Stress tensor and current correlators of interacting conformal field theories in 2+1 dimensions: Fermionic Dirac matter coupled to U(1) gauge field

We compute the central charge $C_T$ and universal conductivity $C_J$ of $N_F$ fermions coupled to a $U(1)$ gauge field up to next-to-leading order in the $1/N_F$ expansion. We discuss implications of these precision computations as a diagnostic for response and entanglement properties of interacting conformal field theories for strongly correlated condensed matter phases and conformal quantum electrodynamics in $2+1$ dimensions.


I. INTRODUCTION
A variety of strongly correlated electron systems at quantum critical points or phases in two spatial dimensions are believed to be described by (interacting) conformal field theories in 2+1 dimensions (CFT 3 's). The workhorse is the Wilson-Fisher CFT 3 , also known as the O(N)-model of a real-valued vector field with N components [1][2][3], which describes, among other things, the Ising model for N = 1 [4,5], superfluid-to-insulator transitions for N = 2 [6,7], and quantum magnetic transitions for N = 3 [8,9]. Especially intriguing are gauge theoretical descriptions of condensed matter systems (e.g.: [10] and references therein for an overview) such as of quantum Hall systems (e.g.: [11,12] and references therein), fractionalized magnets and deconfined critical points in strongly correlated Mott insulators [13][14][15], and effective theories for the cuprates [16][17][18][19]. There, the relevant dynamics is often provided by emergent or effective degrees of freedom not necessarily present in the bare Hamiltonian. These conformal phases of quantum matter in 2+1 dimensions provide a unique interpolation between the better understood CFT's in 1+1 dimensions [20] and much studied gauge theories for high energy vacua in 3+1 dimensions [21,22].
A common feature of CFT 3 's is the absence of quasi-particles and for condensed matter systems it is of particular interest to understand response properties of interacting CFT 3 's to externally applied perturbations such as electromagnetic fields or mechanical forces without invoking a quasiparticle picture. In this paper, we consider N F Dirac fermions minimally coupled to a U(1) gauge field. This theory arises in a variety of condensed matter contexts [10,12,16,17,19]. The Euclidean action, contains Grassmannian two-component fermion fieldsψ α and ψ α , where α is the fermion flavor index, and µ is the spatial and (imaginary) temporal index in 2+1 dimensions. Repeated indices are summed over. γ µ 's are the Dirac matrices that satisfy {γ µ , γ ν } = 2δ µν . We use the same conventions as Kaul and Sachdev for their fermion sector [10].
The gauge field A µ , a conventional spin-1 boson often dubbed as "emergent photon" in the condensed matter context, ensures fulfillment of a local U(1) gauge symmetry at every point (τ, r) in (Euclidean) space-time. A potential, bare Maxwell term 1 2e 2 F µν F µν is not written in Eq. (1) and is unimportant for the universal constants of interest in this paper. The gauge field gets dynamical by integrating the fermion fields in the large N F limit. In Landau gauge, the gauge field propagator at N F → ∞ is purely transverse and takes the characteristic overdamped form (with p = | p|) Model Eq. (1) with a bare Maxwell term is also known as QED 3 and flows to strong coupling in the infrared and shares its propensity to form fermion bound states "mesons" with QCD in 3+1 dimensions [29][30][31]. Deforming QED 3 toward graphene-type models with instantaneous Coulomb interactions are also interesting [32][33][34][35]. It is believed that for sufficiently large N F , Eq. (1) flows to a strongly coupled conformal phase in the infrared, preserving scale invariance [36] (and references therein). This is the regime of interest in the present paper.
B. Key results: central charge C T and C J up to next-to-leading order in 1/N F The main result of this paper is an explicit formula and numerical value of the central charge C T of Eq. (1), defined below as the universal constant appearing in the stress tensor correlator at the interacting conformal fixed point, up to next-to-leading order in the 1/N F expansion: C T (1) comes from one out of nine Feynman graphs in momentum space computed below in Fig. (5) where Li n (z) = ∞ k=1 z k k n is the polylogarithm or Jonquiére's function for n = 2. The sum of other eight diagrams evaluate to the remaining term in the innermost bracket, 4 + 104 15π 2 , in the first line of Eq. (3). We observe from Eq. (3) that 1/N F corrections to the N F → ∞ value remain as large as ≈ 50% down to N F ≈ 8. Similarly large corrections were also observed (for current correlators) in the CP N−1 model and attributed in particular to vertices directly involving the gauge field [28].
It is hard to overestimate the fundamental importance of the central charge in conformal field theory with applications ranging from thermodynamics, quantum critical transport, to quantum information theory [23]. An interesting recent application are explicit formulae for the Rényi entropy for d-dimensional flat space CFT's and we quote here the formula from Perlmutter [24].
The prime denotes a derivative with respect to q of the Rényi entropy S q = 1 1−q log Tr ρ q , ρ a reduced density matrix, and H d−1 the hyperboloid entangling surface. Moreover, precision values of C T may be useful for conformal bootstrap approaches for the 3D-Ising and other models [4] as well as serving as a benchmark for numerical simulations of frustrated quantum magnets [25].
Computations of stress tensor correlators in interacting CFT's (at least without an excessive amount of symmetry such as supersymmetries) in effective dimensionality greater than 2 are extremely scarce and we are not aware of a previous computation of C T for Eq. (1) in 2+1 dimensions. We quote here related works across the quantum field theory universe we are aware of to date: two papers by Hathrell using loop expansions from 1982, one on scalar fields up to 5-loops [40] and one on QED up to 3 loops [41], an -expansion around four dimensions for scalar and gauge theories by Cappelli, Friedan and LaTorre in 1991 [42], and a series of papers on the O(N) vector model from 1994-1996 by Petkou [27,37,43].
In the present paper, we compute C T by direct evaluation of Feynman graphs in momentum space fulfilling and using the relation [38,39], generalizing our recently developed technology [15,28] to Dirac fermions and contractions over stress tensor vertices. We discuss this further in Sec. IV.
The second result of this paper is an (somewhat simpler) computation of the universal constant C J of the two-point correlator of the conserved flavor current of Eq. (1): where T 's are generators of the SU(N F ) group normalized to satisfy Tr(T T m ) = δ m . As the stress tensor T µν , this flavor current is conserved and its two-point correlator depends on one universal For single fermion QED 3 , C J describes the universal electrical conductivity in the collisionless regime ω T , with T being the temperature. Depending on the physical context, however, it may also be related to magnetic or other response functions [16]. Our result for C J to next-to-leading order in 1/N F is (derived in Sec. II) with the analytical expression corresponding to one of the graphs being This result is seemingly in disagreement with the value computed in the Appendix of Ref. 12 and we compare to their value in detail in Sec. II. As a (positive) cross-check, we have repeated a different calculation of the (non-conserved) staggered spin susceptibility in the Appendix of Rantner and Wen [16] using our approach and found the same logarithmically divergent coefficients.
Note that Eq. (1) has a further conserved "topological" current related to the curl of the gauge field [39] but we do not consider it further here.

C. Organization of paper
The remainder of the paper is organized as follows: in Sec. II, we define the Feynman rules for Eq. (1) and the current vertex, and evaluate the 3 graphs renormalizing the current-current correlator. In Sec. III, we briefly recapitulate the main elements of the Tensoria technology for the momentum integrals. In Sec. IV, we define the stress tensor vertex and evaluate the 9 graphs renormalizing the stress tensor correlator. In the conclusions, we summarize and point toward potential future directions where our technology could be applied to.
In this section, we compute the SU(N F ) flavor current-current correlator and compare it to the two previous computations also using the 1/N F expansion that we are aware of [12,16]. We begin by stating the Feynman rules, compute the leading N F → ∞ graph in some detail, and then the more complicated self-energy and vertex corrections at order 1/N F . We will separate the contributions into longitudinal and transverse projections and show that all longitudinal and logarithmically singular corrections mutually cancel as they should for a conserved, transverse quantity.

A. Feynman rules and graphs in momentum space
The Feynman rules for N F Dirac fermions coupled to U(1) gauge field in Eq. (1) contain the relativistic fermion propagator Using the Feynman rules explained above, Fig. 3 exhibits the 3 contractions to the current correlator to order 1/N F . Each of the expressions in Eq. 12 contain a minus sign due to the trace over fermions, a (trivial) trace over flavor indices, a trace over the Dirac matrices, and one (1-loop graph) or two (the two 2-loop graphs) 2 + 1 dimensional momentum integrals k ≡ d 3 k 8π 3 . We get: These expressions are now evaluated in the following way using our "Tensoria" technology [28]: We first perform the trace over the Dirac indices, collecting the contracted expressions in the numerator. Especially for the more complicated expressions it is helpful to automate it and use the Feyncalc MATHEMATICA package for this [50]. Then we replace the integrals of momentum written in components as described in the next section and in the Appendix of Ref. 28. Finally, we separate out the transverse I (i) T and longitudinal I (i) T momentum projections in the following form:

B. Leading order N F → ∞ graph for C J
To illustrate the procedure with a simple example, let us evaluate the leading order graph: The integral over the first term in the numerator 2k 2 δ µν is a power-law divergence in the UV and can be dropped. The second, third, fourth and firth term in the numerator can be integrated using the identities with the abbreviation for the modulus p = | p| and interchangably p 2 = p 2 . The result comes out purely transverse, leading to C We evaluate the vertex correction and self-energy correction graphs (1) and (2) in Eq. 12 algorithmically and the results are in Table I. As expected for a conserved quantity, the log-singularities of each individual graph cancel when taking the sum, so does the longitudinal part. As announced in the Introduction, our result Eq. (9) seems to disagree with Chen et al. [12] who computed C J for QED 3 to order 1/N F . The relevant 1/N F correction is given in Eq. (A17) in the appendix of their paper. Mapping to our conventions we take g = 1 and A = 16 and an overall minus sign. These authors obtained C Chen, et al.
The sign of their 1/N F correction match but the value seem to be different from Eq. (9). Diagram

III. TENSORIA TECHNOLOGY: MINI-RECAP
Before proceeding, let us briefly recapitulate our algorithm to evaluate the tensor-valued momentum integrals as described in more detail in the Appendices of [28,49]. At the heart of the algorithm are Davydychev permutation [47,48] relations to perform integrals of the form: After the Dirac traces, the integrals can all be brought into this form. After the first momentum integration, we temporarily introduce a UV-momentum cutoff that formally breaks symmetries such as conformal invariance. Using this cutoff as a sorter, all power-law divergences are discarded as they would be absent in a gauge-invariant regularization schemes such as dimensional regularization. The remaining finite and logarithmically divergent terms can be integrated analytically graph-by-graph and the log-singularities are seen to cancel exactly.
We close this recap by noting that despite the exact cancellations of the log-singularities as a strong consistency check, and the many additionally performed checks of all sub-routines in Tensoria, at the moment we have no proof that of the exactness to O(1/N F ) of our results and it would be very desirable to compare results to another method.
In this section, we extend our technology to compute the stress tensor correlator of Eq. (1) to next-to-leading order in 1/N F . We first define the stress tensor itself and write down the Feynman rules for the stress tensor vertices. Then, we first illustrate in some detail the calculation of the leading N F → ∞ graph before evaluating the remaining 8 graphs with Tensoria. The two major complications here are: (i) the gauge field can connect directly to the stress tensor vertex leading to a vertex involving 3 lines, and (ii) four 3-loop graphs, including those of the Azlamasov-Larkin type, appear. As in the JJ computation, we explicitly show that all log-singularities cancel when summing all graphs to ensure to conserved nature of T µν in accordance with symmetries.

A. Feynman rules and graphs in momentum space
The stress tensor operator for Eq. (1) depends on both the fermions and the gauge fields via the leading to the stress tensor vertices shown in Fig. 4. The eight graphs and their analytical expressions shown in Figs. 5, 6 contribute to order 1/N F and we again denote their sum by  Table II.
In order to compute the "central charge" C T , we will project it out from the evaluated graphs using the relation Eq. (6): We note here that a number of previous analyses [3,27,37] have been conducted in real space, where the invariance of correlators under the full set of conformal transformations are transparent but the analysis to work out the constants for an interacting CFT is quite involved.
B. Leading order N F → ∞ graph for C T Let us evaluate the leading order graph, the first line in Fig. 6. Including the index permutations described in the caption of the figure, we have where we dropped the second term in the numerator in the second line because it is a powerlaw divergence in the UV, absent in dimensional regularization. We can also check that without immediately contracting the graph, the uncontracted terms fulfill the index structure of Eq. (6).

C. 1/N F corrections for C T and discussion
Tensoria computes the 1/N F corrections algorithmically and Table II collects the results. As before, we observe an exact cancellation of the logarithmic singularities of each graph in accordance with symmetry requirements. Summing the graphs leads to Eq. (3) in the Introduction.
In addition to the discussion in the Introduction, we mention here that the components of the stress tensor correlator also yield the shear viscosity particularly relevant for strongly interacting quantum field theories at finite temperature [51,52]. In order to resolve the collisional physics, however, it is necessary to solve a Boltzmann equation or invoke the AdS-CFT correspondence (see e.g.: Refs. 52, 53 and references therein).

V. CONCLUSIONS
The aim of this paper was to provide precision computations of the "central charge" C T and universal conductivity C J of interacting conformal field theories in 2 + 1 dimensions. We considered N F Dirac fermions coupled to an "emergent photon" motivated by frequent occurrence of this field theory in a variety of condensed matter systems. The low-energy sector is also equivalent to many-flavor QED 3 in the conformal phase.
Our hope is that our results could become a useful diagnostic for numerical evaluations of entanglement properties of CFT 3 's, conformal bootstrap approaches, or application of the AdS-CFT correspondence. Going forward, our technology may also complement explicit computations of conformal correlators in the context of dualities of Large N Chern-Simons Matter Theories [54,55].