the Thirring action in the
-basis (
41), where the rotations (
42) and (
43) take the form
Equation (
41) also looks to be invariant under a U(2) rotation among the tastes indexed by
. Beyond that, in the limit
there are apparently additional symmetries corresponding to all the rotations given in (
2) and (
3), which together with taste rotations would generate a U(4
N) global symmetry broken to U(
U(
by a mass
. Our viewpoint is that this symmetry is not fundamental and can only be recovered in certain limits, such as long wavelength or weak coupling.
We will proceed on the assumption that the geometric description employed in the KDE is more natural, so that after quantum corrections the field expansion of Equation (
6) is modified:
Here, a renormalised field
is defined in terms of bare components
via wavefunction renormalisation constants
which depend on the interaction strength, the renormalisation scale and, crucially in this context, on the form degree
p. This correction is covariant, in the sense that
is insensitive to rotations acting on the spacetime indices specific to
, and the key symmetries (
42) and (
43) continue to be respected by
even with
.
The form of (
54) motivates a more general exploration of possible interaction currrents. In
the space of bilinear currents consistent with the four renormalisation constants
is spanned by
,
,
and
. Transcription to the
-basis for the first two of these is given in (
34) and (
45), and e.g.,
Now, observe the following identities for the components of
:
Recalling
, we deduce a particularly convenient combination:
Here, the second component of the tensor product is a
matrix acting on taste indices. Similarly,
In either case, what emerges is an interaction current which although parity-odd and respecting the U(1)⊗U
(1) symmetries (
42) and (
43) no longer treats fermion tastes as independent degrees of freedom but rather entangles taste and spacetime rotations, contrary to what is expected for particle flavour degrees of freedom. Remarkably, the currents
,
,
and
all feature in equal weight contact interactions in the Thirring model formulated with staggered fermions on a 3
d cubic lattice as derived in a basis with explicit spinor and taste indices using the formalism of [
8], and given in Equation (2.12) of [
6]. In view of the equivalence [
11] between Kähler-Dirac fermions and the formal continuum limit of staggered lattice fermions, this result should not be surprising.
It is now clear that these interactions survive the long-wavelength
limit, where the lattice spacing
a furnishes an explicit UV cutoff. Other terms entangling spinor and taste degrees of freedom, which formally vanish as
, are also present in the lattice formulation [
6]. The current analysis demonstrates that spin/taste entanglement is not a lattice artifact, but is rooted in a continuum action of the form (
40) with U(
U
symmetry. However, it is significant that such terms also emerge from a well-defined regularisation capable of application to strongly interacting dynamics.
8. Discussion
This paper has developed the description of relativistic fermions in the language of differential geometry, originally set out in [
12], to three spacetime dimensions. The principal result is the specification of a continuum field theory sharing the same parity and global U(
U(
N) invariances as the “staggered Thirring model” originally studied numerically using lattice field theory simulations in [
6]. In our view, this puts the staggered Thirring model on a firm footing as an interacting quantum field theory distinct from the U(
)-invariant version based on the action (
1), which is the focus of much recent numerical work [
1]. This result is entirely consistent with Becher and Joos’ demonstration that Kähler-Dirac fermions are the correct continuum limit for staggered lattice fermions [
11]. Beyond the weak-coupling and long-wavelength limits, we have seen that spin/taste entanglement is not merely a lattice artifact, but a genuine feature of an interacting continuum field theory: tastes are not the same as flavours.
An important consequence of regarding the
-basis as more fundamental than the more familiar
-basis is the response to quantum corrections encapsulated in the proposed relation (
54) relating renormalised to bare fields, in which multiplicative renormlisation depends solely on
p, consistent with U(
U(
N) symmetry. This was demonstrated explicitly in
Section 6 through the recovery of interaction currents entangling spin and taste originally found in the staggered Thirring model. However, a more spectacular, if speculative, consequence was worked out in
Section 7, where the assumption of a strong hierarchy of the
arising due to large anomalous scaling dimensions in the vicinity of a renormalisation-group fixed point motivated the investigation of truncated actions retaining just two
p-values. The Lagrangian
(
75) is especially interesting, describing six-component spin-one fermions, with fermions/antifermions being states of well-defined polarisation, and dynamics dominated by the four components lying in the transverse subspace in the UV limit. Could these exotica form the basis for a description of strongly interacting fixed-point dynamics? The answer must await a controlled non-perturbative investigation.
We conclude with a brief discussion of spin and statistics. The Lagrangian (
75) describes spin-one fermions which in the canonical approach to field quantisation would be represented by field operators with the anticommutator
. An immediate concern is the apparent contradiction with the spin-statistics theorem requiring Lorentz-invariant theories of anticommuting fields to be quantised with half-integer spin representations of the Lorentz group. A symptom of the problem is revealed through the ground state expectation of the anticommutator of fields at arbitrary spacetime separation [
18]:
Here,
represents Minkowski space versions of the
-matrices, we have assumed that all states are defined in the transverse subspace, and for field quantisation with the “wrong” statistics, the
theorem dictates the appearance on the RHS of the symmetric solution of the Klein–Gordon equation (or its generalisation):
where for free fields
. Specialising to the case of a spacelike interval
with
, we find
Since the RHS of (
79) does not vanish outside the lightcone, there is a violation of microcausality. This is a general result independent of the detailed form of the dispersion
. For free fields, the asymptotic properties of the modified Bessel functions in (
81) can be used to to find
and
that is, the causality violation is localised to within roughly a Compton wavelength of the lightcone, but diverges as
, although less severely than the
behaviour of 3 + 1
d [
18].
Since microcausality is a desirable property for a fundamental theory, the correct relation between spin and statistics is a necessary ingredient of a complete quantum field theory. By hypothesis, however, the spin-one action (
75) serves only as an effective description of the dynamics near a UV fixed point, in the deep Euclidean regime
very far from the lightcone. The question of whether the spin-statistics linkage compromises the fixed-point description remains open.