Trace anomaly for non-relativistic fermions

We study the coupling of a 2+1 dimensional non-relativistic spin 1/2 fermion to a curved Newton-Cartan geometry, using null reduction from an extra-dimensional relativistic Dirac action in curved spacetime. We analyze Weyl invariance in detail: we show that at the classical level it is preserved in an arbitrary curved background, whereas at the quantum level it is broken by anomalies. We compute the trace anomaly using the Heat Kernel method and we show that the anomaly coefficients a, c are proportional to the relativistic ones for a Dirac fermion in 3+1 dimensions. As for the previously studied scalar case, these coefficents are proportional to 1/m, where m is the non-relativistic mass of the particle.


Introduction
Newton-Cartan (NC) geometry was originally proposed as a covariant formulation of Newtonian gravity (see e.g. [1] for a review). In recent times it raised growing interest for applications to condensed matter systems (see e.g. [2][3][4][5]), such as fermions at unitarity and quantum Hall effect. The background fields of NC gravity provide a natural set of sources for operators in the energy-momentum tensor multiplet of theories with non-relativistic Schrödinger invariance.
Many theoretical difficulties in dealing with these systems are due to the strong coupling nature of the interaction. Strong coupling may drastically change the infrared (IR) degrees of freedom coming from a given ultraviolet (UV) description. Renormalization Group (RG) trajectories may interpolate from weak to strong coupling changing the nature of the physical spectrum and of the degrees of freedom. In relativistic theories there are general results which formalize the intuition that information is lost when coarse graining is implemented from UV to IR, namely Zamolodchikov's c-theorem in d = 2 [6], the Ftheorem in d = 3 [7][8][9] the a-theorem in d = 4 [10][11][12][13][14][15]. For condensed matter applications, it would be interesting to establish similar results in non-relativistic systems.
With these motivations, in the last years a certain amount of work has been devoted to the study of non-relativistic trace anomalies. In general, trace anomalies can be classified into two classes [16]: type A or B depending if they have non-vanishing or vanishing Weyl variation, respectively. The relevant ones for RG constraints are the type A, such as c in d = 2 or a in d = 4 relativistic systems.
In the non-relativistic case, at a fixed point space and time may have different relative scaling, which can be parameterized by the dynamical exponent z: Moreover one may distinguish between Schrödinger and Lifshitz systems, whose main difference relies on the presence of Galilean boost invariace. So far, in all the known cases, the Lifshitz trace anomalies (see [17][18][19][20][21][22][23]) turn out to be of type B and so they do not give interesting candidates for monotonic quantities. In the Schrödinger case, in d = 2 + 1 dimensions and for dynamical exponent z = 2, if one couples the theory to a curved NC background, it exists a type A anomaly [24] 1 . The structure of this anomaly is the same as the trace anomaly for d = 4 relativistic theories, and so it includes a type-A and a type-B part, parameterized by a and c coefficients: In this equation E 4 and W 2 are the Euler density and the Weyl tensor squared of the null reduction metric in eq. (2.4); these quantities are completely determined in terms of 2 + 1dimensional NC geometry data. The use of the extra dimension is a formidable trick to conveniently keep track of the Milne boost symmetry. Cohomological analysis and general properties were studied in [21,[24][25][26]. The first explicit calculation of anomalies for a physical system was performed in [27] with the Heat Kernel (HK) method, for the case of a free scalar. Later this result was confirmed in [28] using Fujikawa approach 2 .
Fermions are a fundamental ingredient in Nature; the purpose of this paper is to study conformal invariance and anomalies for a free non-relativistic spin 1/2 fermion coupled to a generic curved Newton-Cartan background, using null reduction from a 3 + 1 dimensional relativistic action. First of all, we show that it is possible to couple the fermion to the geometry in a Weyl invariant way; this is not trivial, due to the different scaling properties of the components of frame fields, spin connection and dynamical fermionic fields. Our analysis specializes to the case where the gyromagnetic ratio g is twice the spin s; the generic case requires modified Milne boost transformations [5] on the sources and can not be studied by null reduction.
The other issue that we address is the computation of the anomaly coefficients a and c using Heat Kernel. In the bosonic case, these coefficients turn out to be proportional to the corresponding ones in relativistic systems in 3 + 1 dimensions. We find that the same property still persists also in the fermionic case.
The paper is organized as follows. In sect. 2 we derive the fermionic action from the null reduction of the Dirac one and we discuss the gyromagnetic factor. In sect. 3 we show in detail that the action is Weyl invariant. In sect. 4 we compute the trace anomaly using HK method. We conclude in sect. 5, tecnical details are in appendices.

Metric and frame fields
We will consider the coupling of non-relativistic fermions in 2 + 1 dimensions to a background NC geometry. In order to make the implementation of the local version of the Galilean symmetry (Milne boost invariance) more covenient, we use the null-reduction method [30] from an extra-dimensional relativistic 3 + 1 dimensional theory. We will sometimes refer to null-reduction method as Discrete Light-Cone Quantization (DLCQ). Useful references about NC geometry include [31][32][33][34][35][36][37][38]. Galilei invariance for fermions was first studied in [39]. For other approaches to couple non-relativistic theories to background NC geometry see [5,40]. Other applications of null reduction to fermions were discussed in [41].
In our conventions late latin capital indices, like M, N, . . ., correspond to 3 + 1 dimensional curved space-time indices, whereas early latin capital indices like A, B, . . . , correspond to tangent space indices, where the metric is locally flat. The coordinate x − denotes the null direction of the dimensional reduction. The remaining light-cone coordinate, x + , will play the role of time in the lower dimensional non-relativistic theory. Curved space coordinates will be labelled by lower case latin indices i, j, . . ., whereas the tangent space counterparts will be labelled by a, b, . . .. Collectively, space time indices of the lower dimensional theory will be denoted by µ, ν, . . . and α, β, . . . for curved and tangent space coordinates, respectively. Summarizing, DLCQ indices are Since the light-cone indices ± use the same symbols for curved or tangent space indices, we will use the notations ± indicating that they refer to curved (subscript (M )) or tangent space (subscript (A)) lightcone coordinates.
In order to apply the null reduction, we will consider fields of the form and a metric of the form We denote the determinant of the metric as: The metric tensor G M N defines a non degenerate 3 + 1 dimensional metric whose entrees encode the main ingredients of the 2 + 1 dimensional NC geometry: a positive definite symmetric rank 2 tensor h µν , which corresponds to the spatial inverse metric, and a nowhere-vanishing vector n µ (defining the local time direction), with the condition n µ h µα = 0. In order to define a spatial metric with lower indices and a connection, one introduces a velocity field v µ , with the condition n µ v µ = 1. Given (h µν , n µ , v ν ), one can then uniquely define h µν , with: where P µ ν is the projector onto spatial directions. The velocity vector is not unique (it is only required to satisfy n µ v µ = 1) and the ambiguity in the choice of v is related to the last ingredient of the NC geometry: a non-dynamical gauge field A µ , whose presence is necessary to guarantee Milne boost invariance. This gauge field will act as a source for the particle number symmetry. We introduce, for later convenience, the antisymmetric tensors: The null reduction is a useful trick to realise the invariance under the following Milne boost transformations: while n µ and h µν are invariant. Modified Milne transformation may also be considered, but then the null reduction trick can not be used (see e.g. [31]). Since we are dealing with spinors, the covariant derivative also contains the spin connection term and then it is necessary to introduce an orthonormal frame field (vielbein) which relates the metric in the curved spacetime with the flat tangent space. The metric in the flat tangent space is given by (2.9) As usual, the vielbein are defined by the following relations: In order to consider the coupling of fermions to 2 + 1 NC gravity, the dreibein will be defined by dimensional reduction of fierbein. Such operation is not unique. The following choice turns out to be convenient: We can further simplify our expression by using the consistency relations among fielbein which entail the following constraints: v µ e a µ = 0 These relations simplify the vielbein with the inverted indices: (2.14) The following relation is useful: (2.15)

Dirac action
The Dirac operator is expressed as Conventions for gamma matrices with lightcone indices are summarized in appendix A. The covariant derivative takes the form

17)
ω M AB being the spin connection defined in Appendix B. We shall derive the non-relativistic fermion action in 2 + 1-dimensions from the null reduction of the 3 + 1-dimensional Dirac action: The connection in the covariant derivative D M has no torsion term and so the lagrangian in eq. (2.18) can be made hermitian by partial integration.

Flat space-time
We start by considering the simplest flat case: 19) and A µ = 0. The Dirac action is just the flat one. The following notation is used: 20) and the Dirac Lagrangian can be written as (2.21) We find the Euler-Lagrange equations of motion for the various components: As expected for the Dirac action in the massless case, the left and right Weyl spinors decouple. The auxiliary fields χ L,R can be eliminated by the equations of motion and replaced in the Lagrangian; we obtain a set of decoupled Schrödinger equations for the fermions ϕ L,R .

Curved spacetime
In this section we will write eq. (2.18) in a more explicit way, in order to later establish the gyromagnetic factor and show that the action is conformal invariant. The left and right-handed parts of the Dirac spinor decouple: In the remaining part of this section we will consider the action L 1 for just the left component Ψ L (Ψ R is completely analogous): where (2.25) For convenience, we renamed (χ L , ϕ L ) as (χ, ϕ). An explicit calculation gives: (2.26) We can write L 1 as follows: where we introduced derivatives which are covariant with respect to the local U(1) symmetry:D The last two lines are more troublesome and require the explicit knowledge of the components of the spin connection, because its Lorentz indices are contracted with sigma matrices, containing also spinorial indices. We can put the action (2.27) in the following form: In these expressionsD µ denotes a partially covariant derivative which includes just the gauge and the curved space spin connectionω µab built just with the spatial tetrad e a µ ; this derivative acts on the matter fields ϕ and χ as follows: wherẽ (2.32) The auxiliary field χ is determined by the equations of motion as follows: . (2.33) Replacing it into the action in eq. (2.29), we could obtain a cumbersome Lagrangian written only in terms of ϕ. In order to keep our calculations simple, we will later specialize to some specific backgrounds.

Gyromagnetic ratio
Let us compute the gyro-magnetic ratio of the non-relativistic fermion. We consider flat background space-time as in eq. (2.19) and generic gauge field A µ . Specializing the general results in Appendix B, we find the non-zero components of the spin connection: Eliminating χ with the equations of motion, we obtain: Since the charge associated to the magnetic field is m, we find a coupling to the magnetic field with gyromagnetic ratio g = 1: where in our case the charge q = m and S = σ/2 is the spin. This is consistent with the form of the Milne boost transformations which come from null reduction, which are valid for g = 2s [5].

Weyl invariance
In order to study the conformal symmetry of the theory, it is useful to determine the Weyl weights of the fields appearing in the action. Weyl transformations act on the metric in the following way: The action on the frame fields is as follows: It is also useful to know how each element of the spin connection transforms under a Weyl transformation: In the usual relativistic case the components of a Dirac spinor have all the same Weyl weight; this is not true in the null reduction setting that we are considering, because the tetrads have different Weyl weights. The transformation of the (χ, ϕ) components is as follows: χ→e −2σ χ , ϕ→e −σ ϕ .
We note that this Weyl weight choice is crucial in order to assign to the termΨΨ a welldefined Weyl weight. A conformal coupling term such as RΨΨ would have mass dimension 5, spoiling conformal invariance. Promoting σ to a spacetime-dependent function in eq. (3.4), one can then verify the Weyl invariance of the action in eq. (2.27) by direct calculation, using the non-homogeneus part of the variation of the spin connection (see eq. 3.3). One can also check that this is consistent with eq. (2.33): if we insert ϕ→e −σ ϕ, we indeed find that χ→e −2σ χ.

General framework
For a complex field φ, the vacuum functional W is defined by where S D is the classical action specified by a differential operator D. In the bosonic case, the path integral is evaluated in terms of the functional determinant of the operator D as In the fermionic case there are two differencies: first the change of sign in the r.h.s. of (4.2) due to the Berezin functional integration. Second, there is the difficulty that the Dirac operator / D is not elliptic after a Wick rotation. This problem can be bypassed by evaluating the determinant of the square of the Dirac operator and inserting a factor 1/2: In this way the Euclidean version of the squared Dirac operator is elliptic and meets the requirements needed in order to make the heat kernel computation. In fact, using anticommutation rules for the product of totally antisymmetric Dirac matrices we find (see e.g. [42], [43]): We need to compute the HK with the Euclidean version of the operator△ in eq. (4.4). To this purpose we decompose it as the flat part △ plus curved space perturbation δ△:

The flat case
The computation of the HK is performed in Euclidean space. This is realized by the substitutions The HK operator of a general euclidean operatorÔ E is defined aŝ We will denote by KÔ E the matrix elements and by byKÔ E the diagonal matrix elements In the flat non-relativistic case, with operator △ in (4.5): the heat kernel has been evaluated in [27] and its matrix elements read Here a comment is in order: to use the heat kernel machinery with the Schrödinger operator, we use the formal replacement −2im∂ t −→ −2m −∂ 2 t . This, by itself, does not render the Schrödinger operator elliptic, but it makes possible an integral representation in which the exponential of the Schrödinger operator is written as a sum of exponentials of elliptic operators, which is precisely what is needed to compute the heat kernel, namely This trick was first introduced in [44], although in a different context, and used in [27] to evaluate the anomaly in the bosonic case. In its essence, this regularization is not different from the one normally used in the relativistic case to adapt the heat kernel procedure to fermions: to make elliptic the Dirac operator, one first considers its square, perform the heat kernel, and then takes the square root of the resulting determinant.

The curved case
In order to explicitly compute the functional determinant we work in coordinate representation with scalar product: In curved background, the HK can be evaluated as a perturbative expansion around (4.11).
The diagonal matrix elements in the coordinate basis of the heat kernel can be expanded in powers of s as: (4.14) This defines the De Witt-Seeley-Gilkey coefficients a 2k (△) of the problem. In non-relativistic 2 + 1 dimensional theories, the trace anomaly is proportional to the a 4 coefficient [27]. It is convenient to introduce a quantum mechanical space, with flat inner product and, for any operatorÔ, to define the operatorMÔ such that Thus, one introduces and "effective" operatorMÔ that keeps track of the metric in the inner product. In our case, ifÔ =△ = − 1 4 R, then

A specific perturbation of flat spacetime
To proceed, we specialize to a particular perturbation of flat spacetime: Also we remind that the spatial frame field is a Kronecker delta: (4.20) For simplicity, we choose η independent from the time coordinate. The non-vanishing components of the spin connection and Cristoffel symbols are: The euclidean version of operatorMÔ is obtained by using eq. (4.6): We will need the matrix elements ofM in coordinate representation; to this purpose, it is useful to use the following decomposition:

Perturbative expansion
The next task is to obtain the De Witt-Seeley-Gilkey expansion of the HK operator, in order to find the a 4 coefficient and then the trace anomaly. We will splitM in a free part plus a perturbationV : We can expand perturbatively the HK as a Dyson series: where the terms of the sum arê (4.32) Since we are perturbing around flat space, from [27] we have

Single insertion
At the first order the Dyson series is According to eq. (4.23), we can decompose the expression as (4.36) The contribution K 1a i (s) contains an implicit sum over the index i.
Note that we also introduced in the expression the trace operation, since we are now dealing with squared matrices and the heat kernel expansion required a trace over the operator considered.
We can use the following results from Appendix A of [27]: Moreover the contributionK 1a i due to a i is the sum of the trace of various terms proportional to the derivatives of a i ; these terms have all zero trace and soK 1a i = 0.

Double insertion
At the second order the heat kernel expansion is (4.39) K 2 splits into the sum of several contributions: where in each contribution there is an implicit sum over the indices i, j. We can use the following results from Appendix B of [27]: (4.43) For the remaining terms, the calculation is performed in Appendix C: The expressions forK 2a i S ,K 2Sa i involve traces of matrix products of the kind but these all vanish due to the structure of the matrices a i , S, whose entries sit in orthogonal subspaces.

Results
Summing the contribution from the single and double insertions, we find a 4 up to the second order in η, for d = 2: √ ga 4 (△) = 2 m(4π) 2 (4.47) We should then express the result in terms of curvature invariants. Up to the second order in η, the curvature combinations entering the anomaly are given by: In our conventions the Euler density E 4 and the square of the Weyl tensor W 2 are, in term of the Riemann and Ricci tensor of the null reduction metric eq. (2.4): Since we are studying a Weyl-invariant operator, we know from the Wess-Zumino consistency conditions that the R 2 term cannot enter the anomaly. We can then write the result as: The trace anomaly then can be computed as follows: (4.51)

Conclusions
In this paper we checked that the action of a non-relativistic spin 1/2 fermion coupled to NC geometry is Weyl invariant. Then the trace anomaly was computed using the HK method; the result is for a fermionic spin doublet: , (5.2) and the dots stand for possible additional terms, both higher derivatives and of the kind discussed in [28], which violate the Milne boost symmetry. Up to an overall 1/m multiplicative factor, the anomaly coefficients turn out to be proportional to the ones of a relativistic Dirac fermion in 4 dimensions. A similar numerical coincidence happens also in the scalar case [27], where the value of the anomaly coefficients is: where ξ is the parameter multiplying the conformal coupling.
It is natural to conjecture that an analog of the a-theorem may hold for the E 4 coefficient of Schrödinger-invariant theories in 2 + 1 dimensions. For example, in the case where both the elementary and the composite degrees of freedom would be free scalars and fermions with spin 1/2, it would imply that In Galilean-invariant theories the mass is a conserved quantity and the mass of a bound state is equal to the sum of the masses of the elementary constituents: no bound-state contribution to the mass is present as in the relativistic case. As proposed in [27], the 1/m dependence is consistent with the intuition that bound states form in the infrared: as energy is added bound states tend to be broken. Several interesting problems require further investigation: • Some new anomaly terms were computed in [28] using Fujikawa approach; they are present when a non-trivial background U (1) gauge field is added and they violate Milne boost symmetry. Wess-Zumino consistency conditions for these new terms should be studied and the computation should be checked using HK method.
• The relation between the anomaly coefficients and the correlation functions of the energy-momentum tensor multiplet should be clarified. In the case of vacuum correlation function, these correlators have support just at coincident points. It would be interesting to check if the anomaly coefficients can be related to the form of the finite-density correlators evaluated at separated points.
• The relation between the anomaly and the dilaton effective action should be investigated; in the relativistic case, this leads to a proof of the a-theorem [14]. The study of non-relativistic dilaton was initiated in [45].
• It would be interesting to attempt a perturbative proof using Osborn's local renormalization group approach; this was initiated in [26]. The main missing ingredient to the proof is to control the positivity of some anomaly coefficients whose relativistic analog turn out to be proportional to the Zamolodchikov metric.
• In the relativistic supersymmetric case, there is a powerful relation between the trace anomaly coefficients and R-charges [46]; it would be interesting to check if a similar relation exists also in the non-relativistic case. The supersymmetric local RG approach as in [47] might be a convenient way to investigate these issues. Newton-Cartan supergravity was studied in [48].
• The anomaly coefficients for anyons coupled to NC backgrounds should be computed. This may be interesting for condensed matter applications, as the quantum Hall effect.
which gives: • Gamma matrices in 4 dimensions The Lorentz generators are: which gives:

B Spin connection
The explicit expression for the spin connection is:

C Some double insertion contributions to the Heat Kernel
Here we consider contributions of the form K 2X 1 X 2 (s), where whose explicit expression is: The quantity K 2P P was already computed in [27].
We can split the integration as follows: where Ξ X 1 X 2 and Θ correspond to the space and time integrals, respectively. It is useful to Fourier transform: X 1 X 2 , (C.5) and to introduce: The Fourier transforms of the space part of the integrals are: Ξ a i a j = −∂ x,i dx 1 dx 2 − (x 2 − x 1 ) j 2(s 2 − s 1 ) Υa i (k 1 )a j (k 2 ) , (C.7) where P (k) and a i (k) are the Fourier transform of P (x) and a i (x). The two basic integrals give: The expressions forΞ a i P andΞ a i a j can be obtained differentiatingΞ P P andΞ P a j with respect to x i . The time part gives: