Abstract We show that the fermionic and bosonic spectrum of d = 2 fermions at finite density coupled to a critical boson can be determined non-perturbatively in the combined limit k F → ∞, N f → 0 with N f k F fixed. In this double scaling limit, the boson two-point function is corrected, but amandasimpson@example.org. This double scaling limit therefore incorporates the leading effect of Landau damping. The fermion two-point function is determined analytically in real space and numerically in (Euclidean) momentum space. The resulting spectrum is discontinuously connected to the quenched N f → 0 result. For ω → 0 with k fixed the spectrum exhibits the distinct non-Fermi-liquid behavior previously surmised from the RPA approximation. However, the exact answer obtained here shows that the RPA result does not fully capture the IR of the theory. I. INTRODUCTION The robustness of Landau's Fermi liquid theory relies on the protected gapless nature of quasiparticle excitations around the Fermi surface. Wilsonian effective field theory then guarantees that these protected excitations determine the macroscopic features of the theory in generic circumstances [1, 2] . Aside from ordering instabilities, there is a poignant exception to this general rule. These are special situations where the quasiparticle excitations interact with other gapless states. This is notably so near a symmetry breaking quantum critical point. The associated massless modes should also contribute to the macroscopic physics. In d ≥ 3 dimensions this interaction between Fermi surface excitations and gapless bosons is marginal/irrelevant and these so-called quantum critical metals can be addressed in perturbation theory as first discussed by Hertz and Millis [3] [4] [5] [6] . In 2+1 dimensions, however, the interaction is relevant and the theory is presumed to flow to a new interacting fixed point [6] [7] [8] [9] . This unknown fixed point has been offered as a putative explanation of exotic physics in layered electronic materials near a quantum critical point such as the Isingnematic transition. As a consequence, the deciphering of this fixed point theory is one of the major open problems in theoretical condensed matter physics. There have been numerous earlier studies of Fermi surfaces coupled to gapless bosons but to be able to capture their physics one has almost always been required to study certain simplifying limits [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] . I. INTRODUCTION In this article we show that the fermionic and bosonic spectrum of the most elementary d = 2 quantum critical metal can be computed non-perturbatively in the double limit where the Fermi-momentum k F is taken large, k F → ∞, while the number of fermion species N f is taken to vanish, N f → 0, with the combination N f k F held constant. This is an extension of previous work [21] where we studied the purely quenched limit N f → 0 followed by the limit k F → ∞. In this pure quenched N f → 0 limit the boson two-point function does not receive any corrections and the fermion two point function can be found exactly. However, it is well known that for finite N f and k F the boson receives so-called Landau damping contributions that dominate the IR of the theory. These Landau damping corrections are always proportional to N f , and a subset of these are also proportional to k F . I. INTRODUCTION These terms in particular influence the IR as the large scale, low energy behavior should emerge when k F is large. Studying the double scaling limit where the combination N f k F is held fixed gives a more complete understanding of the small N f and/or large k F limit and their interplay. In particular, this new double scaling limit makes precise previous results in the literature on the RPA approximation together with the N f → 0 limit and the strong forward scattering approximation [22] [23] [24] [25] . Importantly, we shall show that the RPA results qualitatively capture the low energy at fixed momentum regime, but not the full IR of the theory in the double scaling limit. The idea of this limit is similar to the limit taken in [26] where they study a similar model, but in a matrix large N limit. In this limit they keep the quantity k d−1 F /N fixed while taking both N and k F large. All the results here refer to the most elementary quantum critical metal. This is a set of N f free spinless fermions at finite density interacting with a free massless scalar through a simple Yukawa coupling. Its action reads (in Euclidean time) I. INTRODUCTION where j = 1 . . . N f sums over the N f flavors of fermions and µ = . We will assume a spherical Fermi surface, meaning k F both sets the size of the Fermi surface, 2πk F , and the Fermi surface curvature, 1/k F . We will study the fermion and boson two-point functions of this theory in the double scaling limit N f → 0, k F → ∞. By this we mean that we take k F → ∞ while keeping the external momenta (measured from Fermi surface), energies, the coupling scale λ 2 and the Fermi velocity v = k F /m fixed. We shall not encounter any UVdivergences, but to address any ambiguities that may arise the usual assumption is made that the above theory is an effective theory below an energy and momentum scale Λ 0 , Λ k , each of which is already much smaller than k F (Λ 0 , Λ k k F ). We do not address fermion pairing instabilities in this work. They have been studied and found for similar models in other limits, outside of the particular double scaling limit studied here [26] [27] [28] [29] [30] . II. REVIEW OF THE QUENCHED APPROXIMATION (N Let us briefly review the earlier results of [21] as they are a direct inspiration for the double scaling limit. II. REVIEW OF THE QUENCHED APPROXIMATION (N Consider the fermion two-point function for the action above, Eq. (1). Coupling the fermions to external sources and integrating them out, and taking two derivatives w.r.t. the source, the formal expression for this two point function is II. REVIEW OF THE QUENCHED APPROXIMATION (N where G(ω, k)[φ] is the fermion two-point function in the presence of a background field φ, defined by II. REVIEW OF THE QUENCHED APPROXIMATION (N In the limit k F → ∞, for external momentum k close to the Fermi surface, we may approximate the derivative part with −∂ τ + iv∂ x . The defining equation for the Green's function can then be solved in terms of a free fermion Green's function dressed with the exponential of a linear functional of φ. In the quenched N f → 0 limit this single exponentially dressed II. REVIEW OF THE QUENCHED APPROXIMATION (N Green's function can be averaged over the background scalar with the Gaussian kinetic term. II. REVIEW OF THE QUENCHED APPROXIMATION (N The result in real space is again an exponentially dressed free Green's function II. REVIEW OF THE QUENCHED APPROXIMATION (N with the exponent I(r, t) given by II. REVIEW OF THE QUENCHED APPROXIMATION (N and r conjugate to momentum measured from the Fermi surface (k x ), not the origin. Here G B (ω, k) is the free boson Green's function determined by the explicit form of the boson kinetic term in the action Eq. (1). This is of course a known function and due to this simple dressed expression the retarded Green's function and therefore the fermionic spectrum of this model can be determined exactly in the limit N f → 0. The retarded Green's function in momentum space reads [21] (here ω is Lorentzian) II. REVIEW OF THE QUENCHED APPROXIMATION (N where σ is the solution of the equation II. REVIEW OF THE QUENCHED APPROXIMATION (N with k x the distance from the Fermi surface, v = k F /m is the Fermi velocity, and → 0 + is an i prescription that selects the correct root. II. REVIEW OF THE QUENCHED APPROXIMATION (N This non-perturbative result already describes interesting singular fixed point behavior: II. REVIEW OF THE QUENCHED APPROXIMATION (N the spectrum exhibits non-Fermi liquid scaling behavior with multiple Fermi surfaces [21] . II. REVIEW OF THE QUENCHED APPROXIMATION (N Nevertheless, it misses the true IR of the theory as the quenched limit inherently misses the physics of Landau damping. This arises from fermion loop corrections to the boson propagator that are absent for N f → 0. Below the Landau damping scale ω < λ 2 N f k F the physics is expected to differ from the quenched approximation. III. LOOP-CANCELLATIONS AND BOSON TWO-POINT FUNCTION It is clear from the review of the quenched derivation that finite N f , i.e. fermion loop corrections, that only change the boson two-point function, can readily be corrected for by replacing the free boson two-point function G B (ω, k) by the (fermion-loop) corrected III. LOOP-CANCELLATIONS AND BOSON TWO-POINT FUNCTION boson two-point function in Eq. (5) (valid at large k F ). This is the essence of many RPA-like approximations previously studied. A weakness is that finite N f corrections will also generate higher-order boson interactions and these can invalidate the simple dressed expression obtained here. III. LOOP-CANCELLATIONS AND BOSON TWO-POINT FUNCTION At the same time, it has been known for some time that finite density fermion-boson models with simple Yukawa scalar-fermion-density interactions as in Eq. III. LOOP-CANCELLATIONS AND BOSON TWO-POINT FUNCTION (1) have considerable cancellations in fermion loop diagrams for low energies and momenta after symmetrization [24, 31, 33] . These cancellations make loops with more than three interaction vertices V ≥ 3 finite as the external momenta and energies are scaled uniformly to zero. We will now argue that this result also means that in the N f → 0, k F → ∞ limit with N f k F fixed, these V ≥ 3 loops vanish. In this limit only the boson two-point function is therefore corrected and only at one loop and we can directly deduce that in this double limit the exact fermion correlation function is given by the analogue of the dressed Green's function in Eq. (4). We comment on the limitations of considering this limit for subdiagrams in perturbation theory later in this section. III. LOOP-CANCELLATIONS AND BOSON TWO-POINT FUNCTION Consider the quantum critical metal before any approximations; i.e. we have a fully rotationally invariant Fermi surface with a finite k F . The Yukawa coupling shows that the boson couples to the density operator ψ † (x)ψ(x). All corrections to the boson therefore come from fermionic loops with fermion density vertices. These loops always show up symmetrized in the density vertices. Consider a fermion loop with a fixed number V of such density vertices, dropping the overall coupling constant dependence, and arbitrary incoming energies and momenta. Such a loop (ignoring the overall momentum conserving δ-function) has energy dimension 3 − V . These fermionic loops are all UV finite so they are independent of the scale of the UV cut-offs. There are only two important scales, the external bosonic energies and momenta ω i , k i , the fermi momentum k F . A symmetrized V -point loop can by dimensional analysis be written as III. LOOP-CANCELLATIONS AND BOSON TWO-POINT FUNCTION Since fermion loops of our theory with V ≥ 3 vertices have been shown to be finite as external energies and momenta are uniformly scaled to 0 [31] , we thus have that f is finite as k F is taken to infinity. This in turn means that I({ω i }, {k i }) scales as k n F with n ≤ 0 for large k F when V ≥ 3. III. LOOP-CANCELLATIONS AND BOSON TWO-POINT FUNCTION 1 Note that the use of the small external energies and momenta limit from [31] was merely a way of deducing the large k F limit. We do not rely on the physical IR scaling to be the same as in [31] , indeed we will find it not to be the same. All single fermionic loops additionally contain a sum over fermionic flavors so are therefore proportional to N f . III. LOOP-CANCELLATIONS AND BOSON TWO-POINT FUNCTION Combining this we see that a fermionic loop with V ≥ 3 density vertices comes with a factor III. LOOP-CANCELLATIONS AND BOSON TWO-POINT FUNCTION where m V ≤ 0 after symmetrizing the vertices. By now considering the combined limit of N f → 0 and k F → ∞ with N f k F constant we see that these V ≥ 3 loops all vanish. III. LOOP-CANCELLATIONS AND BOSON TWO-POINT FUNCTION See Figure 1 . III. LOOP-CANCELLATIONS AND BOSON TWO-POINT FUNCTION We have now concluded that for a fixed set of external momenta, all symmetrized fermion loops vanish in our combined limit, except the V = 2 loop. There is still a possibility that diagrams containing V > 2 loops are important when taking the combined limit after performing all bosonic momentum integrals and summing up the infinite series of diagrams. In essence, the bosonic integrals and the infinite sum of perturbation theory need not commute with the combined N f → 0, k F → ∞ limit. What the IR of the full theory (finite N f ) looks like is not known so taking the N f → 0 limit last is currently out of reach. In [32] the authors show that divergence of fermionic loops does not cancel under a non-uniform low-energy scaling of energies and momenta where the momenta are additionally taken to be increasingly collinear. The scaling they use is motivated by the perturbative treatment 1 Naively corrections to the boson would be expected to scale as k F since it receives corrections from a Fermi surface of size 2πk F . However k F also sets the curvature of the Fermi surface and for a large k F we approach a flat Fermi surface for which V ≥ 3-loops completely cancel. This is shown in more detail in Appendix A. in [10] and if this is the true IR scaling and it persists at small N f , then there will be effects unaccounted for in the above. III. LOOP-CANCELLATIONS AND BOSON TWO-POINT FUNCTION Regardless of the above mentioned caveat, in keeping the V = 2 fermion loops we take the combined limit after performing the fermionic loop integrals and thus move closer than in our previous work [21] to the goal of understanding the IR of quantum critical metals. III. LOOP-CANCELLATIONS AND BOSON TWO-POINT FUNCTION To summarize: the ordered limit we consider is 1. We first perform rotationally invariant finite k F fermionic loop integrals. III. LOOP-CANCELLATIONS AND BOSON TWO-POINT FUNCTION 2. Then we take the limit N f → 0, k F → ∞ with N f k F fixed; this only keeps V = 2 loops. III. LOOP-CANCELLATIONS AND BOSON TWO-POINT FUNCTION 3. Next we perform bosonic loop integrals. III. LOOP-CANCELLATIONS AND BOSON TWO-POINT FUNCTION 4. Finally we sum all contributions at all orders of the coupling constant. III. LOOP-CANCELLATIONS AND BOSON TWO-POINT FUNCTION The result above means the fully quantum corrected boson remains Gaussian in this ordered limit and only receives corrections from the V = 2 loops. A. Boson two-point function We now compute the one-loop correction to the boson two-point function; in our ordered double scaling limit this is all we need. We then substitute the Dyson summed one-loop corrected boson two-point function into the dressed fermion Green's function to obtain the exact fermionic spectrum. A. Boson two-point function The one-loop correction-the boson polarization-in the double scaling limit is given by the large k F limit of the two-vertex fermion loop. This can be calculated using a linearized fermion dispersion: A. Boson two-point function Note from the cos θ dependence in the numerator that we are not making a "patch" approximation. In the low energy limit this angular dependence is the important contribution of the rotationally invariant fermi-surface, whereas the subleading terms of the dispersion can be safely ignored. As stated earlier, the result of these integrals is finite. However, it does depend on the order of integration. The difference is a constant C A. Boson two-point function As pointed out in for instance [19, 20] , the way to think about this ordering ambiguity is that one should strictly speaking first regularize the theory and introduce a one-loop counterterm. This counterterm has a finite ambiguity that needs to be fixed by a renormalization condition. Even though the loop momentum integral happens to be finite in this case, the finite counterterm ambiguity remains. The correct renormalization condition is the choice C = 0. This choice corresponds to the case when the boson is tuned to criticality since a non-zero C would mean the presence of an effective mass generated by quantum effects. A. Boson two-point function A more physical way to think of the ordering ambiguity is as the relation between the frequency (Λ 0 ) and momentum (Λ k ) cutoff. We will assume that Λ k Λ 0 -which means that we evaluate the k integral first and then the frequency k 0 integral. In this case C = 0 directly follows. IV. FERMION TWO-POINT FUNCTION With the single surviving one-loop correction to the boson two-point function in hand, we can immediately write down the expression for the full fermion two-point function. This is the same dressed expression Eq. (5) as in [21] but with a modified boson propagator IV. FERMION TWO-POINT FUNCTION B,0 + Π), with Π the one-loop polarization of Eq. (10) . Substituting this in we thus need to calculate the integral IV. FERMION TWO-POINT FUNCTION At this moment, we can explain clearly how our result connects to previous approaches. A similarly dressed propagator can be proposed based on extrapolation from 1d results [22, 24] . IV. FERMION TWO-POINT FUNCTION An often used approximation in the literature is to now study this below the scale M D , see e.g [6, 23] . This is the physically most interesting limit since in the systems of interest N f is order one and we are considering large k F . In this limit the polarisation term will dominate over the kinetic terms, but since the rest of the integrand in (11) has no k y dependence, it is necessary to keep the k y term in the boson propagator. The ω and k x momenta will suppress the integrand when they are of order λ 2 whereas the k y term will do so once it is IV. FERMION TWO-POINT FUNCTION D . This means that for M D λ 2 , the relevant k y will be much larger than the relevant ω and k x . This argues that we can truncate to the large M D propagator IV. FERMION TWO-POINT FUNCTION This Landau-damped propagator has been used extensively, for instance [6, 23] . In [23] this propagator was used for the type of non-perturbative calculation we are proposing here. We discuss this here, as we will now show that using this simplified propagator has a problematic feature. This propagator only captures the leading large M D contribution but the non-perturbative exponential form of the exact Green's function sums up powers of the propagator which then are subleading in M D . IV. FERMION TWO-POINT FUNCTION Using the large M D truncated boson Green's function the integral I(τ, r) to be evaluated simplifies to IV. FERMION TWO-POINT FUNCTION Writing the cosine in terms of exponentials we can perform the k x integrals using residues by closing the contours in opposite half planes. Summing up the residues gives IV. FERMION TWO-POINT FUNCTION The k y integral can be performed next to yield IV. FERMION TWO-POINT FUNCTION The primitive function to this ω integral is the upper incomplete gamma function, with argument 2/3. Evaluating this incomplete gamma function in the appropriate limits and substituting the final expression for I IV. FERMION TWO-POINT FUNCTION (τ, r) into the expression for the fermion two-point function gives us: IV. FERMION TWO-POINT FUNCTION where the length scale l 0 is given by IV. FERMION TWO-POINT FUNCTION This result has been found earlier in [22] (see also [24] ). However, this real space expression hides the inconsistency of the approach. This becomes apparent in its momentum space representation. The Fourier transform of the real space Green's function IV. FERMION TWO-POINT FUNCTION is tricky, but remarkably can be done exactly. We do so in appendix B. The result is retarded Greens function, G R (ω R , k x ), and from this we can obtain the spectral function IV. FERMION TWO-POINT FUNCTION As it encodes the excitation spectrum, the spectral function ought to be a positive function that moreover equals 2π when integrated over all energies ω R , for any momentum k. This large M D spectral function contains an oscillating singularity at ω R = vk x . We are free to move the contour into complex ω R -plane by deforming ω R → ω R +iΩ where Ω is positive but otherwise arbitrary. Upon doing this it is easy to numerically verify that indeed the integral over ω R gives 2π. However, if we look at the behaviour close to the essential singularity, the function oscillates rapidly and does not stay positive as one approaches the singularity; see Fig. 3 . This reflects that the large M D approximation done in this way is not consistent. Even though the approximation for the exponent I(τ, r) ≡Ĩ IV. FERMION TWO-POINT FUNCTION is valid to leading order in 1/M D , this is not systematic after exponentiation to obtain the IV. FERMION TWO-POINT FUNCTION fermion two-point function IV. FERMION TWO-POINT FUNCTION Reexpanding the exponent one immediately sees that keeping only the leading term in I(τ, r) IV. FERMION TWO-POINT FUNCTION mixes at higher order with the subleading terms at lower order in 1 IV. FERMION TWO-POINT FUNCTION Despite this problematic feature, we will show from the exact result that in the IR G f IV. FERMION TWO-POINT FUNCTION (with a small modification) does happen to capture the correct physics. IV. FERMION TWO-POINT FUNCTION 1. Exact fermion two-point function for large k F with M D fixed; v = 1 IV. FERMION TWO-POINT FUNCTION We therefore make no further assumption regarding the value of M D and we return to the full integral Eq. (11) to determine the real space fermion two-point function. Solving this in general is difficult, and to simplify mildly we consider the special case v = 1. In our previous studies of the quenched M D = 0 limit we saw that this choice for value of v is actually not very special, even though it appears that there is an enhanced symmetry. After setting v = 1 and changing to spherical coordinates we have IV. FERMION TWO-POINT FUNCTION Performing ther integral gives us IV. FERMION TWO-POINT FUNCTION Note that if the signs of both τ and r are flipped, then this is invariant. Changing the sign of only τ , and simultaneously making the change of variable φ → −φ, then the (real) fraction is invariant but the exponent in the prefactor changes sign. Thus, I goes to I * as the sign of either τ or r is changed. Without loss of generality, we can assume that both of them are positive from now on. We further see that the integrand is invariant under φ → φ + π, so we may limit the range of φ to (0, π) by doubling the value of integrand. Similarly we limit θ to (0, π/2) and multiply by another factor of 2. We then make the changes of variables: IV. FERMION TWO-POINT FUNCTION with s ∈ R and u is integrated over the range (0, 1). For convenience we introduce the function IV. FERMION TWO-POINT FUNCTION Now the two remaining integrals can be written as IV. FERMION TWO-POINT FUNCTION After expanding the exponential we can perform the u integral term by term. We are left IV. FERMION TWO-POINT FUNCTION useful at this stage. Instead we once again integrate term by term. Collecting the prefactors and introducing the constant a = τ /r, the n-th term can be written as IV. FERMION TWO-POINT FUNCTION This can be written as IV. FERMION TWO-POINT FUNCTION where w is integrated on (0, ∞). After splitting the integral at s = −a to get rid of the absolute value we can calculate the s integrals in terms of confluent hypergeometric functions 1 F 1 (a, b; z). Adding the two halves s < −a and s > −a of the integral we have IV. FERMION TWO-POINT FUNCTION It may look like we have just exchanged the s-integral for the w-integral, but by writing the hypergeometric functions in series form, IV. FERMION TWO-POINT FUNCTION the w integral can now be performed. The result is IV. FERMION TWO-POINT FUNCTION The sum over m can be expressed in terms of the ordinary hypergeometric function, IV. FERMION TWO-POINT FUNCTION c n (n + 1) (a 2 + 1) IV. FERMION TWO-POINT FUNCTION The space-time dependence in this expression is in a = τ /r and with additional r-dependence in the coefficients c n The result above is the value for both τ and r positive. Using the known symmetries presented above, the solution can be extended to all values of τ and r by appropriate absolute value signs. Then changing variables to IV. FERMION TWO-POINT FUNCTION we have IV. FERMION TWO-POINT FUNCTION with the function f (R, Φ) given by IV. FERMION TWO-POINT FUNCTION This exact infinite series expression for the exponent I(R, Φ) gives us the exact fermion two-point function in real (Euclidean) space (time). We have not been able to find a closed form expression for this final series. Note that f n ∼ 1/n! for large n, and the series therefore converges rapidly. Moreover, numerically the hypergeometric functions are readily evaluated to arbitrary precision (e.g. with Mathematica), and therefore the value of f (R, Φ) can be robustly evaluated to any required precision. IV. FERMION TWO-POINT FUNCTION As a check on this result, we can compare it to the exact result in the quenched M D = 0 limit in [21] , where the exact answer was found in a different way. In the limit where M D → 0 we see that only the first term of this series gives a contribution and the expression for the exponent collapses to IV. FERMION TWO-POINT FUNCTION In Cartesian coordinates this equals IV. FERMION TWO-POINT FUNCTION This is the exact same expression as found in [21] for v = 1. IV. FERMION TWO-POINT FUNCTION There is one value of the argument for which f (R, Φ) drastically simplifies. For r = 0 (Φ = 0, π) we have IV. FERMION TWO-POINT FUNCTION and thus IV. FERMION TWO-POINT FUNCTION Further numerical analysis shows that the real part of f (τ, r) is maximal for r = 0. either sr + τ is small, s is large, or u is small. The first two cases are also unimportant in thẽ R 1 limit. In the first case we restrict the s integral to a small range of order 1/R around −τ /r; this contribution therefore becomes more and more negligible in the limitR 1. In the second case we will have a remaining large denominator in s outside the exponent that also suppresses the overall integral. Thus for largeR, the only appreciable contribution of the exponential term to the integral in I(τ, r) arises when u is small. To use this, we first write the integral as IV. FERMION TWO-POINT FUNCTION We have added and subtracted an extra term to each to ensure convergence of each of the separate terms. Since the important contribution to I IR,exp is from the small u region we can extend its range from (0,1) to (0, ∞). This way, the integrals can then be done IV. FERMION TWO-POINT FUNCTION In total we have for largeR: IV. FERMION TWO-POINT FUNCTION We see that the leading order term in R is the same as was obtained from the large M D approximation of the exponent. The first subleading term is just a constant. This is good news because we already have the Fourier transform of this expression. This result is valid for length scales larger than 1/M D with a bounded error of the order R −1/3 . Defining this approximation as G IR , i.e. IV. FERMION TWO-POINT FUNCTION the error of this approximation follows from: IV. FERMION TWO-POINT FUNCTION Since the exponential in G IR is bounded we have that ∆G IR = O(r −4/3 ). After Fourier transforming this translates to an error of order O(k −2/3 ). IV. FERMION TWO-POINT FUNCTION A. The exact fermion two-point function in momentum space: Numerical method IV. FERMION TWO-POINT FUNCTION Having understood the shortcomings of the naive large M D answer, the way to derive the exact answer in real space, and the correct IR approximation, we can now analyze the behavior of the quantum critical metal at low energies. For this we need to transform to frequency-momentum space. As our exact answer is in the form of an infinite sum, this is not feasible analytically. We therefore resort to a straightforward numerical Fourier analysis. IV. FERMION TWO-POINT FUNCTION To do so we first numerically determine the real space value of the exact Green's functions. IV. FERMION TWO-POINT FUNCTION To do so accurately, several observations are relevant IV. FERMION TWO-POINT FUNCTION • The coefficients f n in the infinite sum for I(τ, r) decay factorially in n so once n is of orderR, convergence is very rapid. IV. FERMION TWO-POINT FUNCTION • The hypergeometric functions for each n are costly to compute with high precision, but with the above choice of polar coordinates the arguments of the hypergeometric functions are independent of R and M D . We therefore numerically evaluate the series over a grid inR and Φ. We can then reuse the hypergeometric function evaluations many times and greatly decrease computing time. IV. FERMION TWO-POINT FUNCTION • The real space polar grid will be limited to a finite size. The IR expansion from Eq. IV. FERMION TWO-POINT FUNCTION (47) can be used instead of the exact series for large enoughR. To do so, we have to ensure an overlapping regime of validity. It turns out that a rather large value ofR is necessary to obtain numerical agreement between these two expansions, i.e. IV. FERMION TWO-POINT FUNCTION one needs to evaluate a comparably large number of terms in the expansion. For the results presented in this paper it has been necessary to compute coefficients up to order 16 000 inR, for many different angles Φ. The function is bounded for large τ andR but each term grows quickly. This means that there are large cancellations between the terms that in the end give us a small value. We therefore need to calculate these coefficients to very high precision in evaluating the polynomial. For these high precision calculations, we have used the Gnu Multiprecision Library [34] . IV. FERMION TWO-POINT FUNCTION • On this polar grid we computed the exact answer forR <R 0 ≈ 1000 and used cubic We then use a standard discrete numerical Fourier transform (DFT) to obtain the momentum space two-point function from this numeric prescription for G(R, Φ). Sampling G(R, Φ) at a finite number of discrete points, the size of the sampling grid will introduce an IR cut-off at the largest scales we sample and a UV-cut off set by the smallest spacing between points. These errors in the final result can be minimized by using the known asymptotic values analytically. Rather than Fourier transforming G(τ, r) as a whole, we Fourier transform G diff (τ, r) = G(τ, r) − G IR (τ, r) instead. Since both these functions approach the free propagator in the UV, the Fourier transform of its difference will decay faster for large ω and k. This greatly reduces the UV artefacts inherent in a discrete Fourier transform. These two functions also approach each other for large τ and x. In fact, with the numerical method we use to approximate f (R, Φ) described above, they will be identical forR IV. FERMION TWO-POINT FUNCTION This means that we only need to sample the DFT within that area. With a DFT we will always get some of the UV tails of the function that gives rise to folding aliasing artefacts. Now our function decays rapidly so one could do a DFT to very high frequencies and discard the high frequency part. This unfortunately takes up a lot of memory so we have gone with a more CPU intensive but memory friendly approach. To address this we perform a convolution with a Gaussian kernel, perform the DFT, keep the lowest 1/3 of the frequencies and then divide by the Fourier transform of the kernel used. This gives us a good numeric value for G diff (ω, k). To this we add our analytic expression for G IR (ω, k). V. THE PHYSICS OF 2+1 QUANTUM CRITICAL METALS IN DOUBLE SCAL-ING LIMIT With the exact real space expression and the numerical momentum space solution for the full non-perturbative fermion Green's function, we can now discuss the physics of the elementary quantum critical metal in the double scaling limit. Let us emphasize right away that all our results are in Euclidean space. Although a Euclidean momentum space Green's function can be used to find a good Lorentzian continuation with a well-defined and consistent spectral function, this function is not easily obtainable from our numerical Euclidean result. We leave this for future work. The Euclidean signature Green's function does not visually encode the spectrum directly, but for very low energies/frequencies the Euclidean and the Lorentzian expressions are nearly identical, and we can extract much of the IR physics already from the Euclidean correlation function. V. THE PHYSICS OF 2+1 QUANTUM CRITICAL METALS IN DOUBLE SCAL-ING LIMIT In Fig. 4 we show density plots of the imaginary part of G(ω, k x ) for different values of V. THE PHYSICS OF 2+1 QUANTUM CRITICAL METALS IN DOUBLE SCAL-ING LIMIT This result is in contradistinction to what happens to the bosons. When the bosons are not affected in the IR, i.e. the quenched limit, the fermions are greatly affected by the boson: there is a topological Fermi surface transition and the low-energy spectrum behaves as critical excitations [21] . However, once we increase M D to realistic values, the bosonic excitations are rapidly dominated in the IR by Landau damping but we now see that this V. THE PHYSICS OF 2+1 QUANTUM CRITICAL METALS IN DOUBLE SCAL-ING LIMIT reduces the corrections to the fermions. As M D is increased for fixed ω, k x , the deep IR fermion two-point function approaches more and more that of the simple RPA result with self-energy Σ ∼ iω 2/3 . V. THE PHYSICS OF 2+1 QUANTUM CRITICAL METALS IN DOUBLE SCAL-ING LIMIT A more careful analysis of the IR reveals that there are several distinct ω → 0 limits of the two-point function: V. THE PHYSICS OF 2+1 QUANTUM CRITICAL METALS IN DOUBLE SCAL-ING LIMIT However in the case of ω 2 ∼ l 0 k 3 x , the full expression for G IR is necessary to describe the low energy limit, V. THE PHYSICS OF 2+1 QUANTUM CRITICAL METALS IN DOUBLE SCAL-ING LIMIT This existence of multiple limits shows in fact that the RPA result is never a good low energy (less than M D ) approximation for any value of M D /λ 2 . We can illustrate this more clearly by studying the self-energy of the fermion Σ(ω, by definition is meant a spectrum with a discontinuity in the zero-temperature momentum V. THE PHYSICS OF 2+1 QUANTUM CRITICAL METALS IN DOUBLE SCAL-ING LIMIT with A(ω R , k x ) the spectral function. As the spectral function is the imaginary part of the retarded Green's functions and the latter is analytic in the upper half plane of ω R we can move this contour to Euclidean ω and use the fact that G(ω, k x ) approaches G 0 (ω, k x ) in the UV to calculate the momentum distribution function from our Euclidean results. In detail V. THE PHYSICS OF 2+1 QUANTUM CRITICAL METALS IN DOUBLE SCAL-ING LIMIT the first integral can be done with the numerics developed in the preceding section. The contour C goes from Λ to −∞ and for large enough Λ this is in the UV and can well be approximated by the free propagator. The resulting momentum distribution n(k) is shown in Fig. 6 . Within our numerical resolution, these curves are continuous as opposed to a Green's function close to the UV. The error-bars are exaggerated by a factor 100. Perturbation theory Exact result FIG. 7. A sketch of the regimes of applicability of various approximations to the exact fermion Green's function of the elementary quantum critical metal in the double scaling limit. The G IR approximation is applicable both in the deep yellow region and the red and green regions. FIG. 7. A sketch of the regimes of applicability of various approximations to the exact fermion finite M D curves approach the quenched result for |k x | > k * x where k * x is the point of the discontinuity of the derivative of the quenched occupation number. At k * the (derivative) FIG. 7. A sketch of the regimes of applicability of various approximations to the exact fermion of the quenched momentum distribution number does have a discontinuity (reflecting the branch cut found in [21] ). VI. CONCLUSION We have presented a non-perturbative answer for the (Euclidean) fermion and boson twopoint functions of the elementary quantum critical metal in the double limit of small N f and large k F with N f k F fixed. This limit was taken order by order in perturbation theory and additionally before performing bosonic momentum integrals. VI. CONCLUSION Our exact results clarify how approximations that have been made in the past hang together. The N f → 0, k F → ∞ theory is characterized by two energy scales λ 2 and M D . VI. CONCLUSION For very large ω/λ 2 and ω/M The quenched result we obtained earlier [21] does not seem to have a useful regime of validity. As the momentum occupation number n(k) indicates, its precise regime of applicability depends discontinuously on the momentum k/λ 2 . The discontinuity is surprising, but it can be explained analytically as an order of limits ambiguity. Although it is hard to capture the deep IR region for very small N f in the full numerics we find from G IR that in the ω-k plane there is a region where the limit N f → 0 and ω, k x → 0 do not commute. We VI. CONCLUSION show an indication of this in appendix C. Thus for scales below λ 2 , the quenched result is not useful. For scales above λ 2 with M D small, we can use perturbation theory so also in this regime the quenched result is not useful. For small ω and k x , below M D , the IR approximation we presented above gives a good description of the physics. We have presented a pictorial overview of how the various approximations are related in Fig. 7 . VI. CONCLUSION As argued before the spectrum of the elementary quantum critical metal in the ordered double scaling limit obtained here is not the definitive one. Taking this limit after the bosonic integrals and the perturbative sum may give a different result. Another estimate for the true IR spectrum of the elementary quantum critical metal has been postulated before based on large N f approximations [25, 35, 36] , but our result at small N f gives a different IR. VI. CONCLUSION Our non-perturbative answer follows from the insight that in the limit of large k F all fermion loops with more than two external boson legs have cancellations upon symmetrization such that their scaling in k F is reduced. This higher loop cancellation has been previously put forward as a result of the strong forward scattering approximation. This approximation is engineered such that the Schwinger-Dyson equations combined with the fermion number VI. CONCLUSION Ward identity collapse to a closed set of equations. The solution to this closed set is the same dressed fermion correlator that we have presented. However the exact connection between the strong forward scattering approximation and the double scaling limit, is not clear. In the double scaling limit here, it is manifest that only the boson propagator is corrected and this in turn implies that the exact fermion Green's function in real space is an exponentially dressed version of the free Green's function. VI. CONCLUSION At the physics level, an obvious next step is therefore to explore the spectrum of the elementary quantum critical metal including corrections in N f that are not proportional to k f . Since this necessarily involves higher-point boson correlations, the role of self-interactions of the boson needs to be considered. These are also relevant in the IR and may therefore give rise to qualitatively very different physics than found here. We leave this for future work. The robustness of the one-loop result in case of linear fermionic dispersion was recognized before under the name of multiloop cancellation [24] . The technical result is that for a theory with a simple Yukawa coupling and linear dispersion around a Fermi surface, a symmetrized fermion loop with more than two fermion lines vanishes. In our context the linear dispersion is a consequence of the large k F limit. In other words all higher loop contributions to the polarization Π should be subleading in 1/k F . This was explicitly demonstrated at two loops in [37] . VI. CONCLUSION We will give here a short derivation of this multiloop cancellation in the limit k F → ∞. VI. CONCLUSION We may assume in this limit that the momentum transfer at any fermion interaction is always much smaller than the size of the initial ( k) and final momenta ( k ) which are of the order of the Fermi momentum, i.e. VI. CONCLUSION Green's function then reflects a linear dispersion VI. CONCLUSION We now Fourier transform back to real space, as multiloop cancellation is most easily shown in this basis. The real space transform of the "linear" free fermion propagator above is VI. CONCLUSION where as before r is the conjugate variable to k = | k| − k F . The essential step in the proof is that real space Green's function manifestly obeys the identity [21] VI. CONCLUSION with z ≡ r + ivτ . Consider then (the subpart of any correlation function/Feynman diagram containing) a fermion loop with n ≥ 2 vertices along the loop connected to indistinguishable scalars (i.e. no derivative interactions and all interactions are symmetrized). The corresponding algebraic expression in a real space basis will then contain the expression VI. CONCLUSION where S n is the set of permutations of the numbers 1 through n. Using the "linear dispersion" VI. CONCLUSION identity Eq. (A3) and the shorthand notation G 0 (z i 1 − z i 2 ) = G 12 we obtain VI. CONCLUSION Next we cyclically permute the indices from 1 to n − 1: 1 → 2, 2 → 3, ..., n − 1 → 1 in the VI. CONCLUSION This gives us VI. CONCLUSION Then since G i,j corresponds to a (spinless) fermionic Green's function, it is antisymmetric VI. CONCLUSION , and we can conclude that F vanishes for n ≥ 3. For n = 2 it is not possible to use the identity Eq. (A3) since we would need to evaluate G(0) which is infinite. VI. CONCLUSION Appendix B: The Fourier transform of the fermion Green's function in the large VI. CONCLUSION We will now show how to perform this Fourier transform. We need to calculate the following integral: VI. CONCLUSION First we note that integrand is τ -analytic in the region {τ ∈ C : min(0, vx) < Im(τ ) < max(0, vx)}. Since the integrand necessarily goes to 0 at ∞ we can thus shift the τ contour, τ → τ + ix/v. We now have VI. CONCLUSION We have allowed ourselves to choose the order of integration, shift the contour, and then change the order. We see that the x integral now is trivial but only converges for VI. CONCLUSION This is fine since we know that the final result is ω-analytic in both the right and left open half-planes. As long as we can obtain an answer valid within open subsets of both of these sets we can analytically continue the found solution to the whole half planes. We thus proceed assuming Re(ω) is in this range. One can further use symmetries of our expression to relate the left and right ω-half-planes, so to simplify matters, from now on we additionally assume ω to be positive. Let us now consider a negative x, we then see that the τ integral can be closed in the upper half plane and since it is holomorphic there the result will be 0. VI. CONCLUSION We can thus limit the x integrals to R + . We then have VI. CONCLUSION where a = ik + ω/v. The integrand has a pole at (iτ ) 2/3 = −1/(al 1/3 0 v 2/3 ). Now break the integral in positive and negative τ and write it as VI. CONCLUSION where VI. CONCLUSION Usng Morera's theorem we can prove that h is holomorphic on C/R − . Consider any closed curve C in C/R − . We need to show that VI. CONCLUSION We do this by rotating the contour slightly counter clockwise. For any curve C there is clearly a small > 0 such that we will still not hit the pole τ + uτ 1/3 = 0. VI. CONCLUSION The piece at ∞ converges without the denominator and thus goes to 0. Since the integral now converges absolutely we can use Fubini's theorem to change the orders of integration VI. CONCLUSION And since also the integrand is holomorphic on a connected open set containing C the proof is finished. VI. CONCLUSION The function h can for 0 < arg(u) < 2π/3 be expressed as a Meijer G-function: VI. CONCLUSION The G-function has a branch cut at R − and because of that we can not easily write h(u) VI. CONCLUSION in terms of it since the argument u appears cubed in the argument of the G-function and we know that h is holomorphic on all of C/R−. We can however express h as an analytic continuation of the G-function past this branch cut, onto the further sheets of its Riemann surface. We will write the G-function as a function with two real arguments, first the absolute value and second the phase of its otherwise complex argument. We will allow the phase to be any real number and when outside the range [−π, π], we let the function be defined by its analytical continuation to the corresponding sheet. We hereafter omit the constant parameters of the G-function. We then have: VI. CONCLUSION In the last step we used the fact that the G-function commutes with complex conjugation. VI. CONCLUSION We see that the Green's function is given by a certain monodromy of the G-function. It is given by the difference in its value starting at a point u 3 0 /4 on the sheet above the standard one and then analytically continuing clockwise around the origin twice to the sheet below the standard one and there return to u 3 0 /4. Since we only need this difference, we might expect this to be a, in some sense, simpler function as would happen for e.g. monodromies of the logarithm. To see how to simplify this we look at the definition of the G-function. It is defined as an integral along L: VI. CONCLUSION There are a few different options for L and which one to use depends on the arguments. VI. CONCLUSION In our case L starts and ends at +∞ and circles all the poles of Γ(b i − s) in the negative direction. Using the residue theorem we can recast the integral to a series. We have double poles at all negative integers and some simple poles in between. The calculation to figure out the residues of all these single and double poles is a bit too technical to present here but in the end the series can be written
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