PHYSICAL REVIEW RESEARCH 2, 023310 (2020)

Landau poles in condensed matter systems

Shao-Kai Jian ,1 Edwin Barnes ,2 and Sankar Das Sarma1 1Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742, USA 2Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA

(Received 12 March 2020; revised manuscript received 7 May 2020; accepted 21 May 2020; published 9 June 2020)

The existence or not of Landau poles is one of the oldest open questions in nonasymptotic quantum field theories. We investigate the Landau pole issue in two condensed matter systems whose long-wavelength physics is described by appropriate quantum field theories: the critical quantum magnet and Dirac fermions in graphene with long-range Coulomb interactions. The critical quantum magnet provides a classic example of a quantum phase transition, and it is well described by the φ4 theory. We find that the irrelevant but symmetry-allowed couplings, such as the φ6 potential, can significantly change the fate of the Landau pole in the emergent φ4 theory. We obtain the coupled β functions of a φ4 + φ6 potential at both small and large orders. Already from the one-loop calculation, the Landau pole is replaced by an ultraviolet fixed point. A Lipatov analysis at large orders reveals that the inclusion of a φ6 term also has important repercussions for the high-order expansion of the β functions. We also investigate the role of the Landau pole in a very different system: Dirac fermions in 2 + 1 dimensions with long-range Coulomb interactions, e.g., graphene. Both the weak-coupling perturbation theory up to two loops and a low-order large-N calculation show the absence of a Landau pole. Furthermore, we calculate the asymptotic expansion coefficients of the β function. We find that the asymptotic coefficient is bounded by that of a pure bosonic φ4 theory, and consequently graphene is free from Landau poles if the pure φ4 theory does not manifest a Landau pole. We briefly discuss possible experiments that could potentially probe the existence of a Landau pole in these systems. Studying Landau poles in suitable condensed matter systems is of considerable fundamental importance since the relevant Landau pole energy scales in particle physics, whether it is quantum electrodynamics or Higgs physics, are completely unattainable.

DOI: 10.1103/PhysRevResearch.2.023310

I. BACKGROUND experiment agreeing certainly up to well over 100 significant digits, although neither experiment nor theory is likely to get Quantum electrodynamics (QED) is the most successful to such a high precision in the foreseeable future. Because theory in physics. The most recent QED calculation up to the QED perturbation theory is asymptotic, it will eventually tenth order in the fine structure coupling constant (i.e., the break down, with the perturbation series eventually diverging fifth order in the QED perturbation theory) involves the ac- at some very high order (137), but this is not a serious curate determination of 389 different high-dimensional in- concern at this stage. The infinite-order perturbative QED tegrals contributed by 6354 Feynman vertex diagrams, with result is thus bound to be incorrect since it gives a divergent a resultant electron anomalous magnetic moment agreeing answer. quantitatively with experiments up to ten significant digits [1]. In addition to the asymptotic nature of the QED pertur- Obviously, we have come a long way (although it took bation theory, there is a second fundamental “problem” with 70 years for this progress) from Schwinger’s ground-breaking QED, first emphasized by Landau [3], and often referred to analytical work of 1948 calculating just the first-order QED as the Landau pole (it also is sometimes called the “Landau correction, which obtained the electron anomalous magnetic ghost” or “Moscow zero”) [4]. The term Landau pole refers moment correct to two significant digits [2]. This amazing to the divergence of a running (or renormalized) coupling agreement between theory and experiment is the most im- constant at a finite energy in a field theory. Landau poles pressive success of quantum mechanics, and the common happen only in field theories that are not asymptotically free, belief is that this astounding success will continue up to where the running coupling increases with increasing energy, many orders in the QED perturbation theory, with theory and in contrast to asymptotically free field theories where the coupling constant decreases with increasing energy. If the Landau pole is a true feature of QED, then the only way to obtain a reasonable theory would be to set the bare charge to Published by the American Physical Society under the terms of the zero, resulting in a theory that is completely trivial (or non- Creative Commons Attribution 4.0 International license. Further interacting). Landau poles have been discussed extensively distribution of this work must maintain attribution to the author(s) in the context of both QED and scalar field theories, such and the published article’s title, journal citation, and DOI. as the φ4 theory, which is relevant for Higgs bosons in the

2643-1564/2020/2(2)/023310(17) 023310-1 Published by the American Physical Society JIAN, BARNES, AND DAS SARMA PHYSICAL REVIEW RESEARCH 2, 023310 (2020) standard model. The problem is, however, that the existence asymptotically free. Another equally important objective of of a Landau pole is theoretically established only within the our work is to motivate experimental work in condensed mat- leading-order perturbative approximation (extended to very ter systems to directly probe the existence or not of Landau high energies) and, therefore, whether the Landau pole is real poles in these two systems, which are described by continuum or an artifact of the perturbation theory is unknown. In the field theories containing Landau poles in the leading-order context of QED, direct numerical simulations have been used perturbative analysis. The question of the existence or not to explore the existence of Landau poles, but whether the of Landau poles is of sufficient fundamental significance that poles have any physical relevance in the sense of imposing anything we can learn from condensed matter systems about triviality remains unclear [5–7]. Unfortunately, approaches the possible presence or absence of Landau poles would be based on lattice simulations are hindered by chiral symmetry valuable for future progress in the subject. breaking, making the parameter regime where a Landau pole occurs inaccessible [5]. Numerical methods have also been II. INTRODUCTION applied to φ4 theory, with the conclusion that the continuum theory is most likely trivial. This finding has in turn been The concept of Landau poles is a long-standing issue in used to impose bounds on the Higgs mass [8,9]. However, quantum field theory that raises questions about fundamental the triviality question has not been fully resolved here either, aspects of the RG, in particular the asymptotic behavior of and a stronger bound on the mass comes from unitarity [10], renormalized couplings [3,12,13]. In a quantum field theory obscuring the role of the Landau pole. It is also unclear that is not asymptotically free, consider the β function that to what extent these findings carry over to effective scalar governs the running of a dimensionless coupling constant λ, theories, where higher-order couplings allowed by symmetry dλ can affect renormalization group (RG) flows at high energies. = β(λ), (1) d log μ It is possible that a fully nonperturbative, strong-coupling theory would be necessary to eventually settle the question where μ is the energy scale of interest. One can integrate over since weak-coupling perturbation theories simply may not both sides to get be applicable in predicting the physics of a divergent renor- ∞ dλ μ∞ malized coupling. The Landau pole problem is intrinsically = d μ. β λ log (2) tied to the asymptotic behavior of the QED perturbation λ(μ1 ) ( ) μ1 expansion and to the nonperturbative effects of instantons. μ∞ is the energy scale at which the running coupling becomes Although there has been recent progress in developing a more infinite. Suppose the β function has the asymptotic behavior rigorous treatment of nonperturbative effects like instantons β(λ) ∝ λa, λ 1, where a is a constant independent of λ; using resurgent trans-series [11], the question of the existence then or not of the Landau pole in QED remains as open today as it ∞ dλ ∞ ∞,  was in the early 1950s when Landau first proposed it. ∝ λ−a λ = a 1 d μ∞ < ∞, > . β λ log μ a 1 One thing is, however, clear. Even if the Landau pole exists λ(μ1 ) ( ) λ(μ1 ) 1 / in QED and or quartic scalar field theories, there is no hope (3) for its experimental manifestation in particle physics because the energy scale for the Landau pole is unphysically huge In the first case, a  1, there is no Landau pole. The running (e.g., way above the Planck scale in QED and the Higgs mass coupling reaches infinity only at infinite energy scales. In the in the φ4 theory). Quite apart from the fact that such large second case, a > 1, however, the coupling diverges at a finite energy scales are experimentally unattainable (not only now, energy scale μ∞. This is a Landau pole. Clearly, the existence perhaps ever), new physics, outside the scope of QED, comes of a Landau pole depends on the asymptotic form of the β in at high energy scales, and the predictions of QED for the function at λ 1. Landau pole become academic since QED itself (quite apart Because Landau poles were first discovered theoretically from a perturbation theoretic analysis of QED leading to the in quantum field theories relevant for high-energy physics, Landau pole) is no longer a correct description of nature at such as QED, research into this issue has remained largely such high energies. within the high-energy physics community [5–7,14–17]. The This is the context of the current theoretical work, where simplest solution to the Landau pole problem is the idea of we investigate the condensed matter analogs of the Landau quantum triviality [6,14–16], in which the pole is avoided by pole in the theories of critical quantum magnets and the setting the coupling to zero at all scales, yielding a trivial, physics of graphene. It is well known that the φ4 scalar noninteracting theory. This is of course unsatisfactory if one field theory describes the long-wavelength critical behavior is interested in the effect of interactions (which are clearly of quantum magnets, and the two-dimensional massless Dirac present in QED experiments), so a more phenomenological theory describes the long-wavelength behavior of graphene. resolution that is often adopted is to argue that the theory Thus, graphene is an example (with suitable modifications) of is incomplete and gets replaced by another theory at high QED, whereas quantum magnets are examples (with suitable energies before the Landau pole is reached. For instance, QED modifications) of the φ4 scalar field theories. Our goal is to is usually believed to be just one part of a more fundamental study the presence or absence of Landau poles in these two electroweak theory. Moreover, the Landau pole in QED can be concrete condensed matter physics examples to motivate fur- estimated to occur at an energy scale on the order of 10286 eV, ther theoretical work that could shed light on the fundamental which is far beyond the Planck scale 1028 eV, suggesting issue of triviality in the quantum field theories which are not that it is a purely academic issue. However, the reliability of

023310-2 LANDAU POLES IN CONDENSED MATTER SYSTEMS PHYSICAL REVIEW RESEARCH 2, 023310 (2020) such estimates is questionable given that the Landau pole is much higher than the inverse lattice spacing energy (within intrinsically tied to the large-coupling regime, whereas most an order of magnitude) so that there is some reason to believe analyses rely on weak-coupling perturbation theory carried that the effective field theory does not completely break down out to first or second order. Many attempts have been made at the Landau pole energy. In such a situation, it is sensible to go beyond small-order perturbation theory starting from to consider condensed matter experiments to investigate the Lipatov’s method [18,19] for calculating large orders in an presence or absence of Landau poles. If the pole happens to be asymptotic series. In fact, a resummation procedure based on at an energy much larger than the natural ultraviolet cutoff for such large-order expansions suggests that the Landau pole the lattice in the system, there is no hope for experimentally does not exist at all in the φ4 theory [20,21]. But the Landau studying the Landau pole in the corresponding condensed pole question is by no means settled since the resummation matter system since the effective field theory is unlikely to technique is essentially a Borel interpolation, and the exact apply at that high-energy scale. strong-coupling theory remains elusive, so one cannot be sure Here, we consider two condensed matter systems: the that the Landau pole does not exist. In particular, numerical critical quantum magnet and Dirac fermions in 2 + 1di- simulations seem to indicate its existence [6]. mensions with long-range Coulomb interactions (graphene). In addition to having wide application in particle physics, Both systems are believed to be well described by effective continuum field theories and the renormalization group are continuum field theories. The critical quantum magnet is also important for condensed matter systems, especially in described by a bosonic φ4 theory as dictated by symmetry. the context of phase transitions, where the universal physics In four dimensions, the φ4 theory has a coupling that grows is controlled by long-wavelength fluctuations at a critical logarithmically with energy scale, eventually leading to a point [22,23]. (In fact, Wilson developed his RG theory moti- Landau pole as determined from perturbation theory [30]. vated by condensed matter considerations in critical phenom- However, this theory is an effective field theory, meaning ena, and the first problem he solved using his momentum-shell that all terms allowed by symmetry, e.g., all even-order terms RG theory is the Kondo problem, a celebrated condensed φn, where n ∈ 2Z, are in principle present in the action. matter problem involving singular spin-flip scattering of elec- Although the φ4 term is the most relevant one deep in the trons in metals from quenched magnetic impurities [24].) infrared, the n > 4 terms can become important at the higher Thus, it is natural to consider the Landau pole problem in energy scales where the Landau pole potentially arises. This condensed matter systems. Unlike most cases in high-energy motivates us to investigate to what extent Landau poles are physics, here the ultraviolet completion is well understood affected by infrared-irrelevant terms in the φ4 effective field as it comes from the lattice, which brings a natural energy theory. To answer this question, we study the β function of scale cutoff set by the inverse lattice constant. The question a φ4 + φ6 potential and find that the Landau pole can be then becomes whether a Landau pole arises above or below strongly affected by the presence of the φ6 potential at both this scale. This question is particularly important in systems small and large orders of the asymptotic expansion. where the coupling constant can be large, as is the case in Dirac fermions in graphene constitute a second important graphene, where the fine structure constant away from the case study because the Fermi velocity approaches zero in Dirac point is α ∼ 1[25], and even more so in synthetic the ultraviolet, enhancing the role of Coulomb interactions at twisted bilayer graphene, where α ∼ 10 [26–29]. The inter- high energies and potentially inducing a Landau pole [25]. actions in these systems are substantially stronger than in The theory is infrared stable because the Fermi velocity QED, where α ≈ 1/137, suggesting that if there is a Landau increases as the energy scale approaches zero, leading to pole, then its energy scale could occur below the cutoff, a decreasing running coupling (which is cut off at some potentially leading to consequences that are experimentally exponentially low energy scale when retardation effects due accessible. We emphasize that in condensed matter physics, to the finite speed of light come into consideration). The all continuum field theories are, by definition, effective field marginal Fermi liquid behavior that results from the logarith- theories since the lattice explicitly breaks the continuum by mic growth of the Fermi velocity near the Dirac point is well providing a short-distance cutoff length scale (or, equivalently, studied [31–33]. We review the result for the small-order β an effective ultraviolet energy or momentum cutoff for the function in graphene [34,35] and show that the Landau pole system which does not have to be put in by hand in the theory). problem is not resolved by either weak-coupling or large- The field theory is valid up to this ultraviolet energy or N perturbation theories at one loop. On the other hand, a momentum scale (approximately the inverse lattice constant), two-loop analysis using either ordinary perturbation theory which is a physical constraint defining the domain of validity or a large-N expansion indicates the absence of a Landau of the effective field theory. Our current understanding of all pole. We also compute the large-order expansion coefficients field theories as effective field theories up to some scale is nonperturbatively, showing that the asymptotic coefficients explicitly obeyed in condensed matter physics and does not are smaller than those of the pure φ4 theory. This indicates have to be artificially introduced as, for example, in lattice that if the pure φ4 theory does not manifest a Landau pole, gauge theories. Of course, one does not know the precise then neither does the Coulomb-interacting graphene field cutoff scale except that it should be set dimensionally by theory. Our main, but tentative, conclusion in this work is the lattice spacing. It is possible that the cutoff is in fact that in all likelihood, Landau poles do not exist in condensed somewhat shorter (longer) than the lattice spacing, in which matter systems at any energy scale, although the leading-order case the effective field theory would apply to energies above perturbative RG theory may imply their existence, often at (below) the inverse lattice spacing. In the context of Landau energy scales beyond which the corresponding effective field poles, the hope is that the pole energy here would not be theory is applicable.

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The paper is organized as follows. We discuss Landau poles in the critical quantum magnet and in graphene in Secs. III and IV, respectively. In each section, we compute both the small- and large-order terms of the β function for the running coupling. While the small-order β function is obtained through the standard perturbation theory, the large orders are evaluated through the saddle-point approximation following the Lipatov method [18,19]. In Sec. V, we briefly discuss possible experiments and conclude the paper. The Appendixes contain technical details and an instructive zero- dimensional toy model to illustrate the steps of the Lipatov method for the φ4 + φ6 potential.

λ λ III. CRITICAL QUANTUM MAGNETS FIG. 1. The flow diagram of 4- 6. The blue arrow indicates the RG flow to higher energies. The black and red dots are the Gaussian In this section, we compute the small- and large-order and the new fixed point in Eq. (7), respectively. The dashed line terms of the asymptotic expansion of the β function for the which separates different flow regimes is given by Eq. (8). We take 4 6 effective theory consisting of both φ and φ terms. We use N = 1 for simplicity. the Wilsonian renormalization group to obtain the one-loop β function. We show that while the pure φ4 theory exhibits a Landau pole, the inclusion of the φ6 term removes the pole The fixed point has two relevant directions at the critical and replaces it with an ultraviolet fixed point. In addition, surface. One should distinguish it from the tricritical point.  we compute the large-order expansion of the β function by For d 3 space-time dimensions, the tricritical point is the using a saddle-point approximation in the regime of negative same as the Gaussian fixed point, because the upper critical φ6 couplings. We find that the expansion coefficients of the φ4 dimension is three for the theory. Moreover, a tricritical coupling λ grow as k! while those of λ grow as (k!)2.From point has only one relevant direction at the critical surface. 4 6 λ λ both small-order and large-order expansion coefficients, we The flow diagram of 4- 6 is shown in Fig. 1, where see that including the φ6 term strongly affects the fate of the the black and red dots are the Gaussian fixed point and the Landau pole compared to a pure φ4 theory. ultraviolet fixed point in Eq. (7), respectively. The dashed curve is given by

A. Small orders + λ3 16(N 26) 4 f (λ4) = ,λ4 > 0. (8) We consider the effective theory, 3(6Nλ4 + 84λ4 + 1) 4 d 1 2 g4 4 g6 6 If one fixes λ6 = 0 to reduce to the pure φ theory, as indicated S[g4, g6; φ] = d x (∂φ) + φ + φ , (4) 2 4 6 by the black line in Fig. 1, λ4 will become infinite at the β φ = ,..., Landau pole. This is consistent with the one-loop function where i, i 1 N,isanO(N) field, and the sum- λ λ 2 N 2 2 for 4. However, in the global flow diagram of both 4 and mation over i is implicit: φ = (φi ) , and (∂φ) = i λ6, the points at λ6 = 0 are not stable at higher energies. N d ∂ φ 2 i μ( μ i ) . Using the standard Wilsonian RG technique Depending on the initial values, the RG flow regimes are (see Appendix A), the perturbative β equations at the critical separated by the dashed line given by Eq. (8). Denote the surfacer ¯ = 0, wherer ¯ is the dimensionless mass, are initial values by λ40 and λ60. On the one hand, at λ40 < 0 ∪ λ > ∩ λ > λ dλ [ 40 0 60 f ( 40 )], it will flow to the new fixed point 4 =− + λ + + λ2, (3N 12) 6 (4N 32) (5) λ = i = , ,... d log μ 4 (if one can fine-tune i 0for 8 10 ). On the other hand, at λ40 > 0 ∩ λ60 < f (λ40 ), λ6 will flow to negative dλ 32N + 832 values, indicating that the theory is not stable because the 6 = 2λ + (12N + 168)λ λ − λ3, (6) d log μ 6 4 6 3 4 energy is not bounded from below. Therefore, if one includes the symmetry-allowed φ6 term, the Landau pole is replaced 2 λ = A3 g4 λ = A3 g6 2 β where 4 π 4 and 6 ( π 4 ) are dimensionless by a fixed point of the one-loop function. Note that even if (2 ) 4 (2 ) 6 φ6 couplings, and we have set d = 4. Ad is the area of a d sphere the term is not included initially, it will be generated by the 4 with unit radius. φ potential. From Eqs. (5) and (6), besides the Gaussian fixed point, there is also an ultraviolet fixed point at B. Large orders φ6 (λ4,λ6) We have shown that the inclusion of the term in the effective theory will replace the Landau pole by an ultravi- + + 3 = − N 8 , (N 8) olet fixed point even in the one-loop calculation. We now 2(N2 + 6N + 128) 3(N + 4)(N2 + 6N + 128)2 extend the analysis to large orders. Namely, we calculate the asymptotic expansion coefficients βk, k 1ofβ functions, 1 2 1 3 β λ = β λk ≈ − + O(1/N ), + O(1/N ) . (7) ( ) k k . We first calculate the asymptotic expansion 2N 3N2 of the correlation function, and then relate it to the β function.

023310-4 LANDAU POLES IN CONDENSED MATTER SYSTEMS PHYSICAL REVIEW RESEARCH 2, 023310 (2020)

In Appendix B, we present a toy model that provides a simple Moreover, the Jacobian that results from the change of inte- illustration of the basic steps we employ. gration variables from the functional measure to x0 and cn is given by ⎡ ⎤ 1. Asymptotic expansions of the correlation function 1/2 d d/2 Consider the n-point correlation function of the massless ⎣ d 2⎦ S0 J = d x(∂μϕ ) = − , (16) φ4 + φ6 theory defined by Eq. (4), c dg μ=1 4 n −S where we have used the spherical symmetry of the saddle- Gn(g4, g6; Xn ) = Dφ φ(x j )e , (9) point solution. The quadratic kernel around the saddle point j=1 is where X = (x ,...,x ) denotes all n space-time arguments, n 1 n δ2S[g , 0; ϕ ]   and in this section we consider N = 1 for simplicity. We M x, x = 4 c = − ∂2 − ξ 2 x δ x − x , ( ) δϕ δϕ x 3 ( ) ( ) are interested in computing the asymptotic expansion of the c(x) c(x ) n-point correlation function Gn(g4, g6) (in the following, we (17) suppress the argument Xn for notational simplicity). In gen- which is independent of g . Note that ∂μϕ(x) is the eigen- eral, the expansion has the following form: 4 function of the kernel with zero eigenvalue. The above or- (n) k (n) k {ϕ } G (g , g ) = A g + B (g )g . (10) thonormal basis j can be chosen to be the eigenfunctions n 4 6 k 4 k 4 6 , k0 k 1 of M(x x ). Putting everything together, the imaginary part of the , In four dimensions, the asymptotic expansion of Gn(g4 0) correlation function under the saddle-point approximation is φ4 as a function of g4 is known for the pure theory with given by = g6 0[18,19]. We first review this calculation and then (n) k 1 − , ϕ generalize it to g > 0. When g = 0, G (g , 0) = A g d S[g4 0; c] 6 6 n 4 k k 4 ImGn(g4, 0) = Jd x0e Dϕ⊥ has a branch cut at g ∈ (−∞, 0). As shown for the toy model 2i 4 ϕc in Eq. (B8), the expansion coefficient is related to the branch n cut by d d × ϕ − d xd x ϕ⊥ (x)M(x,x )ϕ⊥ (x ) (x j )e 0 dz ImG (z, 0) j=1 A(n) = n . (11) k k+1 −∞ π z S d/2 0 S e 4g4 F (X ; ξ ) g < = 0 n n , (18) Since the unstable saddle-point solution at 4 0 will d/2+n/2 1/2 d (−g4 ) i(det M) contribute to the imaginary part, we use the saddle-point  approximation. When g = 0, the saddle-point equation is ξ = d ξ − ∈ ϕ 6 where Fn(Xn; ) d x0 j (x j x0), n 2Z, and ⊥ de- 2 3 notes the functions orthogonal to ∂μϕ. ϕ denotes the sum- −∂ ϕc + g4ϕ = 0. (12) c c mation over the two saddle points. The expansion coefficient ϕ = √ξ (x) ξ ∂2ξ + ξ 3 = Let c(x) − , where is the solution of 0 is g4 = d/ and is independent of g4√.Ind 4 dimensions, there are 2 (n) S0 iFn(Xn ) 2 2 = − k ξ x =± Ak ( 1) / two solutions: ( ) x2+1 (see Appendix C). Note that the d (det M)1 2 saddle-point equation is scale invariant in the sense that if ξ (x) −1 S0 is a solution, then b ξ (x/b) is also a solution for any rescal- 0 4z × dz e ing parameter b. We will remove this scaling redundancy later + / + / + −∞ π (−z)k d 2 n 2 1 on. Because dd x[(∂ϕ )2 + g ϕ4] = 0, the on-shell action is c 4 c d k+ n 2 ξ 2 g S 1 4 iFn(Xn; ) 4 d n , ϕ =− 4 d ϕ4 =− 0 , = (−1)k k + + . S[g4 0; c] d x c (13) π 1/2 4 4g4 d (det M) S0 2 2 d 4 (19) where S0 ≡ d xξ is a positive constant independent of g4. 2 In d = 4 dimensions, S0 = 32π /3. Because M(x, x ) has a unique negative eigenvalue [19], i The fluctuations around the saddle-point solution can be 1/2 is a positive number. {ϕ } (det M) expanded in terms of an orthonormal basis j (x) , We now generalize this procedure to the case of φ4 + φ6 ∞ theory. We first treat the pure φ6 theory nonperturbatively and φ4 φ(x) = ϕc(x − x0 ) + c jϕ j (x). (14) then include perturbative corrections from the term. The 6 j=1 saddle-point equation of a pure φ theory is −∂2φ + φ5 = . Here, x0 is an arbitrary point in space-time. Using this basis, c g6 c 0 (20) one can change the functional integration into a c-number φ = χ(x) −∂2χ − χ 5 = Thesolutionis c(x) − 1/4 , where 0. Be- integration. Note that to make the integration variable linearly ( g6 ) d ∂φ 2 + φ6 = independent, one requires cause d x[( c ) g6 c ] 0, the on-shell action is g S d , φ =− 6 d φ6 = √ 6 , d xϕ j (x)∂μϕc(x) = 0,μ= 1,...,d. (15) S[0 g6; c] d x c (21) 3 3 −g6

023310-5 JIAN, BARNES, AND DAS SARMA PHYSICAL REVIEW RESEARCH 2, 023310 (2020) d 6 d 4 where S6 ≡ d xχ (x) is a positive constant independent where S4 = d xχ is a positive constant independent of g4 of g6. and g6. Then the asymptotic coefficient is As before, fluctuations around the saddle-point solution (n) can be expanded in an orthonormal basis {φ j (x)}, Bk (g4)

∞ d − √S6 + g4 0 S4 S 2 iF (X ; χ ) dz e 3 −z z φ = φ − + φ , = − k 6 n n (x) c(x x0 ) c j j (x) (22) ( 1) / + / + / + d (det M)1 2 −∞ π (−z)k d 4 n 4 1 j=1 d 2k+ n d φ ∂ φ = μ = ,..., 2 3 2 iFn(Xn; χ ) 3 2 d n and d x j (x) μ c(x) 0, for 1 d. In this case, = (−1)k 2k + + 1/2 the Jacobian is π d (det M) S6 2 2   / 1 2 d/2 d n 1 S2 1 d 2 S6 × U k + + , , 6 , (29) J = d x(∂μφc ) = √ . (23) − 4 4 2 36S4 g4 μ d g6 where we have defined The quadratic kernel around the saddle point is   a M , = −∂2 − χ 4 δ − , U(a, b, c) = c U (a, b, c), (30) (x x ) x 5 (x) (x x ) (24) U which is independent of g6. We evaluate the imaginary part of and is Tricomi’s (confluent hypergeometric) function. One the correlation function using the saddle-point approximation, can further expand Tricomi’s function to get the expansion j k which yields coefficients of g4g6, as we will do in the next section. 1 d −S[φc] ImGn(0, g6) = Jd x0e Dφ⊥ 2. β functions at large orders 2i φ c In the previous section, we computed the asymptotic ex- 4 6 n pansion of the n-point correlation function of the φ + φ d d − d xd x φ⊥ (x)M(x,x )φ⊥ (x) × φ(x j )e theory. In this section, we use this result to compute the j=1 high-order expansion of the β function. We define the renormalized couplings as the full n-point S − √6 2 / − p d 2 3 g6 λ =− , = , S e F (X ) vertex functions, n n(¯g4 g¯6; μ2 ), where n 4 6, and = 6 n n , (25) d/4+n/4 1/2 = = μ2 d (−g6) i(det M) g¯4 g4 andg ¯6 g6 are the dimensionless couplings. Here,  p is the momentum at the renormalization point, and μ de- χ = d χ − ∈ where Fn(Xn; ) d x0 j (x j x0 ) and n 2Z.Theex- notes the energy scale. From the equations defining the saddle pansion coefficients are points, Eqs. (12) and (20), we have B(n)(0) k d 3 ξ (x) = d y0(x − y)ξ (y), (31) S d/2 − √6 S iF (X ) 0 dz e 3 −z = (−1)k 6 n n 1/2 k+d/4+n/4+1 d 5 d (det M) −∞ π (−z) χ(x) = d y0(x − y)χ (y), (32) d 2k+ n 2 χ 2 k 2 3 iFn(Xn; ) 3 d n  = 1 = (−1) 2k+ + . where 0(p) 2 is the free propagator. We then have 1/2 p π d (det M) S6 2 2 n (26) d Fn(Xn; s) = d yFs(y) 0(x j − y), (33) The fact that the kernel M has a negative eigenvalue can be j=1 seen by noting that d 3 where s = ξ,χ, e.g., Fξ (y) = d x0ξ (y − x0 ) and Fχ (y) = d χ 5 − dd xdd x χ(x)M(x, x )χ(x ) =−4 dd xχ 6(x) < 0. d x0 (y x0 ). Equation (33) converts the correlation function to the vertex function [18]. There is a subtlety for (27) d = 4 dimensions because the φ4 potential is scale invariant in this case. If ξ (x) is a solution to the saddle-point equation, It remains to be shown that there is an odd number of negative ∂2ξ + ξ 3 = 0, then it follows that b−1ξ (x/b) is also a solution i eigenvalues. We assume this is true, in which case (det M)1/2 is for any rescaling factor b. We can remove this trivial scaling a real number. redundancy by using the following identity: Now we can consider the correction from a finite dd xg φ4 term. The first-order perturbation is given by the x2 4 1 = S d log b2δ d4xξ 4(x)log , (34) correction to the on-shell action, 0 b2 S6 d 4 S6 g4 in conjunction with the rescaling transformation ξ (x) → S[g4, g6; φc] ≈ √ + g4 d xφ = √ − S4, − c − −1ξ / δ 3 g6 3 g6 g6 b (x b). The function√ above fixes the scaling of the so- ξ =±2 2 (28) lution, namely, (x) x2+1 . These modifications are taken

023310-6 LANDAU POLES IN CONDENSED MATTER SYSTEMS PHYSICAL REVIEW RESEARCH 2, 023310 (2020) into account by working with the function where F (p), F˜ (p)(s = ξ,χ) are the Fourier transforms of s s − Fs(x), F˜s(x). 2 d −3 3 y x0 F˜ξ (y) = S d log b d x b ξ . (35) The function in Eq. (30) has the expansion 0 0 b

Putting everything together, we get ∞ (a)n(a + 1 − b)n − p2 U(a, b, c) = (−c) n, (39) g¯ , g¯ ; = A(n)g¯k + B(n)g¯ j g¯k , (36) n! n 4 6 μ2 k 4 j,k 4 6 n=0 k and the asymptotic forms of the coefficients are given by where (a)0 ≡ 1, (a)n ≡ a(a + 1)···(a + n − 1). The two d ∂mU − k 2 ˜ limits of are given by A(n) = ( 1) 4 iFξ (p) m! k π d (det M)1/2    2m m k+ n ∂mU k − 36S4 , 2 m! S2 k m 1 4 d n →  6  × + + , 2k m (40) k (37) m!m − 36S4 , , S 2 2 m! 2 2 m k 1 0 (k!) S6 d − k 2 μ2 B(n) = 2( 1) 3 i Fχ (p) m,k π M 1/2 A(n) d (det ) from which we see that the asymptotic behaviors, k and (n) (n) 2k+ n B , for fixed m, increase factorially as k!, while B , for fixed 3 2 d n ∂mU k m m k × 2k + + , (38) m increases faster as (k!)2, i.e., S6 2 2 m! k (n) 4 d + n − A ∝ − k 2 2 1 , , k ( 1) k k! k 1 (41) S0 ⎧  2k m d+n−3 + ⎨ + 2 2m − k m 3 36S4 k k 2, ( 1) 2 4 (k!) k m 1 (n) S6 S m! B , ∝    6  (42) m k ⎩ + 2k 36S m d+n−3 − k m 3 4 2 k 2k , . ( 1) 2 k 4 m m! m k 1 S6 S6

φ4 + φ6 We are interested in the theory in four dimensions; ∞ λ λ dλ the coupling constants 4 and 6 are 6 =− ∂A(6) k − ∂B(6) j k , k g¯4 j,k g¯4g¯6 ∞ d log μ k=2 j,k λ = − A(4) k − B(4) j k , 4 g¯4 g¯ , g¯ g¯ (43)     k 4 j k 4 6 k = , ≈− ∂A(6) λk + B(4)λ k 2 j k k 4 0,1 6 ∞ k λ =− A(6)g¯k + g¯ − B(6)g¯ j g¯k , (44) +   6 k 4 6 j,k 4 6 − ∂A(6) ( j k)!λ j B(4)λ k k=2 j,k k j!k! 4 0,1 6 j,k from which we can reexpandg ¯n as a function of λn.The − ∂B(6)λ j λk, leading orders are given by j,k 4 6 (48)   = λ + B(4)λ + λ2,λ2,λ λ , k g¯4 4 0,1 6 O 4 6 4 6 (45) where ∂A = ∂ A.   ∂ log μ = λ + λ2,λ2,λ λ . Equations (41), (42), (47), and (48) are the main results of g¯6 6 O 4 6 4 6 (46) this section. We see from these results that the inclusion of the The β functions have the asymptotic expansion φ6 term, which is allowed by symmetry in condensed matter ∞ systems (and, in any case, is automatically generated in the dλ 4 =− ∂A(4)g¯k − ∂B(4)g¯ j g¯k , theory from the φ4 term), dramatically changes the asymptotic d log μ k 4 j,k 4 6 β λ λ ∼ λ k=2 j,k behavior of the function for 4. On the one hand, if 6 4,     the asymptotic coefficient of the β function for λ4 changes ≈− ∂A(4) λk + B(4)λ k 2 k 4 0,1 6 qualitatively from k!to(k!) : k 2k (4) 6 d+1 2 ( j + k)!   B ∝ k 2 (k!) , k 1. (49) − ∂A(4) λ j B(4)λ k 0,k j+k 4 0,1 6 S6 , j!k! j k λ Thus, we see that the inclusion of 6 completely changes (4) j − ∂B λ λk, the asymptotic behavior of the β function for λ4. In light of j,k 4 6 (47) k Eq. (3), this will in turn change the fate of the Landau pole.

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6 On the other hand, even if λ6  λ4, as long as the φ term is by [34] λ > nonzero, 6 0, the asymptotic form of the second term in d−1 Eq. (47) is comparable to the first term of the pure λ4, i.e., d 0 i 0 S = d x ψγ¯ ∂0ψ + vψ¯ γ ∂iψ + igφψγ¯ ψ i=1 k (4) 4 d +1 3 ∂A ∝ k 2 k!, k 1, (50) 1+k 1 4 2 S0 + d x (∂ φ) , (56) 2 i i=1 which can also qualitatively change the behavior of the β where ψ (φ) refers to the four-component Dirac fermion function compared to the pure φ4 theory. In particular, includ- (Hubbard-Stratonovich scalar field), and v (e) refers to the ing the φ term will result in a new asymptotic series, so even 6 Fermi velocity (coupling strength). γ μ, μ = 0,...,d − 1, de- if the pure λ series does not end up with a Landau pole, it is 4 notes the 4 × 4 γ matrices. The effective interaction strength possible that including the φ6 term will lead to a Landau pole. α = Ng2 We are not aware of a technique to resum an asymptotic is given by 16v (also known as the fine structure con- = 2 = 2e2 series with multiple variables, but it is still useful to illustrate stant), where N 2 is the spin degeneracy, and g (1+ε) , 6 0 how the Borel sum of λ4 is affected by the φ potential. Notice where ε is the background dielectric constant (which in that a Borel transform [36] will take the function B(g) with a general can be >1 in solid-state systems), and 0 is the branch cut at −α, vacuum permittivity. (Note that the definition of α used here contains an additional factor of πN/4 compared to that used −β− g 1 in Ref. [35].) B g = + = B gk, ( ) 1 α k (51) As in the previous analysis of critical quantum magnets, we k calculate the behavior of the β function of the fine structure (k + β)! constant at both small and large orders of perturbation theory. B = (−α)k → (−α)kkβ , (52) k k!β! Small-order terms can be calculated using either the standard perturbation theory in α or a large-N analysis in which 1/N is = k the expansion parameter. Here, we review the results obtained to a series W (g) k Wkg whose asymptotic coefficient is given by previously for both methods [34,35,37], and we discuss their implications for the Landau pole problem. To our knowledge, α W → (−α)kkβ k!. (53) the large-order terms of the asymptotic series in had not k been computed prior to the present work. Here, we obtain these by first integrating out the Dirac fermion and then using Comparing Eq. (53)toEqs.(41) and (42), a naive Borel a saddle-point approximation similar to the one employed in sum without considering the φ6 potential is given by the previous section on quantum magnets. −4 S0 B(0)(λ ) ∼ 1 + λ . (54) A. Small orders 4 4 4 The two-loop β equation was calculated in Ref. [35]for the case of graphene (where d = 2 + 1 and we have N = 2 The presence of λ6 leads to a different (naive) Borel sum that at kth order has the form flavors of four-component Dirac fermions), dα −2k−1 = α2 + α3, S2 f1 f2 (57) (k) λ ∼ λk + 6 λ . d log μ B ( 4) 6 1 4 (55) 36S4 α = g2 = 1 = 2 log 2−8/3 where 8v , f1 2π , and f2 π 2 . Importantly, we 3 It is apparent that these Borel sums lead to different asymp- see that f2 < 0, so when α becomes large, the negative α totic behaviors. Nevertheless, we need to emphasize that term dominates and prohibits α from becoming larger. There- instead of having multiple Borel sums, like Eqs. (54) and (55), fore, the Landau pole is replaced by a fixed point at α∗ =−f1 . λ f2 for each power in 6, it is more appropriate to treat the This is similar to the φ4 + φ6 theory in the previous section. λ λ asymptotic expansions of 4 and 6 on equal footing, because (A similar strong-coupling fixed point is found in the direct λ 6 has a vanishing radius of convergence. nonperturbative numerical simulation of the usual (3 + 1)- dimensional QED on the lattice in Ref. [5], where the Landau pole was found to lie in a region of the parameter space made IV. DIRAC FERMIONS WITH COULOMB INTERACTIONS inaccessible by a strong-coupling chiral symmetry-breaking In this section, we consider d-dimensional Dirac fermions transition.) However, evidence for the two-loop critical point with Coulomb interactions. Ultimately, we are interested in in Eq. (57) has not been observed in graphene experiments, Dirac fermions in graphene, where d = 2 + 1, with Coulomb suggesting that the second-order perturbation theory in α may interactions. This is the condensed matter analog of QED in not be reliable, as was argued in Ref. [35]. In particular, the context of Landau poles. To have a local quantum field the existence of such a possible strong-coupling fixed point theory, one can implement a Hubbard-Stratonovich transfor- preempting the Landau pole may necessitate a stability anal- mation to decouple the Coulomb interaction by introducing ysis going to higher orders. For example, if the two terms in a scalar field. The theory under consideration is then given Eq. (57) were the leading terms of a geometric series, then the

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invariance in Eq. (56) corresponding to the transformation ψ → iθ(τ )ψ, φ → φ − 1 ∂ θ τ , e g τ ( ) (60) where θ is an arbitrary function of τ. This residual gauge symmetry leads to the Ward identity, −1 −1 p0 3(p0, p; q0, q) = ig[G(q0 + p0, q) − G(q0, q) ], (61) −1 where 3 is the vertex function, and G is the inverse fermion propagator. This Ward identity is similar to the Ward identity in standard QED where the photon is dynamical, except that here it only involves the temporal component. The FIG. 2. (a) The polarization of the scalar field. The solid line Ward identity relates the charge renormalization factor to the (single dashed line) represents the Dirac fermion propagator (bare anomalous dimension of the scalar field. scalar propagator). (b)–(d) The 1/N-order Feynman diagrams. The While the Dirac fermion is confined to a two-dimensional double-dashed line represents the screened scalar propagator. plane, the scalar field in Eq. (56) can propagate in the full three-dimensional space. Thus, the back-reaction from the result to all orders would be [35] Dirac fermion is restricted within the two-dimensional plane, dα α2 and so it does not lead to an anomalous dimension for the = f 2 , (58) d log μ 1 f − f α scalar field. Technically, this is because the scalar propaga- 1 2 −1 ∼|| tor is nonlocal in the plane D0 (q) q , and thus it does which gives rise to β(α  1) ∝ α2 at small α, and β(α not receive corrections. As a result, according to the Ward 1) ∝ α at large α. This time there is no ultraviolet fixed point identity, the charge does not renormalize. The only parameter in the theory defined by Eq. (58), and the coupling grows in Eq. (56) that does renormalize is the Fermi velocity v,or indefinitely with energy scale. There is still no Landau pole, equivalently, the dynamical exponent z = 1 + γv, where γv but the mechanism by which it is avoided is quite different. A is the anomalous dimension of Fermi velocity. Note that this one-loop theory with just the first term in Eq. (57) clearly has a is not true for the theory in 3 + 1 dimensions, because the Landau pole whereas neither the full Eq. (57) with both terms scalar field can have a finite anomalous dimension in that case, or the resummed theory of Eq. (58) has a Landau pole (albeit leading to two independent parameters in 3 + 1 dimensions. for very different reasons). This illustrates how leading-order In light of the above discussion, the only Feynman diagram one-loop perturbative calculations can be misleading when it we need to evaluate is Fig. 2(b), because the vertex correction comes to determining the existence of a Landau pole in a given can be inferred from the self-energy correction by the Ward theory. Similar trends can occur in large-N calculations, as we identity. Using the results from Ref. [34], we get the fermion now discuss in the case of graphene. self-energy, Besides expanding in the coupling strength, it is often 4 useful to consider a large-N expansion (in the number of (p) = [ f (α)p γ 0 + f (α)p · γ]logμ, (62) π 2 0 0 1 fermion flavors rather than a perturbative expansion in the N coupling constant) instead, especially when the bare coupling where  √ parameter is not small. This is the situation for free-standing − 1−α2 α − + π ,α< α ≈ . α arccos 1 2α 1 graphene, where bare 3 4, suggesting that an expansion in f1(α) = √ √ / = α2−1 α + α2 − − + π ,α>, powers of 1 N with N 2 would be more reliable than an α log( 1) 1 2α 1 α expansion in powers of bare (simply by virtue of the fact that (63) α>1 and 1/N < 1). This was confirmed in Ref. [37]. In the  2 following, we review the large-N calculation for graphene and − √2−α arccos α − 2 + π ,α<1 α = α 1−α2 √ α show that a Ward identity guarantees that only one parameter, f0( ) 2 √α −2 log(α + α2 − 1) − 2 + π ,α>1, the Fermi velocity, renormalizes. Moreover, we show that the α α2−1 α β function obtained from the leading-order large-N calcula- (64) tion does not exhibit a Landau pole. We believe that the same remains true for the next-to-leading-order theory in 1/N also, from which we can extract the dynamical exponent, 1 − z = 4 α − α β which is likely to produce only small quantitative corrections π 2N [ f1( ) f0( )], and the equation, to the leading-order 1/N theory. ∂α = − z α. The leading large-N diagram shown in Fig. 2(a) gives rise ∂ μ (1 ) (65) to the dressed boson propagator, log − We can expand the β function at small and large α: g2N |q|2 1    D(q) = 2|q|+ , (59) 1 1 α2 − 8 α3 + α4 ,α 2 π π 2 O( ) 1 8 q β(α) = N  3  (66) 1 4 α − 4 + O(α−1) ,α 1. 2 = 2 +||2 N π 2 π where q q0 q . The Feynman diagrams at the next order are shown in Figs. 2(b)–2(d). Before calculating these It is apparent that there is no Landau pole because β(α diagrams, it is worth noting that there is a residual gauge 1) ∝ α. Instead, the coupling diverges as the energy scale goes

023310-9 JIAN, BARNES, AND DAS SARMA PHYSICAL REVIEW RESEARCH 2, 023310 (2020) to infinity. However, if we expanded perturbatively in α (and on-shell action is / α3 also in 1 N), and retained terms up to order , we would d − 2 − 2 α = π/ S = |α | d−2 S , (69) erroneously conclude that there is a fixed point at 3 8. on-shell 2d d d This is similar to the fixed point [cf. Eq. (57)] we found from d − 2 d our original perturbation theory in α above. Of course, if we where Sd = d xd d−2 |η| is a positive constant independent α2 α2 β α  = of αd . instead stopped at order , we would obtain ( 1) πN , and we would conclude that there is a Landau pole as in At each time τ, the field configuration can be expressed by ϕ the leading-order perturbative one-loop theory [i.e., Eq. (57) an orthogonal basis j (x): with just the first term on the right-hand side]. Thus, we see φ(τ,x) = φ (x − x ) + f (τ )ϕ (x). (70) that truncating the series at a low order in α can produce c 0 j j misleading results about the Landau pole issue. This is again j an indication that the Landau pole in graphene, which is We can change the functional integration measure to x0 and f j. apparent in the one-loop perturbative RG theory as in the The Jacobian associated with the change of variables is given original QED context [3], may be an artifact of perturbation by     theory. 1/2 3/2 S = 4 ∂ φ 2 = d , J d x( i c ) 2 (71) |α | d−2 B. Large orders i 3 d Now we turn to the large-order expansion of the β function. and the quadratic kernel around the saddle-point solutions is Because of the fermionic nature of Dirac fields, the saddle- δ2S[φ ] point analysis is not directly applicable. We can circumvent M x, x = c ( ) δφ δφ this problem by integrating over the Grassmanian field, which c(x) c(x ) 2 d−2 (4−d) (4) results in a functional determinant. In the strong-coupling = [−∇ − (d − 1)|η| δ (z⊥)]δ (x − x ). limit, the determinant can be evaluated using the quasilocal (72) approximation (see Appendix D), which then yields an ef- fective action involving just the scalar field [19,38]. In the It is straightforward to show that this kernel has a negative following, we start from this effective action and apply a eigenvalue by projecting it onto η in a manner analogous to saddle-point analysis to obtain the large-order expansion of Eq. (27). We again assume that there is an odd number of the β function. negative eigenvalues. The effective action is given by [19,39,40] We are interested in the n-point correlation function, 1 n S = d4x (∇φ)2 + dd xα |φ|d , (67) G (α ; X ) = Dφ φ(x )e−S. (73) 2 d n d n j j=1 α = 2 (−d/2) N 2 d/2 In the saddle-point approximation, the imaginary part of this where d (4π )d/2 vd−1 (g ) . Notice that it involves a dif- ferent combination of g and v because of the different approx- correlation function is imation we implement in the large-order series. However, in 1 − φ α = 3 S[ c] φ 2 + 1 dimensions, since from the previous section we know ImGn( d ) Jd x0e D ⊥ 2i φ that the RG correction to the charge g vanishes and all RG c n effects come from the renormalization of the Fermi velocity v, d d N − d xd x φ⊥ (x)M(x,x )φ⊥ (x) α = 1 | |3 α × φ(x j )e 3 3π v2 g is directly connected to in the weak-coupling expansion approach. The large-order perturbation series for j=1   − Dirac fermions with Coulomb interactions is equivalent to 3 (2 d)Sd 2 exp 2 1 S F (X ; η) −α d−2 that of the effective scalar field theory given by Eq. (67). = d n n 2d( d ) , n / n+3 The potential term in Eq. (67) leads to a branch cut in the d−2 M 1 2 − d 3 i(det ) (−αd ) d 2 n-point correlation function for α ∈ (−∞, 0). This allows us d (74) to use similar techniques as before to evaluate the asymptotic  η = 3 n η  −  expansion of the correlation function. where Fn(Xn; ) d x0 j=1 (x j x0 ). Because the The equation of motion that follows from Eq. (67)is effective action is scale invariant, i.e., invariant under η(x) → b−1η(x/b), the scaling freedom needs to be −∇2φ + α φ |φ |d−1δ(4−d) = , c d d sign( c ) c (z⊥) 0 (68) fixed in the path integral. Using a similar method to what is shown in Eq. (35), we arrive at F˜n(Xn; η) =  − where z⊥ denotes the coordinates perpendicular to the space- 2 3 n −1 xj x0 Sd d log b d x0 = b η( ). The expansion time where the Dirac fermions live. For instance, z⊥ = z j 1 b coefficient is then is the direction perpendicular to the plane where the Dirac 3 fermions are confined in the case of d = 2 + 1 dimen- 1 S 2 iF˜ (X ; η) A(n)(z) = (−1)k d n φ = k n 1/2 sions (graphene). The equation of motion is solved by c d−2 3 (det M) − 1 d | α | d−2 η η   d d , where is a solution to the differential equation (2−d)S − 2 −∇2η − η |η|d−1δ(4−d) = α < 0 dz exp d (−z) d−2 sign( ) (z⊥) 0. Note that here d × 2d π k+1+ n+3 0 since we want to apply the saddle-point technique. The −∞ (−z) d−2

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3 d − S 2 iF X η still remains about the existence of the original theory with = − k 2 d ( n; ) ( 1) n / the Landau pole (e.g., QED): Is it a well-defined interacting π d−2 M 1 2 2 d 3 (det ) theory or is it trivial? − + d 2 k+ 3 n 2d 2 2 d − 2 n + 3 The Landau pole problem becomes a bit more concrete × k + . in condensed matter systems, because the limitations of such (d − 2)S 2 2 d theories are typically known and determined by lattice-scale (75) cutoffs. This raises interesting questions: Is it possible for a Comparing Eq. (75)toEq.(19), we obtain the large-order Landau pole to occur below the lattice-scale cutoff? What expansion of the β equation: would be the experimental implications of this? The ef- fective field theory point of view makes the Landau pole dα = C αk , (76) existence question rather subtle though. The fact that all d log μ k d k symmetry-allowed terms can become important in the high- − d 2 k energy regime of an effective field theory makes it particularly d − k 2d 1 n+1 d 2 C ∝ − k 2 k . challenging to determine if a Landau pole arises. Because the k ( 1) − ! (77) d 2 Sd 2 Landau pole is by definition a high-energy phenomenon, it The kth-order expansion coefficient of the β function is pro- does not matter whether these terms influence the low-energy portional to ( d−2 k)!. It is less than the asymptotic expansion properties of the theory or not. In the critical quantum magnet 2 φ4 φ2n > coefficient ∼k! in the pure φ4 theory in d < 4 dimensions. In described by the theory, (n 2) potentials are present the (2 + 1)-dimensional Dirac fermion theory with Coulomb without question. Although irrelevant to the long-wavelength φ6 interactions, the asymptotic coefficient is proportional to ( k )!. critical phenomena, the appearance of a potential on top of 2 φ4 Since every term in the asymptotic expansion is bounded by the theory can significantly change the fate of the Landau that of a pure φ4 model, we can conclude that if there is no pole at high energies (which is a short-distance rather than Landau pole in pure φ4 theory, then there is no Landau pole in a long-wavelength phenomenon), as we have shown. On the β the Dirac fermion theory with Coulomb interactions. This is a one hand, at small orders of the function, we find that the direct consequence of the fact that the existence of a Landau Landau pole is removed and replaced by a new ultraviolet λ λ pole is fully determined by the asymptotic behavior of the β fixed point because of the coupled RG flow of 4 and 6.On function, as shown in Eq. (3). the other hand, at large orders, the asymptotic expansion of the β function gets a large contribution from λ6, which can also alter the fate of the Landau pole. What happens if other V. DISCUSSION AND CONCLUSION φ2n terms are also included in the analysis? Ultimately, it may Since the seminal works of Wilson [23,24], it has been have to be decided by experiments that probe strong-coupling, understood that quantum field theories are essentially effective high-energy regimes to finally settle the Landau pole issue in descriptions of the low-energy and long-wavelength behav- condensed matter systems. ior of an underlying physical system, and the quest for an We note that there is nothing in principle ruling out the axiomatic foundation of quantum field theories, fashionable possible existence of a Landau pole in condensed matter during the 1960s, is futile and unnecessary. Such an effective systems at energy scales well below the ultraviolet lattice field theory description is of course particularly germane in cutoff scale, making the issue relevant both theoretically and condensed matter systems where the existence of a physical experimentally. In fact, rough estimates suggest that this may lattice imposes a real high-energy short-distance cutoff on any indeed be the case for systems currently under investigation. continuum description. It is remarkable that such theories can The well-studied three-dimensional antiferromagnet TlCuCl3 give quantitatively accurate predictions over a wide variety features an O(3) quantum phase transition realized by tuning of systems and phenomena, ranging from Kosterlitz-Thouless the pressure. From Ref. [30], the quartic interaction strength transitions in Josephson-junction arrays to the tricritical point at the quantum critical point of TlCuCl3 is estimated to be where water and gas can no longer be distinguished. In the around λ4 ≈ 0.23/4π at 1 meV, and accordingly the one-loop context of particle physics, the Landau pole issue is often calculation predicts that the Landau pole occurs at 3.5 meV, viewed as purely academic, because when a pole appears, it is which is far below the lattice scale. For comparison, the mea- typically at an incredibly high energy scale, and one normally sured magnon dispersion at zero pressure (in the disordered assumes that even if the pole is a real feature of the field phase) reaches as high as 7 meV [41], well above the predicted theory, the theory itself will likely become invalid by the time Landau pole energy. In the ordered phase, the reported gap of this energy is reached. It is difficult to make these statements the longitudinal excitation reaches about 1.2 meV [42]. These precise because, in the particle physics context, effective field facts imply that the predicted Landau pole is within the reach theories capture the low-energy limit of a system whose high- of experiments. A possible signature of a Landau pole could energy, microscopic degrees of freedom are often unknown. come from the fact that a diverging quartic interaction should In addition, the continuum is presumably real in particle cause a strong decay of the longitudinal mode to transversal physics since there is no underlying physical lattice providing modes in the ordered phase. This in turn suggests that the a short-distance cutoff. Appealing to the possibility of a decay width should exhibit a significant enhancement if there theory with the Landau pole becoming inapplicable at high is a Landau pole. However, existing data show no evidence of energies (where the pole presumably lies) because some other such an increase in the linewidth (cf. Ref. [43]). On the other theories may control the physics at the higher energy scale hand, there has by no means been an exhaustive search, and is not aesthetically pleasing because the disturbing question a systematic experimental survey at energies approaching the

023310-11 JIAN, BARNES, AND DAS SARMA PHYSICAL REVIEW RESEARCH 2, 023310 (2020) lattice scale is required to settle this issue. We believe that extracted from experiments is not consistent with a one-loop experiments in quantum magnets looking for signatures of calculation, indicating that intriguing strong-coupling effects Landau poles are needed given that the existence of an infinite may be taking place in twisted bilayer graphene. Such strong number of symmetry-allowed field operators (i.e., all the φ2n interaction effects, in light of our discussion, may provide terms) in the theory make a decisive theoretical conclusion important insights on the Landau pole problem as it pertains impossible. to Dirac fermions. On the other hand, if a small gap is opening We also examined a second type of condensed matter up at the Dirac point in the twisted bilayer graphene, then the system (namely, graphene) searching for Landau poles: Dirac Dirac description fails at low energies, and the Landau pole fermions in 2 + 1 dimensions with Coulomb interactions. issue becomes moot. This is the QED analog of the Landau pole (albeit in two It may be useful to emphasize that the detailed analysis spatial dimensions). We argued with the help of a Ward of Ref. [29] focusing on the twisted bilayer graphene ex- identity that the renormalization group flow is controlled by periments of Refs. [27,28] and Ref. [44] came to the con- α ∼ g2 clusion that the measured effective mass and Fermi velocity a single parameter [35], the fine structure constant v (or equivalently the Fermi velocity). Both perturbation theory in near the Dirac cone of low-angle twisted bilayer graphene α and the large-N expansion to leading order suggest that the agrees with strong-coupling nonperturbative theories such as Landau pole is absent in graphene. However, keeping only the resummed Borel-Padé perturbation series and the 1/N the first few orders in perturbation theory is insufficient to expansion, while disagreeing very strongly with the one-loop address the issue. Therefore, we also evaluated the coeffi- perturbative RG theory. In particular, the one-loop theory cients of high-order terms in the asymptotic series using a predicts a very large renormalization of the effective mass nonperturbative approach adapted from the Lipatov method. and the Fermi velocity for the large (α ∼ 10π/2) effective Similarly to the case of relativistic fermions with Yukawa- fine structure constant in the system, which is simply not type interactions [19,38], the asymptotic coefficient is, for observed experimentally. If these preliminary experimental d < 4 dimensions, bounded by that of the four-dimensional results hold in future measurements in flat-band twisted bi- pure φ4 theory, and consequently graphene is free of Landau layer graphene, where the effective interaction strength is poles if the pure φ4 theory does not manifest a Landau pole. very large, one inevitable conclusion is that the Landau pole Moreover, the knowledge of the asymptotic series combined as inferred from the running of the coupling implied by the with a few small-order coefficients can be used as the input one-loop perturbative RG theory does not exist (and is purely into a resummation technique that could potentially lead to a an artifact of the one-loop theory) since the one-loop theory resolution of the Landau pole problem that is independent of seems unable to quantitatively describe the running of the pure φ4 theory in this particular context [20,21]. We leave this coupling at large coupling. These preliminary measurements technically demanding calculation to future work. should be repeated in future experiments for a definitive Finally, we point to another direction for possible future in- resolution of the question of Landau poles in graphene since vestigations. As is apparent from the form of the fine structure its implications extend far beyond graphene all the way to constant, small velocities can yield large coupling strengths. It QED, where α ∼ 1/137, and the RG running of the coupling is worth noting that the Fermi velocity at the charge neutrality all the way to the Landau pole in laboratory experiments is point in twisted bilayer graphene near the magic angle is manifestly impossible. Careful experimental investigation of extremely small [26–29]. This suggests that this system could twisted bilayer graphene near the Dirac point may finally shed be a particularly interesting place to explore the Landau pole light on the 80-year-old question of the existence or not of problem, assuming the continuum Dirac description is still Landau poles in quantum electrodynamics. valid here. We can estimate the Landau pole energy scale β using the√ result for the one-loop function, which yields ACKNOWLEDGMENTS 2π/α EL ≈ h¯vF ne , where n is the carrier density and α is the effective coupling (fine structure constant). Typical values S.-K.J. is supported by the Simons Foundation through 8 12 −2 for these quantities are vF ≈ 10 cm/s and n ≈ 10 cm . the It from Qubit Collaboration. S.D.S. is supported by the For graphene grown on a BN substrate where α ≈ 0.4π/2, Laboratory for Physical Sciences and the Microsoft Corpora- 3 EL ≈ 10 eV, which is several orders of magnitude larger than tion. E.B. acknowledges support from the National Science the lattice cutoff. On the other hand, for suspended graphene Foundation (Grant No. DMR-1847078). where α ≈ 2.2π/2, the Landau pole energy is around EL ≈ 0.4 eV, which is comparable to the lattice scale. Interestingly, APPENDIX A: PERTURBATIVE RENORMALIZATION for twisted bilayer graphene on hBN where α ≈ 10π/2, the GROUP ANALYSIS OF φ4 + φ6 THEORY Landau pole energy is further suppressed to EL ≈ 98 meV. This suggests that the Landau pole could be very likely within Since there are two vertices, the structure of Feynman the reach of experiments. A possible signature could come diagrams is from the fact that the Fermi velocity should be strongly sup- L = I − V − V + 1, (A1) pressed as the Landau pole energy scale is approached due to 4 6 /α the fact that the velocity gets renormalized and scales as 1 . E = 4V + 6V − 2I, (A2) This in turn could lead to interaction-enhanced dispersion 4 6 flattening and strong effective couplings that could persist as where L, I, and E denote the number of loops, internal the system is tuned slightly away from a magic angle. In fact, propagators, and eternal propagators, and V4 and V6 denote it was reported in Ref. [29] that the running coupling constant the number of four-vertices and six-vertices, respectively. For

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FIG. 3. Feynman diagrams contribute to the β equation of the polarization. The solid line represents the boson propagator of the same index, and the wavy line represent short-range interactions (φ4 and φ6 vertices). the RG equation at the one-loop level, the Feynman diagrams contributing to the RG of polarization (V4,V6) = (1, 0), four- point vertex (2, 0), (0, 1), and six-point vertex (1, 1), (3, 0) are shown in Figs. 3–5. The renormalized action coming from the Feynman dia- grams is d 2 δS = d x g4(2N + 4) G(k)φ k FIG. 5. Feynman diagrams contribute to the β equation of the 4 + g6(3N + 12) G(k)φ six-point vertex. k 1 − g2 (8N + 64) G(k)2φ4 2 4 the effective action k 2 1 1 Ad−1  l − g g (12N + 168) G(k)2φ6 S = dd x ∂φ2 + r + g (2N + 4) φ2 4 6 4 π d + k 2 2 (2 ) 1 r¯ 1 2 + 3 + 3φ6 . Ad−1 l g4(64N 1664) G(k) (A3) + g4 + g6(3N + 12) 6 k (2π )d 1 + r¯ 1 A − l Using the momentum shell, the loop integrals are − g2 (8N + 64) d 1 φ4 4 π d + 2 2 (2 ) (1 r¯)  2 1 Ad−1  l G(k) = = , (A4) + − + Ad−1 l k2 + r (2π )d 1 + r¯ g6 g4g6(12N 168) k k (2π )d (1 + r¯)2  2 1 Ad−1 l G(k) = = , (A5) 1 A − l 2 + 2 π d + 2 + g3 (64N + 1664) d 1 φ6 , (A7) k k (k r) (2 ) (1 r¯) 4 d 2 3  6 (2π )  (1 + r¯) 1 A − l G(k)3 = = d 1 , (A6) 2 3 d 2 3 β k k (k + r) (2π )  (1 + r¯) which leads to the equations in Eqs. (5) and (6) by setting = r¯ 0 to restrict to the critical surface.  d  ≡ d k = Ad−1 d−1 > where k  (2π )d (2π )d e−l k dk, l 0 is the run- −2 ning parameter, andr ¯ = r . Ad is the area of the d- APPENDIX B: ZERO-DIMENSIONAL φ4 + φ6 THEORY: dimensional unit sphere. Using these integrals, we arrive at ATOYMODEL Here, we study a toy model [19] to better illustrate the field theory calculation. The following integral is a zero-dimension “toy” version of the φ4 theory: 1 −( 1 x2+ g4 x4 ) Z(g4) = √ dxe 2 4 . (B1) 2π The integral can be done analytically,

1   8g 1 e 4 K 1 4 8g4 Z(g4 ) = √ , (B2) 2 πg4 where K is a modified Bessel function of the second kind. It FIG. 4. Feynman diagrams contribute to the β equation of the is apparent that Z(g4) has a branch cut at g4 ∈ (−∞, 0) if one four-point vertex. analytically continues g4 to the complex plane. As a result, the

023310-13 JIAN, BARNES, AND DAS SARMA PHYSICAL REVIEW RESEARCH 2, 023310 (2020) integral is controlled by the branch cut, namely, the other two saddle points lead to the contribution − ∞ 0 i dz Z(z + i) − Z(z − i) 1 1 − 1 2+ z 4 2 = √ ( 2 xs 4 xs ) x Z(g4) = ImZ(z) e dxe −∞ 2πi z − g 2i 2π i∞ 4 x =± √1 s −z 0 dz ImZ(z) = . (B3) 1 1 =−√ 4z . −∞ π z − g e (B13) 4 2 We are interested in getting the asymptotic expansion at the Thus, we get the coefficient strong-coupling limit, 0 − k dz 1 1 ( 1) k ∞ A =− √ e 4z = √ k k k+ 4 ( ) = k , →∞. −∞ 2π z 1 2π Z(g4) Akg4 k (B4) k k=0 (−1) = √ 4k (k − 1)!. (B14) To get the answer, it is straightforward to expand the integral 2π − k It is precisely the asymptotic expansion shown in Eq. (B6). 1 − 1 x2 ( 1) −k 4k k Z(g ) = √ dxe 2 4 x g , (B5) From this lesson, we know that it is the unstable saddle point 4 π k! 4 2 k giving rise to the imaginary contribution, and consequently to the asymptotic expansion. from which   Now let us consider another integral that is the zero- − k 2k + 1 − k dimensional version of the φ4 + φ6 theory, ( 1) 2 ( 1) k Ak = √ → √ 4 (k − 1)!, (B6) π k! π 1 − 2 Z(g , g ) = √ dxe S(x), (B15) 4 6 π where we have used the Stirling formula and the identity 2 1 g g = 2 + 4 4 + 6 6. 1 (2n)!√ S(x) x x x (B16) n + = π. (B7) 2 4 6 2 4nn! 4 Unlike the zero-dimensional φ integral, Z(g4, g6) is analytic > We see that the asymptotic expansion is not convergent. for all g4 as long as g6 0. The branch cut comes from ∈ −∞, There is another way to get the asymptotic expansion, g6 ( 0). So we can use the same strategy to get the which relates the expansion coefficient to the branch cut of asymptotic expansion coefficient of g6. The asymptotic ex- pansion of Z(g4, g6) is given by g4. To get the expansion coefficient, we use k k 0 0 , → + , 1 ∂k dz ImZ(z) dz ImZ(z) Z(g4 g6) Akg4 Bk (g4)g6 (B17) A = = , > k k k+1 k k 0 k! ∂g −∞ π z − g4 −∞ π z 4 g4=0 where Ak is given by Eq. (B14). Similar to Eq. (B8), the (B8) asymptotic expansion of g6 is given by from which we know the expansion coefficient through the 0 , = dz ImZ(g4 z). imaginary part of Z(g ) at the branch cut. Bk (g4) + (B18) 4 −∞ π zk 1 When g4 is negative, one needs to continue the variable We can use the saddle-point approximation to get the imagi- x to make the integral converged. For g4± ≡ g4 ± , making θ x ≡ ρei ,theφ4 potential in Eq. (B1) is given by nary part of the integral. In the following, we consider g4 as a small perturbation. The saddle-point equation is (g4 = 0, g6 < g ± |g | 4 x4 = 4 ρ4e±iπ+i4θ 0) 4 4 5 x + g6x = 0, (B19) |g4| = ρ4[cos(±π + 4θ ) + i sin(±π + 4θ )]. (B9) < = =± 1 and the saddle points are (for g6 0) x 0, xc 1/4 . 4 (−g6 ) The quadratic fluctuation near the saddle-point solution is |x| g ± Thus, we can use a different contour at 1for 4 : given by π π 3 g 1 g4 = g4+, − <θ<− , (B10) δ ≈− 4 + √ − δ 2, 8 8 S( x) 2 x (B20) 4g6 3 −g6 π 3π = , <θ< . g4 g4 g4− (B11) where we include x4 as a perturbation. As expected from 8 8 4 the unstable solution, the fluctuation has a negative mass. Instead of evaluating the integral directly, we can use the This contributes to the imaginary part of integral; namely, the saddle-point approximation, which can be extended to the imaginary part of the integral is given by field theory. The saddle-point equation is −i∞ 1 1 −S(δx) 3 ImZ(g4, g6 ) = √ dδxe x + g4x = 0, (B12) 2i 2π i∞ x=xc = =±√ 1 where the saddle points are x 0 and x − . Appar- g4 − √1 g4 1 4g − = =− e 6 3 g6 , (B21) ently, x 0 does not contribute to the imaginary part. And 2

023310-14 LANDAU POLES IN CONDENSED MATTER SYSTEMS PHYSICAL REVIEW RESEARCH 2, 023310 (2020) from which we can get the asymptotic expansion and analyze the large-y behavior. As y →∞, we require → − k f (y) 0 because we need the action at the saddle point, ( 1) −k 1 1 d 4 B (g ) = g (2k)U k, , , (B22) S0 = d xξ (x), to be finite so that it makes a non-negligible k 4 π 4 2 9g4 contribution to the correlation function. In this limit, the f 3(y) where U is the confluent hypergeometric function. term either decays more quickly than the other two terms in Thus, the theory has at least two fates. On the one hand, if Eq. (C2) or it is comparable to them. If it is comparable, then the solution behaves as g4 g6, the asymptotic expansion coefficient of g6 vanishes: √ − lim B (g ) = 0. (B23) d 3 →∞ k 4 f (y) →± √ as y →∞. (C6) g4 y The expansion is controlled by A in Eq. (B14) as expected. k This implies that ξ (x) ∼ 1/|x| as |x|→∞, which in turn On the other hand, if g  g , we recover the asymptotic 4 6 leads to a divergent action, S →∞, for dimension d  4. expansion of the pure φ6 integral, 0 Therefore, we discard such solutions and instead consider the (−1)k (−1)k62k case where the f 3(y) decays faster than the other terms in lim B (g ) = 9k (2k) → (k!)2. (B24) k 4 3/2 Eq. (C2). In this case, the asymptotic behavior of f (y)is g4→0 π 2(πk) 1−d/2 To check this, it is straightforward to calculate the coefficient f (y) → cy as y →∞, (C7) directly: where c is a constant. This yields ξ (x) ∼|x|2−d as |x|→∞,   − 1 k ∞ and the action is finite for d  3. When d = 4, this asymptotic 6 1 6k − 1 x2 Bk (0) = √ dxx e 2 behavior is of course the same as that of Eq. (C5). The k! 2π −∞ c     constant in Eq. (C7) is again a reflection of the scale k − 4 3k + 1 (−1)k62k invariance of the solutions, and so we conclude that the only = √ 3 2 → (k!)2. (B25) two solutions with finite action are those given in Eq. (C5). π + π 3/2 (k 1) 2( k) The saddle-point equation for pure, massless φ6 theory is We can see that the results from the saddle-point approx- ∂2χ + χ 5 = 0. (C8) imation exactly reproduce the asymptotic expansion. More importantly, different from the φ4 integral, the asymptotic We again assume that χ(x) is spherically symmetric, in which coefficient of the φ6 integral increases as (k!)2, which is case Eq. (C8) reduces to 4 qualitatively different from that of the φ theory. 4yg (y) + 2dg (y) + g5(y) = 0, (C9)

APPENDIX C: SADDLE-POINT SOLUTIONS where g(y) = χ(x). We require that the solutions vanish as →∞ = d χ 6 < ∞ 5 OF φ4 AND φ6 THEORIES y so that S6 d x (x) .Ifg (y) is comparable to the other two terms in Eq. (C9)asy →∞, then the φ4 The saddle-point equation for pure, massless theory is asymptotic behavior is ∂2ξ + ξ 3 = . / 0 (C1) (2d − 5)1 4 g(y) →± √ . (C10) ξ 1/4 We assume that (x) is spherically symmetric; i.e., it only 2y depends on x2 = d x2, where d is the dimension of the √ i=1 i This implies that χ(x) ∼ 1/ |x| as |x|→∞, which in turn (Euclidean) space-time. Writing f (y) = ξ (x) where y ≡ x2, leads to a divergent S for d  3. We therefore conclude Eq. (C1) becomes 6 that the g5(y) term decays faster than the other two terms 4yf (y) + 2df (y) + f 3(y) = 0. (C2) in Eq. (C9), which then yields the following asymptotic behavior: In d = 4 dimensions, the solution to this equation is − / √ g(y) → cy1 d 2 as y →∞. (C11) c + 2 c c + 2 f (y) = sn √ log(y/y0 ) , (C3) Because Eq. (C8) has a scale invariance such that if χ(x)is y 2 2 c a solution then so is b−1χ(x/b2 ), we understand that c is a where sn(w, m) is the Jacobi elliptic function, and y0 and c are redundant scale factor. Therefore, there are only two distinct integration constants. For c ≈ 0, this reduces to solutions of Eq. (C8) that contribute to correlation functions, √ and these two solutions are related by an overall minus sign. 16 cy f y = 0 . ( ) + (C4) cy 32y0 APPENDIX D: QUASILOCAL APPROXIMATION If we then set c =±2 and y0 =±1/16, this becomes OF A DETERMINANT √ 2 2 First consider a determinant [19] f (y) =± . (C5) 1 + y det(− + m2 + gV (x))(− + m2 )−1, (D1) The fact that we can set c =±2 in the last step and still obtain d 2 where = ∂μ. A useful formula is a valid solution is a consequence of the scale invariance of μ ∞ Eq. (C1). Does Eq. (C3) contain other valid solutions beyond dt Tr log AB−1 =−Tr (e−At − e−Bt ). (D2) Eq. (C5)? To answer this question, we return to Eq. (C2) 0 t

023310-15 JIAN, BARNES, AND DAS SARMA PHYSICAL REVIEW RESEARCH 2, 023310 (2020)

μ μ Since we are interested in the large-g behavior, the integral where ∂/ = γ ∂μ, and γ is the 4 × 4 Dirac matrix. Here to is dominated by the small-t regime. We can use the quasilo- regularize the determinant, we introduce a mass m. We set it to cal [19,38] approximation: zero in the final step to retain the gapless Dirac dispersion. To   evaluate the determinant, note that, under the parity symmetry, −(− +m2+gV (x))t −t(− +m2 ) −tgV(x) e ≈ e e . (D3) − D(e) → det(−∂/ − ieφγ 0 + m)(−∂/ + m) 1. (D6) Now we have As a result, we can evaluate the square of determinant, i.e., D 2 = − + 2φ2 + 2 − + 2 −1, Tr log(− + m2 + gV (x))(− + m2 )−1 log (e) Tr log( e m )( m ) (D7) ∞  φ dt − − + 2 − where we have assumed the field is large and smooth to =−Tr e t( m )(e tgV(x) − 1) γ t neglect the correction from derivatives and used the fact the 0 matrix is traceless. From Eq. (D4), we note ∞ d dt d p − 2+ 2 − =− e t(p m ) dd x(e tgV(x) − 1) 2 (−d/2) t (2π )d D(e) = exp − dd x[(m2 + e2φ2 )d/2 − md . 0 π d/2 ∞   (4 ) 1 dt 1 2 2 =− dd x e−t(m +gV (x)) − e−tm d/2 d/2 (D8) (4π ) 0 t t Now we can safely send m to zero to get the effective field (−d/2) / =− dd x[(m2 + gV (x))d 2 − md ]. (D4) theory in Eq. (67). Including the Fermi velocity and the (4π )d/2 fermion species N, the final answer is modified as One can also evaluate the correction to the above approx- 2 (−d/2) D(e) = exp − imation orderby order using the Baker-Campbell-Hausdorff (4π )d/2 formula. But we do not consider these corrections here. N Now we include spin, and consider the determinant of the × dd x[(m2 + e2φ2 )d/2 − md ] . Dirac operator, vd−1 (D9) D(e) = det(∂/ + ieφγ 0 + m)(∂/ + m)−1, (D5)

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