Exact critical exponents for the antiferromagnetic quantum critical metal in two dimensions

Unconventional metallic states which do not support well defined single-particle excitations can arise near quantum phase transitions as strong quantum fluctuations of incipient order parameters prevent electrons from forming coherent quasiparticles. Although antiferromagnetic phase transitions occur commonly in correlated metals, understanding the nature of the strange metal realized at the critical point in layered systems has been hampered by a lack of reliable theoretical methods that take into account strong quantum fluctuations. We present a non-perturbative solution to the low-energy theory for the antiferromagnetic quantum critical metal in two spatial dimensions. Being a strongly coupled theory, it can still be solved reliably in the low-energy limit as quantum fluctuations are organized by a new control parameter that emerges dynamically. We predict the exact critical exponents that govern the universal scaling of physical observables at low temperatures.


I. INTRODUCTION
One of the cornerstones of condensed matter physics is Landau Fermi liquid theory, according to which quantum many-body states of interacting electrons are described by largely independent quasiparticles in metals [1]. In Fermi liquids, the spectral weight of an electron is sharply peaked at a well defined energy due to the quasiparticles with long lifetimes. On the other hand, exotic metallic states beyond the quasiparticle paradigm can arise near quantum critical points, where quantum fluctuations of collective modes driven by the uncertainty principle preempt the existence of well defined single-particle excitations [2][3][4][5]. In the absence of quasiparticles, many-body states become qualitatively different from a direct product of single particle wavefunctions. Due to strong fluctuations near the Fermi surface, the delta function peak of the electron spectral function is smeared out, leaving a weaker singularity behind. The resulting non-Fermi liquids exhibit unconventional power-law dependences of physical observables on temperature and probe energy [6].
Antiferromagnetic (AF) quantum phase transitions arise in a wide range of layered compounds [22][23][24]. Despite the recent progress made in field theoretic and numerical approaches to the AF quantum critical metal [25-33], a full understanding of the non-Fermi liquid realized at the critical point has been elusive so far. In two dimensions, strong quantum fluctuations and abundant lowenergy particle-hole excitations render perturbative theories inapplicable. What is needed is a non-perturbative approach which takes into account strong quantum fluctuations in a controlled way [20].
In this article, we present a non-perturbative field theoretic study of the AF quantum critical metal in two dimensions. Although the theory becomes strongly coupled at low energies, we demonstrate that a small parameter which differs from the conventional coupling emerges dynamically. This allows us to solve the strongly interacting theory reliably. We predict the exact critical exponents that govern the scaling of dynamical and thermodynamic observables.

II. LOW-ENERGY THEORY AND INTERACTION-DRIVEN SCALING
The relevant low-energy degrees of freedom at the metallic AF critical point are the AF collective mode and electrons near the hot spots, a set of points on the Fermi surface connected by the AF wavevector. In the presence of the four-fold rotational symmetry and the reflection symmetry in two spatial dimensions, there are generically eight hot spots, as is shown in Figure 1. Following Ref.
The component of the Fermi velocity parallel to Q AF at each hot spot is set to have unit magnitude.
v measures the component of the Fermi velocity perpendicular to Q AF . Φ(q) = 3 a=1 φ a (q)τ a is a 2 × 2 matrix boson field that represents the fluctuating AF order parameter, where the τ a 's are the generators of the SU(2) spin. c 0 is the velocity of the AF collective mode. g is the coupling between the collective mode and the electrons near the hot spots.n represents the hot spot connected to n via Q AF :1 = 3,2 = 4,3 = 1,4 = 2. u is the quartic coupling between the collective modes.
In two dimensions, the conventional perturbative expansion becomes unreliable as the couplings grow at low energies. Since the interaction plays a dominant role, we need to include the interaction up front rather than treating it as a perturbation to the kinetic energy. Therefore, we start with an interaction-driven scaling [20] in which the fermion-boson coupling is deemed marginal.
Under such a scaling, one cannot keep all the kinetic terms as marginal operators. Here we choose a scaling that keeps the fermion kinetic term marginal at the expense of making the boson kinetic term irrelevant. This choice will be justified through explicit calculations. It reflects the fact that the dynamics of the boson is dominated by particle-hole excitations near the Fermi surface in the low-energy limit, unless the number of bosons per fermion is infinite [34]. The marginality of the fermion kinetic term and the fermion-boson coupling uniquely fixes the dimensions of momentum and the fields under the interaction-driven tree-level scaling, Here, the fermion-boson coupling is set to be proportional to √ v by rescaling the boson field. The Yukawa coupling is replaced with √ v because the interaction is screened such that g 2 becomes O(v) in the low-energy limit [30]. Although g and v can be independently tuned in the microscopic theory, they rapidly flow to a universal line defined by g 2 ∼ v at low energies [35]. Eq. (3) should be understood as the minimal theory that captures the universal physics at low energies, where the dynamics of the collective mode is dominated by particle-hole excitations rather than the bare kinetic term, and v is the only dimensionless parameter. In the small v limit, g also vanishes because a nested Fermi surface provides a large phase space for low-energy particle-hole excitations with momentum Q AF that screen the interaction. Even when g, v are small, this is a strongly interacting theory because g 2 /v ∼ 1 is the expansion parameter in the conventional perturbative series. With g 2 /v ∼ 1, the leading boson kinetic term which is generated from particle-hole excitations is O(1), as will be seen later. The triangle represents the fully dressed vertex.

III. SELF-CONSISTENT SOLUTION
Naively the theory is singular due to the absence of a boson kinetic term. However, particle-hole excitations generate a self-energy which provides non-trivial dynamics for the collective mode.
The Schwinger-Dyson equation for the boson propagator (shown in Figure 2) reads Here D(k), G(k) and Γ(k, q) represent the fully dressed propagators of the boson and the fermion, and the vertex function, respectively. m CT is a mass counter term that is added to tune the renormalized mass to zero. The trace in Eq. (4) is over the spinor indices. It is difficult to solve the full self-consistent equation because G(k) and Γ(k, q) depend on the unknown D(q). One may use v as a small parameter to solve the equation. The one-loop analysis shows that v flows to zero due to emergent nesting of the Fermi surface near the hot spots [25,26,28,31]. This has been also confirmed in the ǫ expansion based on the dimensional regularization scheme [30,35]. Of course, the perturbative result valid close to three dimensions does not necessarily extend to two dimensions.
Nonetheless, we show that this is indeed the case. Here we proceed with the following steps: 1. we solve the Schwinger-Dyson equation for the boson propagator in the small v limit, 2. we show that v flows to zero at low energies by using the boson propagator obtained under the assumption of v ≪ 1.
We emphasize that the expansion in v is different from the conventional perturbative expansion in coupling. Rather it involves a non-perturbative summation over an infinite series of diagrams as will be shown in the following.
We discuss step 1) first. In the small v limit, the solution to the Schwinger-Dyson equation is where the 'velocity' of the strongly damped collective mode is given by We begin by estimating the magnitude of general diagrams, assuming that the fully dressed boson propagator is given by Eq. (5) with Eq. (6) in the small v limit. In general, the integrations over loop momenta diverge in the small v limit as fermions and bosons lose their dispersion in some directions. In each fermion loop, the component of the internal momentum tangential to the Fermi surface is unbounded in the small v limit due to nesting. For a small but nonzero v, the divergence is cut off at a scale proportional to 1/v, and each fermion loop contributes a factor of 1/v. Each of the remaining loops necessarily has at least one boson propagator. For those loops, the momentum along the Fermi surface is cut off by the energy of the boson which provides a lower cut-off momentum proportional to 1/c for c ≫ v. Therefore, the magnitude of a general L-loop diagram with V vertices, L f fermion loops and E external legs is at most where V = 2L + E − 2 is used. Higher-loop diagrams are systematically suppressed with increas-  For v ≪ c, the leading order contribution for the boson self-energy (E = 2) is generated from in v. However, this is not enough because the one-loop diagram gives D(q) −1 = |q 0 |, which is independent of spatial momentum. One has to include the next order diagram (Figure 3(b)) which generates a dispersion. Therefore, Eq. (4) is reduced to Here m ′ CT is a two-loop mass counter term. We can use the free fermion propagator G So far, we have assumed that v is small to obtain the self-consistent dynamics of the AF collective mode. Now we turn to step 2) and show that v indeed flows to zero in the low-energy limit. According to Eq. (7), the leading quantum corrections to the local action in Eq. (3) are the one-loop diagrams for the fermion self-energy and the vertex function. However, the momentumdependent one-loop fermion self-energy happens to be smaller than what is expected from Eq. (7) by an additional power of c ∼ √ v. This is because the dependence on the external momentum is suppressed in the small c limit for the one-loop self-energy. As a result, we include the fermion self-energy up to two loops in order to capture all quantum corrections to the leading order in v. All other higher-loop diagrams are negligible in the small v limit. The self-energy and vertex correction are logarithmically divergent in a UV cut-off. Counter terms are added such that the renormalized quantum effective action becomes independent of the UV cut-off. The full details on the computation of the counter terms and the beta function can be found in Appendix C. The bare action that includes the counter terms is obtained to be where ε B Here Λ is a UV cut-off above which non-linear terms in the fermionic dispersion become important. µ is the scale at which the physical propagators and vertex function are expressed in terms of v through the renormal- . By requiring that the bare quantities are independent of µ, we obtain the beta function β v ≡ dv d log µ , which dictates the dependence of the renormalized velocity on the scale, As a function of the energy scale µ, v is renormalized according to If v is initially small, Eq. (11) is reliable. It predicts that v becomes even smaller and flows to zero in the small µ limit. The way v flows to zero in the low-energy limit does not depend on the initial value of v. This completes the cycle of self-consistency. Eq. (5) obtained in the small v limit becomes asymptotically exact in the low-energy limit within a nonzero basin of attraction in the space of v whose fixed point is v = 0. The dynamical critical exponent and the anomalous dimensions are given by to the leading order in v. Here z sets the dimension of frequency relative to momentum. η φ , η ψ are the corrections to the interaction-driven tree-level scaling dimensions of the boson and fermion, respectively. The critical exponents are controlled by w ≡ v/c(v), which flows to zero as in the low-energy limit. This confirms that the scaling dimensions in Eq. (2) become asymptotically exact in the low-energy limit. This is compatible with the fact that an inclusion of higher-loop corrections in the ǫ-expansion reproduces z = 1, irrespective of

IV. PHYSICAL OBSERVABLES
Although z − 1, η ψ and η φ vanish in the low-energy limit, the sub-logarithmic decay of w with energy introduces corrections to the correlation functions at intermediate energy scales, which are weaker than power-law but stronger than logarithmic corrections [38]. The retarded Green's function for the hot spot 1+ takes the form, in the small ω limit with the ratio k ω Fz(ω) fixed. Here ω is the real frequency. F ψ (ω) and F z (ω) are functions which capture the contributions from η ψ and z at intermediate energy scales. In the small ω limit, they are given by F ψ and F z only contribute as sub-leading corrections instead of modifying the exponents. However, they are still parts of the universal data that characterizes the critical point [28]. The additional logarithmic suppression in the dependence of k x is due to v which flows to zero in the low-energy limit. The local shape of the Fermi surface is deformed as k y ∼ kx log 1/kx log log 1/kx . The scaling form of the Green's function at different hot spots can be obtained by applying a sequence of 90 degree rotations and a space inversion to Eq. (14). The spectral function at the hot spots exhibits a power-law decay with the super-logarithmic correction as a function of frequency, The retarded spin-spin correlation function is given by in the small ω limit with fixed q ω Fz(ω) . F φ (ω) is another universal function that describes the super-logarithmic correction of η φ , in the small ω limit. The factor of log 1 ω −1/2 in the momentum-dependent term is due to the boson velocity which flows to zero in the low-energy limit. Due to the strong Landau damping, the spin fluctuation is highly incoherent. It will be of great interest to test the scaling forms in Eqs. (14) and (16) from angle resolved photoemission spectroscopy and neutron scattering, respectively. Now we turn to thermodynamic properties. The total free energy density can be written as where Π, Σ are the self-energies of the boson and fermion respectively, and Φ 2 includes the two particle irreducible diagrams [39]. Here, the traces sum over three momenta and flavors. To the leading order in v, The dominant fermionic contribution comes from electrons away from the hot spots, f F ∼ k F T 2 , where k F is the size of the Fermi surface. Naively, the bosonic contribution is expected to obey hyperscaling, because low-energy excitations are confined near the ordering vector. However, the free energy of the mode with momentum p is suppressed only algebraically as T 2 c(|px|+|py|) at large momenta, in contrast to the exponential suppression for the free boson. The slow decay is due to the incoherent nature of the damped AF spin fluctuations, which have a significant spectral weight at low energies even at large momenta. As a result, In the presence of the irrelevant local kinetic term, , we obtain f B ∼ΛT 2 F z (T ) in the low temperature limit. Remarkably, the bosonic contribution violates the hyperscaling, and it is larger than the fermionic contribution at low temperatures. In this case, the power-law violation of the hyperscaling is a consequence of the z = 1 scaling rather than the fact that v, c flow to zero [40]. The free energy gives rise to the specific heat which exhibits the T -linear behavior with the super-logarithmic correction, It is noted the deviation from the T -linear behavior is stronger than a simple logarithmic correction because F z (T ) includes all powers of log 1 T . If the system is tuned away from the critical point, the boson acquires a mass term, , where λ is a tuning parameter. Due to the suppression of higher-loop diagrams, the scaling dimension of Φ 2 is −4 in momentum space. This implies that ν = 1 in the low-energy limit, which is different from the mean-field exponent. The power-law scaling of the correlation length ξ with λ is modified by a super-logarithmic correction, where F ξ (δλ) is a universal function which embodies both the anomalous dimension of the boson and the vertex correction for the mass insertion. The former dominates close to the critical point, and F ξ (δλ) is the same as F φ (δλ) to the leading order in small δλ. The derivation of the scaling forms of the physical observables is available in Appendix D.
The scaling forms of the physical observables discussed above are valid in the low energy limit.
At high energies, there will be crossovers to different behaviors. The first crossover is set by the scale below which the dynamics of the collective mode is dominated by particle-hole excitations, and therefore Eqs. (16) and (18) hold. It is determined by the competition between Eq. (5) and the irrelevant local kinetic term for the collective mode in Eq. (1).
, the terms linear in frequency and momentum dominate, whereΛ is an energy scale associated with the irrelevant kinetic term. The details on the crossover are described in Appendix B. In the small v limit with c 0 ∼ 1, this crossover scale for the boson goes as E * b ∼ c 2Λ . The second crossover scale, denoted as E * f , is the one below which the behavior of the fermions at the hot spots deviates from the Fermi liquid one. For a small but non-zero v, the leading order self-energy correction to the fermion propagator is 3 Since v flows to zero only logarithmically, the flow of v can be ignored for the At sufficiently low temperatures The first logarithm is from the usual BCS mechanism. The second logarithm is from the gapless spin fluctuations, where E * b ∼ c 2Λ is the energy cut-off for the spin fluctuations in the small c limit as is shown in Appendix B. This gives There is a hierarchy among the energy scales, E * f ≪ T c ≪ E * b in the small v limit. This suggests that the system undergoes a superconducting transition before the fermions at the hot spots lose coherence. On the one hand, this is similar to the nematic quantum critical point in two dimensions where the system is prone to develop a superconducting instability before the coherence of quasiparticles breaks down [48,49]. On the other hand, even without superconductivity, the fermions are only weakly perturbed by the spin fluctuations in the present case. It is the collective mode that is heavily dressed by quantum effects. For the collective mode, there is a large window between T c and E * b within which the universal scaling given by Eq. (5) is obeyed. The size of the energy window for the critical scaling is non-universal due to the slow flow of v, and it depends on the bare value of v. Our prediction is that there is a better chance to observe the z = 1 critical scaling above T c , and the enhancement of T c by AF spin fluctuations is rather minimal [50] in materials whose bare Fermi surfaces are closer to perfect nesting near the hot spots.

V. SUMMARY AND DISCUSSION
In summary, we solve the low-energy field theory that describes the antiferromagnetic quantum critical metal in two spatial dimensions. We predict the exact critical exponents which govern the universal scaling of physical observables at low temperatures. Finally, we comment on earlier theoretical approaches, and provide a comparison with experiments.  16) is that the width of the incoherent peak scales linearly with energy upto a super-logarithmic correction in the low energy limit. However, it is hard to make a quantitative comparison due to the limited momentum resolution in the experiment. In Nd 2−x Ce x CuO 4±δ (NCCO), inelastic neutron scattering suggests that the magnetic correlation length ξ scales inversely with temperature near the critical doping[53]. Furthermore, ξ measured at the pseudogap temperature diverges as (x − x c ) −1 . If interpreted in terms of the clean AF quantum critical scenario, which may be questionable due to disorder, this is consistent with z = 1 and ν = 1. Angle resolved photoemission spectroscopy (ARPES) for NCCO shows a reduced quasiparticle weight at the hot spots [54,55]. This is in qualitative agreement with the prediction of Eq. (14), which implies that the quasiparticle weight vanishes at the hot spots, as compared to the region away from the hot spots where quasiparticles are well defined. Although the spectroscopic measurements are in qualitative agreement with the theoretical predictions, we believe that more experiments are needed to make quantitative comparisons. On the theoretical side, transport properties need to be better understood, for which electrons away from hot spots In this section, we prove the upper bound in Eq. (7), assuming that the fully dressed boson propagator is given by Eqs. (5) and (6) 7) holds for an example to illustrate the idea that is used for a general proof in the following subsection.

Example
The diagram in Figure 1(a) is a fermion self-energy with one fermion loop and three other loops, which we call 'mixed loops'. For simplicity, we set the external momentum to zero. This does not affect the enhancement factors of 1/c and 1/v which originate from large internal momenta. We label the loop momenta as shown in Figure 1 (a) The seven internal fermion propagators whose energies are denoted as E l with 1 ≤ l ≤ 7.

1(a) is written as
where p r is the set of internal three-momenta, and E i represents the energy of the fermion in the i-th fermion propagator as denoted in Figure 1(c), Since frequency integrations are not affected by v and c, we focus on the spatial components of momenta from now on. Our aim is to change the variables for the internal momenta so that the enhancement factors of 1/v and 1/c become manifest. As our first three new variables we choose The last five variables are chosen to be p ′ l+3 ≡ E l with 1 ≤ l ≤ 5. The transformation between the old variables, written as {vp i,x , p i,y }, and the new variables is given by whereÃ andṼ are written as and I 3 is the 3 × 3 identity matrix. For non-zero v, c, the change of variables is non-degenerate, and the Jacobian of the transformation is (2c 3 v) −1 . We show in the following section that such a non-degenerate choice is always possible for general diagrams. An easy mnemonic is that each fermion loop contributes a factor of 1/v because of nesting in the small v limit, while each mixed loop contributes a factor of 1/c because of the vanishing boson velocity.
In the new coordinates, the momentum integration in Eq. (A1) becomes whereR[p ′ ] includes the propagators that are not explicitly shown. Now, we can safely take the small c limit inside the integrand, because every momentum component has at least one propagator which guarantees that the integrand decays at least as 1/p ′ j in the large momentum limit. Therefore, the integrations are UV convergent up to potential logarithmic divergences. To leading order in small v, the diagram scales as up to potential logarithmic corrections.

General upper bound
Here we provide a general proof for the upper bound, by generalizing the example discussed in the previous section. We consider a general L-loop diagram that includes fermions from patches 1, 3, With this choice of loops, Eq. (A5) is written as Here, frequency is suppressed, and IR divergences in the integrations over spatial momenta are understood to be cut off by frequencies. Our focus is on the UV divergence that arises in the spatial momentum integrations in the limit of small v and c.
for diagrams with E > 0. We express p ′ j ≡ cp j,x and E l (p) in terms of vp r,x , p r,y , . .
Here I a is the a × a identity matrix. require that E l = 0 for all l. Therefore, there cannot be a non-trivial P that satisfies V P = 0. This implies that the column vectors in V must be linearly independent. Since whereÃ is a (2L − L m ) × L m matrix made of the collection of the l k -th rows of A with k = 1, 2, .., (2L − L m ). The Jacobian of the transformation is given by | detṼ| is a constant independent of v and c, which is nonzero becauseṼ is invertible.
In the new variables, Eq. (A6) becomes Every component of the loop momenta has at least one propagator which guarantees that the integrand decays at least as 1/p ′ l in the large momentum limit.R[p ′ ] is the product of all remaining propagators. Therefore, the integrations over the new variables are convergent up to potentially logarithmic divergences. Using L = 1 2 (V + 2 − E), one can see that a general diagram is bounded by The one-loop quantum effective action of the boson generated from Fig. 3(a) is written as where and the bare fermion propagator is G and dk ≡ d 3 k (2π) 3 . The integration of the spatial momentum gives Π 1L (q) = − 1 2 dk 0 (k 0 +q 0 )k 0 |k 0 +q 0 ||k 0 | . The k 0 integration generates a linearly divergent mass renormalization which is removed by a counter term, and a finite self-energy, Since the one-loop self-energy depends only on frequency, we have to include higher-loop diagrams to generate a momentum-dependent quantum effective action, even though they are suppressed by powers of v compared to the one-loop self-energy. According to Eq. (7), the next leading diagrams are the ones with L − L f = 1. Among the diagrams with L − L f = 1, the only one that contributes to the momentum-dependent boson self-energy is shown in Figure 3(b). In particular, other two-loop diagrams that include fermion self-energy insertions do not contribute.
Since the two-loop diagram itself depends on the unknown dressed boson propagator, we need to solve the self-consistent equation for D(q) in Eq. (8). Here, we first assume that the solution takes the form of Eq. (5) with v ≪ c ≪ 1 to compute the two-loop contribution, and show that the resulting boson propagator agrees with the assumed one. The two-loop self-energy reads Here c.c. denotes the complex conjugate. Straightforward integrations over k and k 0 give Since the frequency-dependent self-energy is already generated from the lower order one-loop graph in Figure 3(a), we focus on the momentum-dependent part. This allows us to set the external frequency to zero to rewrite Eq. (B5) as After subtracting the linearly divergent mass renormalization, where F 1L(n) (p 0 , p, q; v) = (p 2 0 + ε 2 n ( p))(p 2 0 + ε 2 n ( p))(ip 0 − εn( p + q))(ip 0 − ε n ( p − q)) − (p 2 0 + ε 2 n ( p + q))(p 2 0 + ε 2 n ( p − q))(ip 0 − εn( p))(ip 0 − ε n ( p)).

(B8)
Now we consider the contribution of each hot spot separately. For n = 1, the dependence on q x is suppressed by v compared to the q y -dependent self-energy. Therefore, we set q x = 0 for small v. Furthermore, the p y dependence in D(p) can be safely dropped in the small c limit because ε 1 ( p) and ε 3 ( p) suppress the contributions from large p y . Rescaling the momentum as (p 0 , p x , p y ) → |q y |(p 0 , p x /c, p y ) followed by the integration over p y , we obtain the contribution from the hot spot n = 1, |q y | dp 0 dp where w ≡ v/c. In the integrand, we can not set w = 0 because the integration over p x is logarithmically divergent in the small w limit, Finally, the integration over p 0 gives In the small w limit, the first term dominates. Hot spot 3 generates the same term, and the contribution from hot spots 2, 4 is obtained by replacing q y with q x . Summing over contributions from all the hot spots, we obtain The two-loop diagram indeed reproduces the assumed form of the self-energy which is proportional to |q x | + |q y | to the leading order in v. The full Schwinger-Dyson equation now boils down to a self-consistent equation for the boson velocity, c is solved in terms of v as This is consistent with the assumption that v ≪ c ≪ 1 in the small v limit. energy scaling dynamical critical exponent  The full propagator of the boson which includes the bare kinetic term in Eq. (1) is given by whereΛ is a UV scale associated with the coupling. Depending on the ratio between c and c 0 , which is determined by microscopic details, one can have different sets of crossovers.
For c 0 > c, one has a series of crossovers from the Gaussian scaling with z = 1 at high energies, to the scaling with z = 2 at intermediate energies and to the non-Fermi liquid scaling with z = 1 at low energies. In the low energy limit, the system eventually becomes superconducting. For c 0 < c, on the other hand, the z = 2 scaling is replaced with a scaling with z = 1 2 at intermediate energies. This is summarized in Tables B1 and B2. According to Eq. (7), the leading order fermion self-energy is generated from Fig. C3 in the small v limit. The one-loop fermion self-energy for patch n is given by where the dressed boson propagator is D(p) = 1 |p 0 |+c(v)(|px|+|py|) . We first compute Σ 1L(n) (k) for n = 1. The quantum correction is logarithmically divergent, and a UV cut-off Λ is imposed on p y , which is the momentum perpendicular to the Fermi surface for n = 1 in the small v limit. However, the logarithmically divergent term is independent of how UV cut-off is implemented. To extract the frequency-dependent self-energy, we set k = 0 and rescale (p 0 , p x , p y ) → |k 0 |(p 0 , p x /c, p y ) to rewrite Σ 1L(1) (k 0 , 0) = iγ 0 k 0 3πv 2c dp where w = v c . Under this rescaling, the UV cut-off for p y is also rescaled to Λ 0 = Λ/|k 0 |. The p 0 integration gives (C3) The logarithmically divergent contribution is obtained to be in the small v limit. The self-energy for other patches is obtained from a series of 90-degree rotations, and the frequency-dependent part is identical for all patches. In order to remove the cut-off dependence in the quantum effective action, we add the counter term, where µ is the scale at which the quantum effective action is defined in terms of the renormalized velocity v. The counter term guarantees that the renormalized propagator at the scale µ is expressed solely in terms of v in the Λ/µ → ∞ limit.

Momentum-dependent fermion self-energy
To compute the momentum-dependent fermion self-energy, we start with Eq. (C1) for n = 1 The integration over p 0 results in Σ 1L(1) (0, k) = Σ 1L(1) ( k) term 1 We first compute the first term. After performing the p x integration, we rescale p y → |ε 3 ( k)|p y to where Λ 3 = Λ |ε 3 ( k)| . The remaining p y integration gives to the leading order in v up to terms that are finite in the large Λ limit.
The second term can be computed similarly in the small v limit, up to UV-finite terms. It is noted that the second term is dominant for small v.  Figure C3.
According to Eq. (7), the upper bound for the one-loop fermion self-energy is v/c. However, Eq. (C12) is strictly smaller than the upper bound. The extra suppression by c arises due to the fact that the external momentum in Figure C3 can be directed to flow only through the boson propagator, and the diagram becomes independent of the external momentum in the small c limit.
Since this suppression does not happen for higher-loop diagrams in general, the one-loop diagram becomes comparable to some two-loop diagrams with L − L f = 2. Therefore, we have to include the two-loop diagrams for the self-energy in order to capture all leading order corrections. The rainbow diagram in Figure 4(a) is smaller for the same reason as the one-loop diagram. Three and higher-loop diagrams remain negligible, and only Figure 4(b) contributes to the leading order. The two-loop self-energy for patch n is given by Σ 2L(n) (k 0 , k) = 3π 2 v 2 4 dpdq [γ 1 Gn(k + q)γ 1 G n (k + q + p)γ 1 Gn(k + p)γ 1 ] D(q)D(p).