Low energy excitations and singular contributions in the thermodynamics of clean Fermi liquids
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Low energy excitations and singular contributionsin the thermodynamics of clean Fermi liquids
Hendrik Meier , Catherine P´epin , , and Konstantin B. Efetov , Institut f¨ur Theoretische Physik III, Ruhr-Universit¨at Bochum, 44780 Bochum, Germany IPhT, CEA-Saclay, L’Orme des Merisiers, 91191 Gif-sur-Yvette, France International Institute of Physics, Universidade Federal do Rio Grande do Norte, 59078-400 Natal-RN, Brazil (Dated: June 15, 2018)Using a recently suggested method of bosonization in an arbitrary dimension, we study the anoma-lous contribution of the low energy spin and charge excitations to thermodynamic quantities of atwo-dimensional (2D) Fermi liquid. The method is slightly modified for the present purpose suchthat the effective supersymmetric action no longer contains the high energy degrees of freedom butstill accounts for effects of the finite curvature of the Fermi surface. Calculating the anomalouscontribution δc ( T ) to the specific heat, we show that the leading logarithmic in temperature cor-rections to δc ( T ) /T can be obtained in a scheme combining a summation of ladder diagrams andrenormalization group equations. The final result is represented as the sum of two separate termsthat can be interpreted as coming from singlet and triplet superconducting excitations. The lattermay diverge in certain regions of the coupling constants, which should correspond to the formationof triplet Cooper pairs. PACS numbers: 71.10.Ca, 71.10.Ay, 71.10.Pm
I. INTRODUCTION
At low temperatures, thermodynamic properties offermions with a repulsive interaction bear strong re-semblance to those of an ideal Fermi gas. This is thequintessence of the Landau theory of the Fermi liquid .It is assumed in this theory that interaction effects merelyrenormalize quantities such as the fermion mass or thedensity of states. In fact, this renormalization can belarge for a strong interaction, thus making perturbativemethods inapplicable. However, such obstacles are al-ways overcome once the renormalized quantities are re-placed by phenomenological effective parameters.Following the similarity to the ideal Fermi gas, onecould expect that quantities like g ( T ) = c ( T ) T , (1.1)where c ( T ) is the Fermi liquid’s specific heat, or its spinsusceptibility χ ( T ) had to be analytic functions of T /ε F with ε F being the Fermi energy and T the tempera-ture. [In the leading order in T , one should have a fi-nite g (0) considered as a phenomenological Fermi liquidparameter.] However, several studies revealed for three-dimensional Fermi liquids the existence of corrections tothe specific heat c ( T ) of the order T ln T , which is in-compatible with ideal Fermi gases. In three dimen-sions, there are also logarithmic contributions | q | ln | q | to the non-homogeneous spin susceptibility χ ( q ), where q is the wave vector. In two dimensions (2D), non-analytic corrections tothe quantities g ( T ), Eq. (1.1), and χ ( T ) are strongerand, in the lowest order in interaction, have been foundto be proportional to T . These anomalous contribu-tions were attributed to one-dimensional backscat-tering processes imbedded in the two-dimensional mo- mentum space. The linear in T correction to g ( T ) hasbeen verified experimentally in a He fluid monolayer. The problem of evaluating the anomalous contribu-tions was reconsidered in Ref. 15 with the help of asupersymmetric field theory especially designed to de-scribe those low energy bosonic excitations which are re-sponsible for the anomalous contributions to the ther-modynamics. It was found that earlier calculations had not been complete and so far unforeseen logarith-mic contributions to δc ( T ) /T d were discovered for di-mensions d = 2 ,
3. Similar anomalous contributions havebeen found for the spin susceptibility χ ( T ) using eitherthe supersymmetric method mentioned above or theconventional diagrammatic technique . Both the meth-ods led to identical results for the spin susceptibility in2D models.In dimension d = 1, the supersymmetric approach tofind the spin susceptibility reproduced the results ofearlier theoretical works . As to the specific heat ofa one-dimensional Fermi gas, the result of the supersym-metric field theory of Ref. 15 can be mapped on knownresults for the Kondo model or for the XXZ spin- chain , showing agreement. A more recent study con-firmed the supersymmetry approach using the conven-tional diagrammatic technique.In spite of the agreement between the results obtainedby these different methods in one dimension, a direct di-agrammatic computation of the anomalous specific heatcarried out up to the third order in the fermion-fermioninteraction by Chubukov and Maslov (CM) for the 2DFermi liquid led to a result that did not coincide withthe one obtained by the supersymmetric approach inRef. 15. Both of them contained logarithmic correctionsto the anomalous contributions. However, while in theframework of the supersymmetry method of Ref. 15, thenon-trivial contributions originated purely from spin ex-citations, CM obtained contributions from both spin andcharge excitations. They attributed the difference be-tween the results to the fact that not all effects of thefinite curvature of the Fermi surface had been properlytaken into account in the approach of Ref. 15.Of course, curvature effects are absent in one dimen-sion and charge excitations do not influence the spin sus-ceptibility χ ( T ) in any dimension in the non-logarithmiclowest order in the interaction. As a result, no discrep-ancy could be seen in these cases. Nevertheless, since ev-idently certain effects of the finite curvature of the Fermisurface were neglected in the supersymmetric method ofRef. 15, it is important to find the correct way of cal-culations. At the same time, approaches based of theconventional diagrammatic expansions for fermions be-come inapplicable beyond some low orders of perturba-tion theory. Indeed, CM performed a standard perturba-tion theory to third order and treated higher orders byplausibility arguments. In this paper, we revise the approach of Ref. 15 for the2D Fermi liquid taking into account all the necessary ef-fects of the curvature of the Fermi surface. As a result, weare able to sum up all leading logarithms, thus correct-ing the previous result for the function g ( T ), Eq. (1.1).To third order in the interaction potential, our resultagrees with the conventional perturbative calculation .Moreover, our result in all orders in the large loga-rithm ln( ε F /T ) shares the same asymptotic behavior asthe conjecture suggested by CM. In principle, our methodand results are applicable for both repulsion and attrac-tion unless one reaches a singularity in the final formulas.We argue that the singularities, if existing, correspondto the singlet or triplet Cooper superconducting pairing.Remarkably, the final formula for the function g ( T ) con-tains a sum of separated spin singlet and spin tripletexcitations. It is important to emphasize that the mod-ification concerns the dimensions d > d = 1 to the same results as thoseobtained in Ref. 15.The calculations are performed using a modification ofa recently suggested bosonization scheme of Refs. 23,24.In contrast to these previous works, we derive an ef-fective supersymmetric action describing only low-lyingmodes. This is achieved by singling out the slowly vary-ing pairs of the fermionic field in the interaction term.Subsequently, we decouple this interaction by means ofHubbard-Stratonovich auxiliary fields slowly varying inspace and imaginary time — similarly to what was donein Ref. 15. Here, however, this decoupling is followed bythe derivation of equations of motion using the methodof Refs. 23,24. In contrast to the equations of Ref. 15,the present equations preserve all necessary effects of thecurvature of the Fermi surface.The solution of the equations of motion is representedin a form of an integral over superfields Ψ, which do notonly depend on conventional coordinates r and imaginarytime τ but also on anticommuting variables θ, θ ∗ . Thisintegral representation allows to average over the auxil- iary fields before we obtain the final effective field theoryfor the low energy bosonic charge and spin excitations.Such a representation, suggested in Refs. 23,24, differsfrom the supervector representation used in Ref. 15 andis considerably more convenient for explicit calculations.Although the general calculational scheme based onthis superfield action shares certain similarities with thatof Ref. 15, the finite curvature of the Fermi surface sup-presses several otherwise logarithmic contributions. Con-sequently, different final results are obtained as a resultof a different calculational procedure. For instance, thequartic part of the action can be renormalized by sum-ming ladder diagram series instead of solving renormal-ization group equations.The calculations performed here can be important notonly from the point of view of finding the complete pic-ture about the anomalous contributions to the thermo-dynamics of the 2D Fermi liquid but also as a demonstra-tion of how the higher-dimensional bosonization schemesuggested in Refs. 23,24 can be used as a method in an-alytical studies. The experience gained on this compara-bly simple example may become important for attackingmore difficult and more interesting problems of stronglycorrelated systems.The paper is organized as follows: In Sec. II, we de-rive the effective low energy field theory for the anoma-lous thermodynamic contributions. Starting from a gen-eral model of repulsive interaction, we discuss and singleout the relevant soft modes and bosonize the microscopicfermion model in the low energy limit.Section III discusses the leading perturbative correc-tions to both the thermodynamic potential and the ver-tices of the low energy field theory on one-loop level. Weidentify the logarithmically divergent one-loop diagramsthat are important for the subsequent renormalizationgroup analysis. This analysis is presented in Sec. IV,in which we derive and solve the flow equations for thecoupling constants of the low energy field theory.In Sec. V, we apply the bosonic technique and theresults of the renormalization group analysis to evalu-ate the anomalous contribution to the specific heat be-yond the T -term obtained from second order perturba-tion theory. First performing an explicit perturbationexpansion to third order in order to check once more ourbosonic approach, we eventually include the completelyrenormalized vertices and find the non-analytic contribu-tion to the specific heat in all orders in ln( ε F /T ).Concluding remarks are found in Sec. VI. II. LOW ENERGY FIELD THEORY
In this section, we formulate the microscopic modelfor the interacting fermions and derive the low energyfield theory that catches the non-trivial physics of thelow-lying bosonic excitations. The derivation does notrequire to specify the dimension d of the system and weassume d to be arbitrary here. A. Microscopic fermion model
We consider a gas of spin- fermions described by theHamiltonian ˆ H = ˆ H + ˆ H int , (2.1)where ˆ H is the kinetic energy,ˆ H = X σ Z c † σ ( r ) [ ε ( − i ∇ r ) − µ ] c σ ( r ) d d r . (2.2)In Eq. (2.2), r and σ = ± c † σ ( r ) [ c σ ( r )] are creation (annihila-tion) field operators, and µ is the chemical potential.In the simplest case, ε ( p ) = p / m with m being thefermion mass. In this case, the Fermi surface is a ( d − ε ( p ),the Fermi surface has a more complex shape but thisdoes not lead to a qualitatively different physical pictureas long as the Fermi surface remains smooth and there isno nesting.The second term in Eq. (2.1) stands for the fermion-fermion interaction and takes the standard form:ˆ H int = 12 X σσ ′ Z c † σ ( r ) c † σ ′ ( r ′ ) V ( r − r ′ ) c σ ′ ( r ′ ) c σ ( r ) d d r d d r ′ . (2.3)At the moment, we do not specify the form of the function V ( r − r ′ ) except for its positivity, guaranteeing repulsiveinteraction.Equations (2.1)–(2.3) constitute the model in theHamiltonian form. It is more convenient for our purposesto use a functional integral representation, in which thepartition function Z is written as Z = Z exp {−S − S int } D ( χ ∗ , χ ) . (2.4)Herein, the Euclidean action is given by S = X σ Z β Z χ ∗ σ ( r , τ ) × [ ∂ τ + ε ( − i ∇ r ) − µ ] χ σ ( r , τ ) d d r d τ , (2.5) S int = 12 X σσ ′ Z β Z χ ∗ σ ( r , τ ) χ ∗ σ ′ ( r ′ , τ ) χ σ ′ ( r ′ , τ ) χ σ ( r , τ ) × V ( r − r ′ ) d d r d d r ′ d τ . (2.6)In Eqs. (2.4)–(2.6), β = 1 /T is the inverse temperatureand χ , χ ∗ are Grassmann fields which are antiperiodic inimaginary time τ , χ ( τ + β ) = − χ ( τ ).Equations (2.4)–(2.6) are the starting point for ouranalysis. B. Low-lying modes
The spin and charge excitations at low temperatures,which we are interested in, correspond to the low-lying
FIG. 1: Decomposition of the interaction S int , Eq. (2.6),into slow modes, | q | . q ≪ p F . (a), (b), and (c) are theHartree, the Fock, and the Cooper vertices, respectively. Inour model, the Cooper vertex should be omitted to avoiddouble-counting. modes of the microscopic model, Eqs. (2.4)–(2.6). Effec-tively, only those fermions contribute that are energeti-cally close to the Fermi energy ε F . From this constraint,we obtain the three relevant vertices shown in Fig. 1 de-scribing scattering processes with momenta p , p ′ locatednear the Fermi surface. We single out these vertices as-suming that the momenta q are small, | q | . q , where q is a phenomenological momentum cutoff which is muchsmaller than the Fermi momentum p F . Vertices (a) and(b) describe soft interactions in the particle-hole chan-nel, whereas (c) is the Cooper vertex. The well-knownHartree-Fock approximation is obtained using the ver-tices (a) and (b) and, thus, they can be referred to asthe Hartree and Fock vertex, respectively. The Coopervertex enters the ladder diagrams leading to the BCSsuperconducting instability in case of attractive interac-tion.In principle, all three vertices are important whencalculating physical quantities. However, we are hereinterested in the anomalous contributions to the ther-modynamics, which originate from (quasi-)one-dimensional processes. As we will see later, the maincontribution to the anomalous terms comes from small q , which is also seen from the conventional perturba-tion theory. Of course, assuming that all the verticesin Fig. 1 are different forbids the regions of the essentialmomenta attributed to them to overlap.A quick glance, however, reveals that this is not thecase. For example, there is a clear overlap between theregions of the momenta in the vertices (a) and (b) atsmall | p − p ′ | ∼ | q | . In order to avoid double-counting,one would have to consider only one of these vertices inthis region. In Ref. 15, e.g., it was chosen to remove theregion | p − p ′ | ∼ | q | from the Fock vertex. Fortunately,the contribution coming from this region of the momentacan be neglected at the low temperatures considered here.As a result, we can consider the vertices Fig. 1(a) and (b)as practically different vertices for | q | . q ≪ p F .At the same time, the Hartree-Fock (a,b) and theCooper (c) vertices do overlap in important momentumregions. As to the role of the Cooper vertex, buildingladders out of it, logarithmic divergencies appear for ar-bitrary scattering angles d pp ′ . The contributions comingfrom large angles are not reproduced by using the Fockvertices (b) instead. However, scattering angles essen-tially different from 0 or π are less important for theanomalous contributions we wish to calculate. Consid-ering only angles d pp ′ close to 0 or π means focusing onalmost one-dimensional scattering and this is where theanomalous contributions emerge.This region of the momenta attributed to the Coopervertex Fig. 1(c), however, fully overlaps with that forthe Hartree-Fock vertices (a,b). Any diagram containingCooper loops with small angles d pp ′ can be representedin an equivalent way using particle-hole loops built fromvertices (a) and (b). Several examples of this equivalencecan be found in Ref. 15. Therefore, taking into accountall the vertices Fig. 1(a,b) and (c) would imply double-counting. In order to avoid it, one should choose betweeneither the Hartree-Fock vertices or the Cooper ones, butnot take into account all of them.In the present study, we choose as in Ref. 15 theHartree-Fock route. This is in contrast to the approachof Ref. 17 where the Cooper channel representation wasused. Our choice will turn out more convenient for sin-gling out the anomalous contributions.As a result, we write the effective interaction describingthe low energy physics as˜ S int = 12 X P P ′ Q,σσ ′ n χ ∗ σ ( P + Q ) χ σ ( P ) V q χ ∗ σ ′ ( P ′ ) χ σ ′ ( P ′ + Q ) − χ ∗ σ ( P + Q ) χ σ ′ ( P ) V p − p ′ χ ∗ σ ′ ( P ′ ) χ σ ( P ′ + Q ) o (2.7)instead of Eq. (2.6). In Eq. (2.7), the fermionic fields arerepresented in Fourier space and four-momentum nota-tions P = ( ε, p ) , Q = ( ω, q )are used. Herein, ε and ω are fermionic and bosonicMatsubara frequencies, respectively. In the “fermionic”summations P P ( ′ ) ( . . . ) = T P ε ( ′ ) R ( . . . )[d d p ( ′ ) / (2 π ) d ], itis understood that p , p ′ are of order p F . On the con-trary in the “bosonic” summation P Q , we introduce afunction f ( q ) which cuts off the momentum q beyond q ≪ p F , X Q ( . . . ) = T X ω Z ( . . . ) f ( q ) d d q (2 π ) d . (2.8)The function f ( q ) can for instance be modeled as f ( q ) =Θ( q − | q | ) with Θ denoting the Heaviside function. Infinal formulas, we may choose a different form which al-lows to conveniently perform the terminal integrations.For the moment, we do not specify it more than that it isassumed to fulfill f ( q = 0) = 1 and decay fast beyond q . C. Bosonization
Our choice of the effective interaction, Eq. (2.7), leadsafter bosonization to a field theory representing particle-hole -type bosonic excitations of the Fermi gas. The routeto obtain the effective low energy field theory for theseexcitations follows the higher-dimensional bosonizationscheme introduced in Refs. 23 and 24. It is done in fivesteps:1. Decouple the effective interaction employing aHubbard-Stratonovich transformation.2. Integrate out the fermionic degrees of freedom.3. Derive the effective equation of motion for thebosonic field subject to the random Hubbard-Stratonovich auxiliary field.4. Write the solution of this equation in form of afunctional integral over superfields thus obtaininga closed supersymmetric field theory.
5. Average over the auxiliary field.In what follows, each of the steps is presented in a sepa-rate section.
1. Hubbard-Stratonovich transformation
In order to decouple the effective interaction ˜ S int byan integration over the auxiliary Hubbard-Stratonovichfield, we recast the spin structure in the Fock channel bymeans of the relation2 δ σ σ δ σ σ = X µ =0 σ µσ σ σ µσ σ , (2.9)where ( σ µ ) = ( σ , σ ) with σ = and σ = ( σ , σ , σ )is the vector of Pauli matrices. In what follows, we usethe convention that Greek upper indices appearing twiceimply the summation from 0 to 3.We thus obtain˜ S int = 12 X Q n V q n ( Q ) n ( − Q ) − X pp ′ V p − p ′ S µ p ( Q ) S µ p ′ ( − Q ) o (2.10)where S µ p are components of the four-component vector( S p , S p , S p , S p ) and n ( Q ) = X P,σ χ ∗ σ (cid:0) ε, p + q (cid:1) χ σ (cid:0) ε − ω, p − q (cid:1) , (2.11) S µ p ( Q ) = T X ε,σσ ′ χ ∗ σ (cid:0) ε, p + q (cid:1) σ µσσ ′ χ σ ′ (cid:0) ε − ω, p − q (cid:1) . The zero component S p is related to the particle density n ( Q ) as X p S p ( Q ) = n ( Q ) (2.12)whereas the other components form the spin density vec-tor S ( Q ), X p S p ( Q ) = S ( Q ) (2.13)with S p = ( S p , S p , S p ).Assuming that the interaction V ( r ) decays sufficientlyfast, the interaction amplitudes V entering Eq. (2.10) canbe written as V q ≃ V , V p − p ′ ≃ ˜ V (cid:16) p F sin (cid:0) d pp ′ (cid:1)(cid:17) (2.14)with d pp ′ denoting the angle between p and p ′ . InEq. (2.14), V is a positive constant and ˜ V is a func-tion of the angle between the vectors p and p ′ located atthe Fermi surface. As the main contribution will come from momenta p near the Fermi surface and small q ,Eq. (2.14) is a good approximation. For contact interac-tion, ˜ V ≡ V .In principle, we are now ready to decouple the quar-tic interaction, Eq. (2.10), by means of a Hubbard-Stratonovich (HS) transformation. Considering the lowenergy theory, we have not said anything about high en-ergies so far. Actually, one can first integrate out thehigh energy degrees of freedom. Following the philoso-phy of the Landau Fermi liquid theory, integrating outthe high energies results in a renormalization of the origi-nal spectrum, Eq. (2.2), of the interaction, Eq. (2.3), andof the ground state energy. Neglecting fluctuations, onewould obtain a partition function containing the renor-malized constants to be considered as phenomenologicalparameters.As a result of the integration over the high energies,one obtains instead of the quadratic forms nn and S µ S µ written in Eq. (2.10) expressions of the type ( n −h n i )( n −h n i ) and ( S µ − h S µ i )( S µ − h S µ i ), where h . . . i denotesthe averages with respect to the high energy part of theHamiltonian. Applying the HS transformation to thiseffective interaction yieldsexp (cid:8) − ˜ S int (cid:9) = Z exp n i X Q ϕ ( − Q ) (cid:2) n ( Q ) −h n ( Q ) i (cid:3) + X Q X p h µ p ( − Q ) (cid:2) S µ p ( Q ) −h S µ p ( Q ) i (cid:3)o W h [ ϕ ] W f [ h µ ] D ϕ D h µ .(2.15)The Gaussian weights entering Eq. (2.15) are given by W h [ ϕ ] = exp n − V X Q ϕ ( Q ) ϕ ( − Q ) o , (2.16) W f [ h µ ] = exp n − X Q X pp ′ h µ p ( Q ) (cid:2) ˆ V − (cid:3) pp ′ h µ p ′ ( − Q ) o , and D ϕ D h µ is the measure normalized such that R W h [ ϕ ] W f [ h µ ] D ϕ D h µ = 1. The HS fields ϕ ( Q )and h µ ( Q ) depend on bosonic Matsubara frequencies ω ,corresponding in imaginary time representation to peri-odicity, ϕ ( τ + β ; q ) = ϕ ( τ ; q ) and h µ p ( τ + β ; q ) = h µ p ( τ ; q ).The constant V has been introduced in Eq. (2.14) andˆ V − is the inverse of the operator ˆ V p which acts on afunction g ( p ) as[ ˆ V g ]( p ) = X p ′ V p − p ′ g ( p ′ ) . Actually, V p − p ′ is essentially the function ˜ V fromEq. (2.14) for momenta p close to the Fermi surface. Byconstruction, the momentum q in the HS fields ϕ ( Q )and h µ p ( Q ) is restricted by the cutoff q ≪ p F and thecoupling of the HS fields to the fermions displaces thefermion momentum p of order p F only locally.
2. Integration over the fermionic fields
After the HS transformation, Eq. (2.15), the partitionfunction is given by Z = Z Z Z [Φ] W [Φ] D Φ (2.17)where Φ is the 2 × p ( Q ) = i ϕ ( Q ) + h µ p ( Q ) σ µ , (2.18)while W [Φ] = W h [ ϕ ] W f [ h µ ] and D Φ = D ϕ D h µ . Thefunctional Z [Φ] is essentially the formal partition func-tion of non-interacting fermions in the field Φ, Z [Φ] = Z − exp (cid:8) −F [Φ] (cid:9) Z exp (cid:8) − ˜ S [Φ] (cid:9) D ( χ ∗ , χ )(2.19)with the action ˜ S [Φ] given by˜ S [Φ] = S − Z β X pq χ † (cid:0) τ, p − q (cid:1) Φ p ( τ, q ) χ (cid:0) τ, p + q (cid:1) d τ. (2.20)Herein, the bare action S has been defined in Eq. (2.5),the field Φ is written in the imaginary time τ representa-tion, and the spinor notation χ = ( χ ↑ , χ ↓ ) is used. Thefunctional F [Φ] in Eq. (2.19) is given by F [Φ] = Z β X pq D χ † (cid:0) τ, p − q (cid:1) Φ p (cid:0) τ, q (cid:1) χ (cid:0) τ, p + q (cid:1)E d τ (2.21)= Z β X pq tr (cid:2) Φ p ( τ, q ) G [0] p + q , p − q ( τ, τ +0) (cid:3) d τ .Herein, G [0] is the Green’s function of the ideal Fermi gasdescribed by the action S , Eq. (2.5), and tr denotes thetrace over spins.In order to integrate out the fermion fields inEq. (2.19), we follow the route suggested in Refs. 15,23and recast the fermion determinant as det ≡ exp Tr ln.The logarithm is then replaced by an inverse using anadditional integration over a variable u of the formln( x + φ ) = ln x + Z φ ( x + uφ ) − d u .This yields Z [Φ] = exp (cid:8) −F [Φ] (cid:9) (2.22) × exp n Z Z β X pq tr (cid:2) Φ p ( τ, q ) G [ u Φ] p + q , p − q ( τ, τ +0) (cid:3) d τ d u o where G [ u Φ] is the 2 × u Φ, i.e. thepropagator of the action ˜ S [ u Φ], Eq. (2.20). It satisfiesthe equation − (cid:8) ∂ τ + ε (cid:0) p + q (cid:1) − µ − u Φ p ( τ, q ) (cid:9) G [ u Φ] p + q , p − q ( τ, τ ′ )= δ ( τ − τ ′ ) δ q , . (2.23)Thus, in order to study the thermodynamics of the low-lying excitations, we need to determine the Green’s func-tion of non-interacting fermions in a random external HSfield at equal times. This quantity G [ u Φ] ( τ, τ +0) is a com-plex matrix function being periodic in imaginary time τ , G [ u Φ] ( τ + β, τ + β +0) = G [ u Φ] ( τ, τ +0). This observationmotivates the introduction of the complex 2 × A pp ′ = A [ u Φ] pp ′ as A pp ′ ( τ ) = G [0] pp ′ ( τ, τ +0) − G [ u Φ] pp ′ ( τ, τ +0) . (2.24)The field A pp ′ ( τ ) obeys bosonic periodicity in time τ , A pp ′ ( τ + β ) = A pp ′ ( τ ), and captures the entire physicsof the low-lying excitations.By this definition, the functional Z [Φ], Eq. (2.22),takes the form of a functional of the boson field A pp ′ ( τ ),Eq. (2.24), Z [Φ] = exp n − Z Z β X pq tr (cid:2) Φ p ( τ, q ) A p + q , p − q ( τ ) (cid:3) d τ d u o . (2.25)Using Eqs. (2.17) and (2.25) one can compute the parti-tion function Z provided the function A pp ′ ( τ ) is knownfor any configuration of Φ. Of course, one could solveEq. (2.23) for the Green’s function and find A pp ′ ( τ )from Eq. (2.24) but this would lead to the conventionaldiagrammatic expansion for the fermions. Here, we fol-low a different route deriving a closed equation for thefield A pp ′ ( τ ).
3. Dynamics of excitations
The derivation of the equation of motion for the bosonfield A pp ′ follows the route presented in detail in Refs. 23and 24: We start with the differential equations for theGreen’s functions G [0] ( τ, τ ′ ) and G [ u Φ] ( τ, τ ′ ), Eq. (2.23).Subtracting them according to the definition of A pp ′ ,Eq. (2.24), yields a differential equation as an interme-diate result. Next, we repeat the same steps with theequations for G [0] ( τ, τ ′ ) and G [ u Φ] ( τ, τ ′ ) while this timethe operator ∂ τ + ˆ H − u Φ acts from the right. Theequation obtained in this way is then subtracted fromthe intermediate result. Eventually putting τ ′ = τ + 0,we find that the dynamics of A p + k , p − k is governed by theequation h ∂ τ + ( p · k ) m i A p + k , p − k − u X q h Φ p + k + q ( q ) A p + k + q , p − k − A p + k , p − k − q Φ p − k + q ( q ) i = − u Φ p (cid:0) − k (cid:1)(cid:2) n p − k − n p + k (cid:3) . (2.26)Herein, n p = [exp( ξ p /T ) + 1] − with ξ p = ε ( p ) − ε F isthe Fermi distribution function of the ideal Fermi gas.In Eq. (2.26), the relevant momenta k in A p + k , p − k are small and do not exceed the cutoff momentum q ≪ p F . This can easily be seen from Eq. (2.25) because thefields Φ p ( τ, k ) are by construction non-zero only if themomentum k is small. The fact that the two momenta p + k / p − k / A p + k , p − k are essentiallyclose to each other significantly simplifies the analyticalstudy as compared to the general formulation studiedperturbatively in Ref. 24 as a check of the bosonizationprocedure.In order to obtain a feasible low energy theory, thestill present degrees of freedom on the scale p F shouldbe integrated out from Eqs. (2.25) and (2.26). Since theright-hand side of Eq. (2.26) contains the combination n p − k − n p + k , the dependence of the function A p + k , p − k on | p | near p F is for small k sharper than the depen-dence of all other functions entering the left-hand sideof Eq. (2.26) or the exponent in Eq. (2.25). As a result,when integrating both sides of Eq. (2.26) or the integrandin the exponent of Eq. (2.25) over the variable ξ p , it isjustified to approximate p ≃ p F n with n = 1 in all otherfunctions depending smoothly on | p | .Proceeding in this way, we rewrite the exponent in theexpression for Z [Φ], Eq. (2.25), as X pq Φ p ( q ) A p + q , p − q (2.27) ≃ X q Z Φ n ( q ) (cid:18) ν Z A p + q , p − q d ξ p (cid:19) d n , thus separating the radial and angular integration of themomentum vector p = [2 m ( ε F + ξ p )] / n . In Eq. (2.27), ν is the density of states at the Fermi surface and theintegration R d n is done over the ( d − S d − . We normalize the integration over n bythe convention Z S d − d n = 1 , (2.28)and write the function Φ p ( q ) as Φ n ( q ) since p ≃ p F n .Equation (2.27) shows us that we need the integralover ξ p of A p + k , p − k rather than this function itself. Thisobservation motivates us to introduce the quasiclassical field a n ( k ) = ν Z A p + k , p − k d ξ p (2.29) and to construct the low energy theory for this field.For this purpose, we integrate the equation of motionfor A p + k , p − k , Eq. (2.26), in a similar manner term byterm. For the first term of Eq. (2.26) we find ν Z h ∂ τ + ( p · k ) m i A p + k , p − k d ξ p ≃ h ∂ τ + v F ( n · k ) i a n ( k ) , (2.30)where v F is the Fermi velocity.For the first expression in the second term, we obtain ν Z Φ p + k + q ( q ) A p + k + q + q , p − k + q + q d ξ p (2.31) ≃ Φ n + k ⊥ + q ⊥ pF ( q ) a n + q ⊥ pF ( k + q ) ,where k ⊥ is the component of the vector k perpendicularto the vector n , k ⊥ = k − ( n · k ) n . (2.32)In Eq. (2.31), we made the same approximation as inEq. (2.27) using the fact that the field Φ is a slow functionof the variable | p | . The radial integration in Eq. (2.31)absorbs the normal component of q and that is why onlythe tangent component q ⊥ enters the result of the inte-gration. The combination n + q ⊥ / (2 p F ) in Eq. (2.31)for small vectors q ⊥ corresponds to a rotation of n . Ac-counting for these rotations is very important becausethey capture the essential effects of the curvature of theFermi surface arising in d > n p − k − n p + k ≃ m − ( p · k ) δ ( ξ p ). Finally, we re-duce the functional Z [Φ], Eq. (2.17), to the form Z [Φ] = exp n − Z Z β Z X q tr (cid:2) Φ n ( q ) a n ( q ) (cid:3) d n d τ d u o , (2.33)where a n ( k ) is the solution of the equation h ∂ τ + v F ( n · k ) i a n ( k ) − u X q h Φ n + k ⊥ + q ⊥ pF ( q ) a n + q ⊥ pF ( k + q ) − a n − q ⊥ pF ( k + q )Φ n − k ⊥ + q ⊥ pF ( q ) i = − uνv F ( n · k )Φ n ( − k ) .(2.34)Let us discuss the analytical properties of thefield a n ( k ) = a n ( u, τ, k ) and its equation of motionEq. (2.34). By construction, the field a n ( τ, k ) describ-ing the low energy excitations is bosonic: It is a complex2 × a n ( τ + β, k ) = a n ( τ, k ).The momentum k entering the field a , | k | . q ≪ p F , determines the scale of the bosonic excitations while theargument n of the field a n ( k ) determines the position onthe Fermi surface. The dependence of a n ( k ) on n and k contains the full information needed to describe the spinand charge excitations in a higher-dimensional Fermi gas.In dimension d = 1, transverse momenta q ⊥ , k ⊥ donot exist. If we neglected these momenta in higher di-mensions, we would come to the low energy model ofRef. 15. In this approximation, Eq. (2.34) describes one-dimensional processes and all effects of the Fermi surfacecurvature in d > a n ( k ) inthe form a n ( k ) = ̺ n ( k ) σ + s n ( k ) · σ , one can see that,as a result of this approximation, the charge ̺ n ( k ) andspin s n ( k ) parts of the field decouple from each other.Eventually, one obtains a model of non-interacting chargeexcitations ̺ k ( n ) and a non-trivial field theory for the in-teracting spin excitations s n ( k ). All the results of Ref. 15have been obtained in this way, implying their validity inone dimension but requesting reconsideration for d > d >
1, Eq. (2.34) shows that the spin and charge excita-tions cannot be treated separately but interact with eachother. Moreover, the charge modes also interact them-selves similarly to the spin modes. The interaction ofspin and charge modes is what constitutes the major dif-ference between the physics of the fermion gases in d = 1and d >
1. Furthermore, we will see that there are classesof diagrams in a perturbative analysis of the final bosontheory that are logarithmic in d = 1, but due to thepresence of the q ⊥ -terms become regular in dimensions d > a n ( ω, k ) ≃ a (0) n ( ω, k ) with a (0) n ( ω, k ) = νu v F ( n · k )i ω − v F ( n · k ) Φ n ( ω, − k ) . (2.35)Inserting this zero order approximation a (0) n ( ω, k ) intothe functional Z [Φ], Eq. (2.33), we reduce the partitionfunction Z , Eq. (2.17), to a Gaussian integral over thefield Φ = i ϕ + h µ σ µ , Eq. (2.18). It is not difficult to under-stand that this limit yields the contributions obtained bysumming certain ladder series in the conventional fermiondiagrammatics. Considering only the HS field ϕ whileneglecting h µ , we obtain the contribution of the ringsbuilt from polarization bubbles, i.e. reproduce the ran-dom phase approximation (RPA). Alternatively, keepingonly h µ the contribution of the particle-hole ladder ringis reproduced. Keeping both ϕ and h µ , we obtain thecontribution of all particle-hole ring diagrams.However, the interesting logarithmic contributions tothe non-analytic terms arise from the fluctuations on topof these ladders. Considering the ladders as propagatorsof elementary bosonic excitations, one can say that thelogarithmic contributions arise as a result of interactionbetween these excitations. In order to treat the fluctua-tions properly, we need a more efficient route of solvingthe equation of motion for a n ( ω, k ), Eq. (2.34), includingthe Φ-term in the left-hand side.
4. Superfield representation
In this section, we represent the solution of the equa-tion of motion for the bosonic field a n ( k ), Eq. (2.34), foran arbitrary Φ in the form of a functional integral oversuperfields. We begin by noting a remarkable symmetryin the left-hand side of Eq. (2.34).Being linear in a n ( k ), the left-hand side can be for-mally represented as [ L a ] n ( k ). We observe that the oper-ator L is antisymmetric with respect to the inner productgiven by (cid:0) f † , a (cid:1) = Z Z β Z X k tr (cid:2) f † n ( − k ) a n ( k ) (cid:3) d n d τ d u . (2.36)Herein, f † is a field having the same structure like a . Theantisymmetry condition ( f † , L a ) = − ( a, L f † ) is straight-forwardly checked using the definitions of the operator L and the inner product Eq. (2.36). We will see shortlythat this antisymmetry of L leads to an important sim-plification of the theory.Since the remaining of the derivation of the low en-ergy field theory is purely formal, we will use short-handnotations in this section. We define R ≡ R n ( u, τ, k ) = − uνv F ( n · k )Φ n ( − k ) (2.37)as short-hand notation for the right-hand side ofEq. (2.34) and use the notation L introduced above forthe antisymmetric operator in the left-hand side. As aresult, Eq. (2.34) can be written in the compact form L a = R . (2.38)Both the operator L and the inhomogeneity term R de-pend (linearly) on the HS field Φ, Eq. (2.18). WithΦ being a Gauss-distributed random field, Eq. (2.38) istechnically a stochastic differential equation for the bo-son field a n ( u, τ, k ). In the context of stochastic fieldequations, a well-known method of analysis is the Becchi-Rouet-Stora-Tyutin (BRST) transformation whichbrings the problem of solving the stochastic equation intothe form of a supersymmetric field theory. The latter for-mulation allows for a study by means of standard fieldtheory techniques.We now apply the BRST map on our problem. First,we rewrite Eq. (2.33) with a satisfying Eq. (2.34) in aform of a functional integral over fields aZ [Φ] = Z δ [ L a − R ] (cid:12)(cid:12)(cid:12) det δ L δa (cid:12)(cid:12)(cid:12) Z [Φ; a ] D a † D a . (2.39)Herein, the integration with respect to the measure D a † D a is performed over all complex fields a whichdo not necessarily satisfy Eq. (2.34). The functional Z [Φ; a ] in Eq. (2.39) denotes formally the functional fromEq. (2.33), yet the field a is here an unspecified com-plex field and not the solution to the constraint equa-tion Eq. (2.34). The equivalence of the representation byEq. (2.39) and the original one, Eqs. (2.33) and (2.34), for Z [Φ] is easily seen as the constraint (2.34) is enforced bythe integration over the functional δ -function in the inte-grand of Eq. (2.39). The determinant in Eq. (2.39) arisesas a consequence of changing variables from a to L a .Our goal is to integrate the functional Z [Φ], Eq. (2.39),over the fields Φ and obtain a field theory for the inter-acting bosonic excitations. This can be achieved repre-senting the δ -function as a Fourier integral, δ [ L a − R ] (2.40) ∝ Z exp n i2 (cid:0) f † , L a − R (cid:1) + i2 (cid:0) [ L a − R ] † , f (cid:1) o D f † D f , and the determinant as an integral over Grassmann vari-ables,det δ L δa (2.41) ∝ Z exp n i2 (cid:0) ρ † , L σ (cid:1) + i2 (cid:0) [ L σ ] † , ρ (cid:1) o D σ † D σ D ρ † D ρ . In Eqs. (2.40) and (2.41), f is a complex field, while σ and ρ are Grassmann fields of the same structure as a .The brackets ( . . . , . . . ) denote the inner product definedin Eq. (2.36). All the fields in Eqs. (2.40) and (2.41)are assumed to be periodic in imaginary time τ in or-der to reproduce the bosonic boundary condition of thesolution a of Eq. (2.34).Substituting Eqs. (2.40) and (2.41) into Eq. (2.39),we come to the representation of the functional Z [Φ] in terms of a Gaussian integral over the fields a , a † , f , f † , σ , σ † , ρ , ρ † . However, as the functional Z [Φ; a ]contains only the field a and not a † , the integral overthe fields a † , f, σ † , ρ can immediately be calculated giv-ing unity. Then, we are left with an integral only overthe fields a, f † , σ, ρ † .Instead of writing all these fields separately we unifythem into one superfield Ψ which we define asΨ( θ, θ ∗ ) = aθ + f † θ ∗ + σ + ρ † θ ∗ θ . (2.42) θ and θ ∗ are additional Grassmann anticommuting vari-ables. By construction, Ψ is an anticommuting field.This, however, does not mean that it describes fermionsas the periodicity in imaginary time,Ψ ( τ ) = Ψ ( τ + β ) , (2.43)guarantees the boson statistics.The antisymmetry of the operator L , Eq. (2.38), im-plies the remarkable and important relation Z (Ψ , L Ψ) d θ d θ ∗ = 2 (cid:0) f † , L a (cid:1) + 2 (cid:0) ρ † , L σ (cid:1) . (2.44)Using additionally the relations a = − Z Ψ θ ∗ d θ d θ ∗ , f † = Z Ψ θ d θ d θ ∗ , (2.45) we can express the entire field theory solely in terms ofthe superfield Ψ.As a result, we write the partition function Z of thelow energy field theory for the excitation modes of theinteracting Fermi gas, Eq. (2.17), in the form of a func-tional integral over the superfield Ψ and the auxiliaryfield Φ, Z = Z Z exp (cid:8) − S S − S B (cid:9) W [Φ] D Φ D Ψ . (2.46)The action S S with S S = i2 Z h −
12 (Ψ , L Ψ) + (Ψ , R ) θ i d θ d θ ∗ (2.47)is invariant under the BRST symmetry transformationΨ Ψ + δ Ψ with the variation given by δ Ψ = η ∂ Ψ ∂θ ∗ . (2.48)Herein, the transformation parameter η is a Grassmannvariable. The action S B , which derives from the func-tional Z [Φ; a ], takes the form S B = − Z Z β Z X q tr (cid:2) Φ n ( q )Ψ n ( q ) (cid:3) θ ∗ d θ d θ ∗ d n d τ d u .(2.49)In contrast to S S , Eq. (2.47), the action S B is not invari-ant when varying the superfield Ψ according to Eq. (2.48)and, thus, breaks the BRST symmetry.
5. Final form of the low energy model
Since the action S S + S B is linear in the auxiliaryfield Φ, the integral over Φ in Eq. (2.46) is Gaussian andcan easily be performed. This yields the final form ofthe low energy field theory for the bosonic excitations ofthe interacting Fermi gas. All the interesting physics isdescribed by the 2 × Z = Z Z exp (cid:8) − S bare − S − S − S (cid:9) D Ψ . (2.50)Herein, the bare action S bare is in Fourier representationgiven by S bare = − i4 Z X K tr (cid:2) Ψ n ( − K, κ ) (2.51) × {− i ε + v F ( n · k ) } Ψ n ( K, κ ) (cid:3) d n d κ . K = ( ε, k ) , X K ( . . . ) = T X ε Z ( . . . ) d d k (2 π ) d ,δ K, − K ′ = δ ε, − ε ′ T δ k , − k ′ ,Q = ( ω, q ) , X Q ( . . . ) = T X ω Z ( . . . ) d d q (2 π ) d , where ε and ω denote bosonic Matsubara frequencies.Also, we use the following short-hand notations in theremaining of our analysis: κ = ( u, θ, θ ∗ ) , (2.52)d κ = d θ d θ ∗ d u ,δ ( κ − κ ′ ) = δ ( u − u ′ )( θ ∗ − θ ′∗ )( θ − θ ′ ) . Whenever we integrate over u , integration over the inter-val (0 ,
1) is implied.Averaging quadratic forms with respect to the bareaction S bare , Eq. (2.50), is done as D Ψ σ ˜ σ n ( K, κ )Ψ σ ′ ˜ σ ′ n ′ ( K ′ , κ ′ ) E = 2i g n ( K ) (2.53) × δ σ ˜ σ ′ δ ˜ σσ ′ δ ( κ − κ ′ ) δ ( n − n ′ ) δ K, − K ′ . In Eq. (2.53), g n ( K ) = 1i ε − v F ( n · k ) (2.54)is the bare Green’s function for the bosonic modes.Higher moments of the field Ψ are reduced to secondmoments, Eq. (2.53), using Wick’s theorem.Let us now have a look at the interaction vertices inEq. (2.50). The quartic interaction term S is given by S = − ν Z d n d˜ n d κ d˜ κ X K ˜ KQ u ˜ u γ c c n ˜ n f ( q ) (2.55) × tr h Ψ n ( − K, κ )Ψ n + q ⊥ pF (cid:0) K + Q, κ (cid:1)i × tr (cid:20) Ψ ˜ n ( − ˜ K, ˜ κ )Ψ ˜ n − q ˜ ⊥ pF (cid:0) ˜ K − Q, ˜ κ (cid:1)(cid:21) − ν Z d n d˜ n d κ d˜ κ X K ˜ KQ u ˜ u γ s c n ˜ n f ( q ) × tr h Ψ n ( − K, κ ) σ k Ψ n + q ⊥ pF (cid:0) K + Q, κ (cid:1)i × tr (cid:20) Ψ ˜ n ( − ˜ K, ˜ κ ) σ k Ψ ˜ n − q ˜ ⊥ pF (cid:0) ˜ K − Q, ˜ κ (cid:1)(cid:21) , where the vector q ˜ ⊥ is the projection of q onto the planeperpendicular to ˜ n , i.e. q ˜ ⊥ = q − ˜ n (˜ n · q ). The amplitudesfor the spin γ s c n ˜ n and charge γ c c n ˜ n channel are expressed atweak interaction in terms of the interaction potential V ,Eq. (2.14), as γ s c n ˜ n = − ν V (2 p F sin c n ˜ n , γ c c n ˜ n = ν V + γ s c n ˜ n , (2.56) respectively. If the interaction is not weak these ampli-tudes can be considered as effective coupling constantsof the Fermi liquid. The cutoff function f ( q ) introducedfor the soft modes in Eq. (2.8) is from now on written ex-plicitly in the formulas. The cubic interaction S reads S = 12 ν Z d n d˜ n d κ d˜ κ X KQ u γ c c n ˜ n f ( q ) (2.57) × tr h Ψ n ( − K, κ )Ψ n + q ⊥ pF (cid:0) K + Q, κ (cid:1)i × n − ˜ uνv F (˜ n · q )˜ θ + 2i˜ θ ∗ o tr [Ψ ˜ n ( − Q, ˜ κ )]+ 12 ν Z d n d˜ n d κ d˜ κ X KQ u γ s c n ˜ n f ( q ) × tr h Ψ n ( − K, κ ) σ k Ψ n + q ⊥ pF (cid:0) K + Q, κ (cid:1)i × n − ˜ uνv F (˜ n · q )˜ θ + 2i˜ θ ∗ o tr (cid:2) σ k Ψ ˜ n ( − Q, ˜ κ ) (cid:3) and, finally, the quadratic action S takes the form S = − ν Z d n d˜ n d κ d˜ κ X Q γ c c n ˜ n f ( q ) (2.58) × { uνv F ( n · q ) θ + 2i θ ∗ } tr [Ψ n ( Q, κ )] × n − ˜ uνv F (˜ n · q )˜ θ + 2i˜ θ ∗ o tr [Ψ ˜ n ( − Q, ˜ κ )] − ν Z d n d˜ n d κ d˜ κ X Q γ s c n ˜ n f ( q ) × { uνv F ( n · q ) θ + 2i θ ∗ } tr (cid:2) σ k Ψ n ( Q, κ ) (cid:3) × n − ˜ uνv F (˜ n · q )˜ θ + 2i˜ θ ∗ o tr (cid:2) σ k Ψ ˜ n ( − Q, ˜ κ ) (cid:3) . Diagrammatically, perturbative calculations within thelow energy boson model can be conveniently representedusing the building blocks shown in Fig. 2. In the di-agrammatic representation and explicitly in Eqs. (2.55)and (2.58), it is evident that the building blocks consti-tuted by S and S are invariant under vertical reflection.On the other hand, it is important to note that for d > S nor S are symmetric with respect to horizon-tal flipping.The quartic interaction S , Eq. (2.55), is fully ob-tained from the Φ-average of the BRST-symmetric ac-tion S S , Eq. (2.47). Since S inherits this BRST sym-metry, there is no (perturbative) contribution to thermo-dynamics originating purely from S . In contrast, thecubic and quadratic interactions S and S , Eqs. (2.57)and (2.58), are formed using also the symmetry breakingaction S B , Eq. (2.49), when averaging over the auxiliaryfield Φ. That is why diagrams contributing to a phys-ical thermodynamic quantity necessarily contain S or S among their building blocks. For example, consid-ering only the terms S bare + S we reproduce the RPAand particle-hole ladder rings because this is equivalentto neglecting the auxiliary field Φ in the left-hand sideof Eq. (2.34), cf. the discussion at the end of Sec. II C 3.Blocks of S may additionally decorate diagrams built1 FIG. 2: Diagrammatic building blocks of our low energy fieldtheory: (a) the propagator g n ( K ), Eq. (2.54), and (b) theinteraction vertices S , S , and S from Eqs. (2.55), (2.57),and (2.58). from S or S and the contribution may consequentlyacquire logarithmic renormalizations.In summary, Eqs. (2.50)–(2.58) specify our effectivelow energy model for the interacting Fermi gas in d > S , the cubic term S and the quar-tic term S . The bare coupling constants are writtenin Eq. (2.56). In principle, one can immediately startperturbative studies of the model using the contractionrule, Eq. (2.53), and Wick’s theorem. A possible dia-grammatic representation is shown in Fig. 2. Althoughthe effective field theory may look somewhat complex, itallows to conveniently treat the low energy limit, identi-fying the interesting logarithms and summing them. Thisis what the next sections are devoted to. III. PERTURBATION THEORY
The bosonized model, Eqs. (2.50)–(2.58), is not triv-ial and the perturbation theory in the coupling constants γ s c n ˜ n , γ c c n ˜ n , Eq. (2.56), yields logarithmic contributions di-verging in the limit T →
0. In this section, we iden-tify the relevant classes of logarithmic one-loop diagrams.Later in Sec. IV, these logarithmic contributions will besummed up to infinite order by means of a one-loop renor-malization group scheme.In one dimension, such a procedure would essentiallyrepeat the steps from Ref. 15. The peculiarity of higherdimensions, d >
1, appears in form of the “rotations” n + q ⊥ / p F of the angular arguments in the interact-ing superfields, cf. Eqs. (2.55)–(2.58). Consequently, therunning momentum Q in a one-loop diagram affects atthe same time the (actual) momentum K and the di-rection n of the propagators. As a result, we will findthat logarithms which certain classes of diagrams feature FIG. 3: Backscattering diagrams for the thermodynamic po-tential Ω: Diagram (a) is the second order diagram containingthe leading backscattering contribution for n ∼ − ˜ n while di-agram (b) represents an exemplary logarithmic renormaliza-tion to diagram (a), cf. Sec. III C. Finally, diagram (c) givesfor n ∼ − n ∼ − n ∼ n a backscattering contributionof higher order in the interaction and also includes a renor-malizing building block S . For weakly interacting fermions,diagram (c) can be neglected. in d = 1 dimension are suppressed in dimensions d > q ⊥ along the Fermisurface. Eventually, the effects of the finite Fermi sur-face curvature lead to renormalization group equationsdifferent from the ones obtained in one dimension.Before studying the one-loop vertex corrections, we be-gin the perturbative analysis of this section as we dis-cuss the relevant diagrams for the thermodynamic po-tential. These diagrams describe physical backscatteringprocesses .While the boson model, Eqs. (2.50)–(2.58), has beenderived for an arbitrary dimension d , we consider fromnow on the most interesting case of a two-dimensionalFermi liquid, d = 2. A. Backscattering diagrams
In the second order in the interaction, only diagramFig. 3(a) describes a contribution to the thermodynamicpotential Ω relevant for studying the backscattering ef-fects. All other second order diagrams cannot containtwo boson propagators g n ( K ) and g ˜n ( K ′ ) with n ∼ − ˜ n .Figure 3(b) shows an exemplary diagram that renormal-izes the bare diagram Fig. 3(a) while Fig. 3(c) representsa backscattering contribution of higher order in the in-teraction. Considering the limit of weak interaction, weare safe to neglect such higher order diagrams becausethey do only describe high energy renormalizations ofthe coupling constants.Working with the effective low energy theory, we haveto be sure that the main contribution to the physicalquantities of interest indeed comes from the low energiesnot exceeding T . Whether this is the case or not, itshould be checked for each quantity under investigation.In fact, the low energy contributions are not the most2important for a perturbative correction ∆Ω ( T ) to thethermodynamic potential and, thus, we cannot compute∆Ω ( T ) using the low energy limit only. However, themain contribution to the difference δ Ω( T ) = ∆Ω( T ) − ∆Ω( T = 0) (3.1)does come from the low energies. In order to determinesuch quantities as the specific heat, the quantity δ Ω ( T )contains all the necessary information and the low energybosonized model, Eqs. (2.50)–(2.58), becomes useful.Formally, the quantity ∆Ω ( T ) will be representedin terms of sums over Matsubara frequencies such as T P ω n ψ ( ω n ). The corresponding expression for δ Ω( T )consequently takes the form T X ω n ψ ( ω n ) − Z ψ ( ω ) d ω π ≃ X l =0 Z ψ ( ω )e − i ωl/T d ω π . (3.2)Equation (3.2) follows from the Poisson summation for-mula. It shows that, when calculating δ Ω ( T ), essential ω in the function ψ ( ω ) are of order T provided the function ψ ( ω ) decays sufficiently at | ω | → ∞ .Using the developed formalism, we can start the cal-culation of thermodynamic quantities. As a first exam-ple, we are going to compute the correction δ Ω (2) ( T ) inthe second order in the coupling constants γ s c n ˜ n and γ c c n ˜ n .Logarithmic contributions are taken into account later byreplacing the bare coupling constants γ s c n ˜ n and γ c c n ˜ n witheffective amplitudes obtained from the summation of log-arithmic contributions. This computational procedure isjustified by the fact that logarithmic contributions comefrom energies exceeding T with the logarithms cut frombelow by max(2 πT, v F q | n + ˜ n | ). Therefore with loga-rithmic accuracy, one may replace the energies in the ef-fective amplitudes by the temperature T and treat themas constants when calculating Matsubara sums.The second order contribution is given by the dia-gram Fig. 3(a). Analytically, we obtain for ∆Ω (2) ( T ) = − ( T / (cid:10) S (cid:11) the expression∆Ω (2) ( T ) = − T ν Z d n d˜ n d κ d˜ κ X Q f ( q ) × n (cid:2) γ c c n ˜ n (cid:3) + (cid:2) γ s c n ˜ n (cid:3) tr[ σ k σ l ]tr[ σ k σ l ] o × { uνv F ( n · q ) θ + 2i θ ∗ } {− uνv F ( n · q ) θ + 2i θ ∗ }× n − ˜ uνv F (˜ n · q )˜ θ + 2i˜ θ ∗ o n ˜ uνv F (˜ n · q )˜ θ + 2i˜ θ ∗ o × (2i) T g n ( Q ) g ˜ n ( − Q ) , (3.3)where g n ( Q ) is the bosonic Green’s function, Eq. (2.54).Performing the remaining integrations over the Grass-mann variables, evaluating the spin traces, and usingEq. (3.1), we obtain the relevant low energy second order correction δ Ω (2) ( T ) as δ Ω (2) ( T ) = T X ω − Z d ω π ! Z d q (2 π ) d n d˜ n f ( q ) × γ c n ˜ n v F ( n · q ) v F (˜ n · q )[i ω − v F ( n · q )][ − i ω + v F (˜ n · q )] . (3.4)The scattering amplitudes for the charge and spin chan-nels, γ c c n ˜ n and γ s c n ˜ n , enter Eq. (3.4) as γ c n ˜ n = 4([ γ c c n ˜ n ] + 3[ γ s c n ˜ n ] ) . (3.5)The evaluation of the integral in Eq. (3.4) follows withsome minor deviations the steps of a similar calculationin Ref. 15. This calculation is not trivial and we presenta possible route of how to carry it out in Appendix A.As a result, we find δ Ω (2) ( T ) = ζ (3) πv F γ π T . (3.6)By virtue of the thermodynamic relation δc = − T ∂ δ Ω /∂T , the anomalous correction to the specificheat δc is in the second order in the interaction obtainedas δc (2) ( T ) = − ζ (3) πv F γ π T . (3.7)We see that only the backscattering amplitude [ c n ˜ n = π ]enters δc (2) ( T ), Eq. (3.7).Equation (3.7) gives the well-known anomalous lowestorder specific heat contribution. It is quadratic in T ,which contrasts what one would expect from the Som-merfeld expansion for the Fermi gas of weakly-interactingquasiparticles. Thus, Eq. (3.7) confirms the equivalenceof the perturbative calculations in both the conventionalapproach and the bosonization one which we are study-ing here. In the remaining of this paper, we investigatethe logarithmic renormalizations to the backscatteringcontribution δc (2) ( T ), Eq. (3.7), and thus refine this in-termediate result. B. One-loop corrections to S The second order result Eq. (3.6) for the thermody-namic potential cannot provide the full qualitative pic-ture of the non-analytic corrections because logarithmiccontributions arise in higher orders in the coupling con-stants. At sufficiently low temperatures, they becomelarge for an arbitrarily weak interaction. In this and thenext section, we study the logarithmic divergencies in theleading one-loop order.Considering first the quartic action S , Eq. (2.55), theone-loop order of the expansion in the coupling constantsyields the diagrams shown in Fig. 4. We want to showthat only diagram (a) is important and leads to the loga-rithmic divergency in the limit T → n → − ˜n , whereas3the contribution of the other diagrams remains finite inthis limit and does not contain large logarithms. For thispurpose, we need to focus on the momentum structureonly while the spin structure present in S in the γ s -termhas nothing to say about the existence of a logarithmicdivergency. Therefore for the sake of a simpler presenta-tion, we only consider the γ c -term for the moment.The logarithmic contributions come from runningbosonic frequencies with | ω | ≫ πT , which are in the fo-cus of the following considerations. Contributions fromthe region | ω | . πT produce terms of higher order in γ c and, therefore, only give perturbative corrections to thecoefficients in front of a large logarithm.The standard diagrammatic technique based on thecontraction rule, Eq. (2.53), yields for diagram Fig. 4(a)the one-loop vertex correction δ S (a)4 = − ν Z d n d˜ n d κ d˜ κu ˜ u X KQ Q − δγ c c n ˜ n × tr (cid:20) Ψ n + q , ⊥ pF ( − K − Q , κ )Ψ n + q , ⊥ pF (cid:0) K + Q , κ (cid:1)(cid:21) × tr (cid:20) Ψ ˜ n − q , ˜ ⊥ pF ( − K + Q , ˜ κ )Ψ ˜ n − q , ˜ ⊥ pF (cid:0) K − Q , ˜ κ (cid:1)(cid:21) . (3.8)This vertex δ S (a)4 reproduces the analytical form of thequartic action S , Eq. (2.55), where a correction − δγ c c n ˜ n / γ c c n ˜ n f ( q − q ).This correction — a function of u , ˜ u , n , ˜ n , q − q , and K — is determined by the integral over the running four-momentum.We find δγ c c n ˜ n = 4 u ˜ uν [ γ c c n ˜ n ] X Q f ( q − q ) f ( q − q ) × g n + q ⊥ pF (cid:0) K + Q (cid:1) g ˜ n − q ˜ ⊥ pF (cid:0) K − Q (cid:1) . (3.9)This integral is conveniently calculated introducing theangular variables ¯ n = 12 ( n − ˜ n ) , (3.10) δ n = 12 ( n + ˜ n ) , and their projections of the momentum vector q ,¯ q k = (¯ n · q ) , (3.11)¯ q ⊥ = q − ¯ q k ¯ n . Calculating the integral in Eq. (3.9) with logarithmic ac-curacy and keeping in mind that the essential K in thefinal integration, e.g. in the diagram in Fig. 3(b), will beof order T , which is the lower cutoff of the logarithms,we can safely put K = 0 in Eq. (3.9). Thus, δγ c c n ˜ n can be written as δγ c c n ˜ n = 4 u ˜ uν [ γ c c n ˜ n ] Z d¯ q ⊥ π f (¯ q ⊥ − q ) f ( q − ¯ q ⊥ ) × T X ω Z d¯ q k π ω − v F ¯ q k − v F ( δ n · ¯ q ⊥ ) − m ¯ q ⊥ × − i ω − v F ¯ q k + v F ( δ n · ¯ q ⊥ ) − m ¯ q ⊥ . (3.12)As to the radial component ¯ q k , it will be sufficient toknow the cutoff’s order of magnitude while the preciseform of the cutoff function f ( q ) is irrelevant to it. Asa result, it is justified to restrict its dependence to thetransverse momenta ¯ q ⊥ — as has been done in Eq. (3.12)— while keeping in mind that v F ¯ q k . Λ with Λ being theupper boundary of the bosonic spectrum.Integrating over the radial component ¯ q k , we find δγ c c n ˜ n = [ γ c c n ˜ n ] Z d¯ q ⊥ q f (¯ q ⊥ − q ) f ( q − ¯ q ⊥ ) λ c n ˜ n (3.13)with λ c n ˜ n = 4 u ˜ uνv F q π Z Λ2 πT d ω π ωω − [ v F ( δ n · ¯ q ⊥ )] . (3.14)Remarkably after the integration over ¯ q k , theterm ¯ q ⊥ / (2 m ), which is of order Λ and, thus, inprinciple large, has dropped out. Therefore, the subse-quent integration over the frequencies ω leads in the limit T, | δ n | → d = 1 or d > any dimension d . The term v F ( δ n · ¯ q ⊥ ) ∼ v F q | δ n | ,though being dependent on the transverse momenta,vanishes in the limit δ n → d . Thishowever is exactly the limit in which λ c n ˜ n eventuallyenters physical quantities such as the thermodynamicpotential δ Ω, Eq. (3.6). Explicitly, we find λ c n ˜ n = 4 u ˜ u ν ∗ ν ln (cid:18) Λmax(2 πT, v F q | δ n | ) (cid:19) (3.15)with the constant ν ∗ defined as ν ∗ = 12 π q v F . (3.16)By construction, the transverse momentum only varieson a small arc with a length of order 2 q ≪ p F on theFermi circle, but in final results this arc should extend toa semicircle, corresponding to q ∼ ( π/ p F or ν ∗ ∼ ν/ q ⊥ in Eq. (3.13) remains tobe done. Since v F ¯ q k ∼ ¯ q ⊥ / (2 m ) ∼ Λ in the region ofthe logarithm, the transverse momenta are much largerthan the parallel ones, ¯ q ⊥ ≫ ¯ q k . As this statement re-mains true for the leading terms in all logarithmic orders,4 FIG. 4: One-loop diagrams for the quartic action S : Diagram (a) is logarithmic independently from the dimensionality of thesystem whereas diagrams (b) and (c) are negligible in dimensions d > q k -integration for any d and, finally, diagram (f) vanishes due to supersymmetry. The contributions of diagrams (a’)and (a”), each of which is logarithmic on its own, cancel each other. we may neglect in the relevant limit of δ n → q , k in the cutoff functions in Eq. (3.13).Then, the remaining integral becomes Z d¯ q ⊥ q f (¯ q ⊥ − q , ⊥ ) f ( q , ⊥ − ¯ q ⊥ ) . (3.17)This is nothing but a convolution [ f ∗ f ]( q , ⊥ − q , ⊥ ), whichbecomes a product after employing a Fourier transforma-tion, f ( r ⊥ ) = Z f ( q ⊥ ) e i r ⊥ q ⊥ /q d q ⊥ q . (3.18)The form of the Fourier transform in Eq. (3.18) has beenchosen such that both r ⊥ and f ( r ⊥ ) are dimensionless.The value r ⊥ = 1 corresponds to the minimal length ofthe theory, which is given by 1 /q . In Fourier represen-tation, the vertex correction δγ c c n ˜ n takes the final form δγ c c n ˜ n ( r ⊥ ) = [ γ c c n ˜ n f ( r ⊥ )] λ c n ˜ n (3.19)with the function λ c n ˜ n , Eq. (3.15), containing the loga-rithm. From Eq. (3.19), we understand that it is actuallythe quantity γ c c n ˜ n f ( r ⊥ ) which flows during a renormaliza-tion procedure.In conclusion, the diagram Fig. 4(a) logarithmicallyrenormalizes the backward scattering amplitude γ cπ ofthe quartic action S and this logarithmic contributioncomes independently of the dimension d . Diagram (a)corresponds in conventional fermion diagrammatics to arung of the particle-particle ladder. We discuss this cor-respondence in Appendix C.Let us now turn our attention to diagram Fig. 4(b).After the integration over the internal momenta and fre-quencies, this diagram also reproduces the structure of S but in contrast to diagram (a), it is not logarithmic fordimensions d > δ S (b)4 , which has the sameform as δ S (a)4 , Eq. (3.8), except that λ c n ˜ n , Eq. (3.14), is replaced by the function λ (b) c n ˜ n ∝ X Q g n + q ⊥ pF (cid:0) K + Q (cid:1) (3.20) × g ˜ n + q ˜ ⊥ pF − q , ˜ ⊥ + q , ˜ ⊥ pF (cid:0) K + Q − ( Q + Q ) (cid:1) . In order to estimate the integral over Q in Eq. (3.20),we put all external momenta and frequencies equal tozero. Also, the precise form of the cutoff functions f ( q )is not needed for this estimate and therefore, we do notwrite them here for simplicity. Then, in the frame ofthe angular coordinates from Eqs. (3.10) and (3.11) weobtain λ (b) c n ˜ n ∝ X Q g n + q ⊥ pF (cid:0) Q (cid:1) g ˜ n + q ˜ ⊥ pF (cid:0) Q (cid:1) (3.21) ≃ T X ω Z d¯ q k π d¯ q ⊥ π ω − v F ¯ q k − m ¯ q ⊥ ω + v F ¯ q k − m ¯ q ⊥ . Similar to the case of diagram Fig. 4(a), the unit vec-tors n and ˜ n need to be close to anticollinearity if wewant to achieve the largest value of the integral. Passingfrom the first to the second line in Eq. (3.21), this hasalready been assumed. Integrating over ¯ q k yields λ (b) c n ˜ n ∝ T X ω Z | ω | + i sgn ω m ¯ q ⊥ d¯ q ⊥ π . (3.22)This intermediate result for λ (b) c n ˜ n already demonstrateswhat makes diagram Fig. 4(b) essentially different fromdiagram (a): Here, the transverse term ¯ q ⊥ / (2 m ) doesnot drop out. Moreover, nothing prevents the momen-tum ¯ q from being large and the energy ¯ q ⊥ / (2 m ) fromtaking values of the order of the cutoff Λ. As a re-sult, logarithms analogous to those that appeared fromthe diagram Fig. 4(a) are suppressed by the presence ofthe transverse term. Since the latter exists for dimen-sions d > d > d = 1, butnot in higher dimensions d > n in both the internal propagatorsare necessarily close to each other. Therefore, the in-tegration contour for ¯ q k can be closed without residuesinside and the integral equals zero. Diagram (f) containsa closed loop of bosonic propagators. Since this diagram-matic substructure is fully supersymmetric in the senseof Eq. (2.48), the contribution of the entire diagram van-ishes.For the study of the remaining diagrams, the discus-sions of the diagrams Fig. 4(a) and (b) can rather easilybe extended. The reasons why a logarithmic divergencyappears in diagrams Fig. 4(a’) and (a”) while it is sup-pressed in diagram (c) in dimensions d > S forfixed external momenta. Once more neglecting the cutofffunctions for the moment, we obtain for (a’) and (c) theanalytical expressions δ S (a’)4 = − ν Z d n d˜ n d κ d˜ κ X KQ Q u ˜ u Z d q ˜ ⊥ πq − (cid:2) γ c c n ˜ n (cid:3) λ c n ˜ n × tr (cid:20) Ψ n + q , ⊥ pF ( − K − Q , κ )Ψ n − q , ⊥ pF + q ˜ ⊥ pF (cid:0) K + Q , κ (cid:1)(cid:21) × tr (cid:20) Ψ ˜ n − q , ˜ ⊥ pF ( − K + Q , ˜ κ )Ψ ˜ n + q , ˜ ⊥ pF − q ˜ ⊥ pF (cid:0) K − Q , ˜ κ (cid:1)(cid:21) (3.23)and δ S (c)4 = − ν Z d n d˜ n d κ d˜ κ X KQ Q u ˜ u Z d q ˜ ⊥ πq − (cid:2) γ c c n ˜ n (cid:3) λ c n ˜ n × tr (cid:20) Ψ n + q , ⊥ pF ( − K − Q , κ )Ψ n − q , ⊥ pF (cid:0) K + Q , κ (cid:1)(cid:21) × tr (cid:20) Ψ ˜ n − q , ˜ ⊥ pF ( − K + Q , ˜ κ )Ψ ˜ n + q , ˜ ⊥ pF − q ˜ ⊥ pF (cid:0) K − Q , ˜ κ (cid:1)(cid:21) . (3.24)The contributions δ S (a’)4 and δ S (c)4 , Eqs. (3.23)and (3.24), contain the logarithmic function λ c n ˜ n ,Eq. (3.15). This function λ c n ˜ n is sensitive to deviationsof n from − ˜ n with n ∼ − ˜ n being the only region of im- portance in our consideration. The structure of the ac-tion S is formally not reproduced by δ S (a’)4 and δ S (c)4 be-cause of the presence of the momentum q ˜ ⊥ in Eqs. (3.23)and (3.24).However, one can easily see that the momentum q ˜ ⊥ enters the vertex corrections δ S (a’)4 and δ S (c)4 quite dif-ferently. As concerns δ S (a’)4 , the additional rotationof the vectors n and ˜n represented by q ˜ ⊥ does notchange the direction of n and ˜ n with respect to eachother. In other words, if n = − ˜ n , then n + q ˜ ⊥ /p F = − [˜ n − q ˜ ⊥ /p F ] + O [( q ˜ ⊥ /p F ) ]. This makes the presenceof the momentum q ˜ ⊥ in δ S (a’)4 unimportant. One shouldsimply keep in mind that eventually we are to calculatethe thermodynamic potential correction δ Ω. Here, δ S (a’)4 can enter in the leading order correction as in Fig. 3(b)or be a part of a larger ladder containing other S -blocks.It will turn out that the terms including q ˜ ⊥ drop out af-ter the integration over the parallel momenta ¯ q k in theadditional loops. Formally, it is therefore legitimate insuch diagrams to simply put q ˜ ⊥ = 0, thus making thediagram (a’) give the same logarithmic contribution as δ S (a)4 , Eq. (3.8).In contrast to δ S (a’)4 , the vertex δ S (c)4 is entered by q ˜ ⊥ in an asymmetric way changing the mutual direction of n and ˜ n . As a consequence, if we attach another S -block tothe right of δ S (c)4 , a curvature term of order Λ containing q ˜ ⊥ will necessarily survive the parallel momentum inte-gration, resulting similarly to the scenario of diagram (b)in the cancellation of the otherwise emerging logarithm.Inserting δ S (c)4 into the perturbation series for the ther-modynamic potential, Fig. 3(b), smears — due to thepresence of q ˜ ⊥ in one of the propagators — the impor-tant region around n = − ˜ n . The expression of the formEq. (3.4) in Sec. III A will accordingly be no longer suffi-ciently sensitive to the backscattering limit, thus result-ing in the suppression of all backscattering logarithms.For these reasons, the vertex δ S (c)4 can be excluded fromthe class of the important one-loop diagrams.The final one-loop contribution to be dealt with is thediagram Fig. 4(a”). Its evaluation at small external mo-menta yields the exactly same analytical form as δ S (a’) ,Eq. (3.23), but due to a necessary transposition of theGrassmann fields with opposite sign , δ S (a”)4 = − δ S (a’)4 . (3.25)Thus renormalizing the quartic action using the renor-malization group, diagrams (a’) and (a”) cancel eachother. One straightforwardly checks that this cancella-tion still prevails when we consider both charge and spinchannel of the quartic action S , Eq. (2.55), at the sametime.As a result, the first-loop analysis of the quartic ac-tion S clearly demonstrates the existence of logarith-mic divergencies arising due to the interaction of the col-lective excitations of the Fermi gas. These divergencies6 FIG. 5: Logarithmic one-loop corrections to the quadraticaction S , Eq. (2.58).FIG. 6: Diagrams (a) and (a’) represent the logarithmic one-loop corrections to the cubic action S , Eq. (2.57). originate from just one of the various one-loop diagramsshown Fig. 4, namely diagram (a). C. One-loop corrections to S and S In the previous section, we have identified the relevantlogarithmic one-loop corrections to the quartic interac-tion S . In a general leading logarithmic diagram of or-der ln n (Λ /T ) with n being a large integer, nearly all loga-rithmic factors will be due to S S -loops. The quadraticand cubic parts S and S of the action, Eqs. (2.58)and (2.57), that are needed to break the BRST sym-metry in diagrams for a thermodynamic quantity, serveas the “abutments” of the big S S -loop structure.In principle, the analysis of the one-loop diagrams forthe quadratic and cubic vertices is very similar to thatperformed for S in Sec. III B. Let us begin with thequadratic action S : One-loop corrections to S comefrom diagrams built from S S and S S . As in therenormalization of the quartic action, most diagrams arenegligible since they vanish due to supersymmetry [likediagram Fig. 4(f)], as a result of integration over ¯ q k incases when the integrand is an odd function of ¯ q k [like di-agram Fig. 4(d)], or due to higher-dimensional curvatureeffects as in the case of diagrams Fig. 4(b) and (c). Theonly diagrams yielding in fact a logarithmic contributionproportional to λ c n ˜ n , Eq. (3.15), in dimensions d > S appear in form of theone-loop diagrams built from S S . Logarithmic renor-malizations come from the diagrams shown in Fig. 6(a)and (a’).One-loop diagrams different from these two do not con-tribute logarithmically. This is once more a consequenceof symmetry aspects and curvature effects. Discussingthe cubic one-loop vertices, one should also bear in mindthat they finally affect the thermodynamic potential onlyvia the effective quadratic vertex in Fig. 5(a). Since thearguments follow the same reasoning as for the quartic interaction in Sec. III B, we refrain from an explicit dis-cussion. IV. RENORMALIZATION GROUP
With the perturbative analysis from the preceding sec-tion, all the relevant logarithmic one-loop diagrams areat hand and we are ready to apply a one-loop renormal-ization group (RG) scheme. At the end of the day, theenergy scales above T will be integrated out of the fieldtheory and we will obtain the specific heat in terms of thebasic backscattering diagram Fig. 3(a) with renormalizedcoupling constants.It is important to mention here that the renormaliza-tion of the quartic term S differs from the renormaliza-tions of S and S . The former can be obtained both us-ing the RG scheme and summing ladder diagrams, whilethe latter ones do not allow for a study based on simplesummations of ladder diagrams. A. Generalized action
The actions S , S , and S , Eqs. (2.55)–(2.58), con-tain various subterms which in general have a specificflow behavior under the RG action. In order to facili-tate the RG procedure, we generalize the action a prioriand introduce proper coupling constants. As a result, wewrite the quartic interaction as S = − ν Z d n d˜ n d κ d˜ κ X K ˜ KQ u ˜ u Γ c (¯ q ⊥ ) (4.1) × tr h Ψ n ( − K, κ )Ψ n + q ⊥ pF (cid:0) K + Q, κ (cid:1)i × tr (cid:20) Ψ ˜ n ( − ˜ K, ˜ κ )Ψ ˜ n − q ˜ ⊥ pF (cid:0) ˜ K − Q, ˜ κ (cid:1)(cid:21) − ν Z d n d˜ n d κ d˜ κ X K ˜ KQ u ˜ u Γ s (¯ q ⊥ ) × tr h Ψ n ( − K, κ ) σ k Ψ n + q ⊥ pF (cid:0) K + Q, κ (cid:1)i × tr (cid:20) Ψ ˜ n ( − ˜ K, ˜ κ ) σ k Ψ ˜ n − q ˜ ⊥ pF (cid:0) ˜ K − Q, ˜ κ (cid:1)(cid:21) , S = 12 ν Z d n d˜ n d κ d˜ κ X KQ u (4.2) × tr h Ψ n ( − K, κ )Ψ n + q ⊥ pF (cid:0) K + Q, κ (cid:1)i × n ˜ uνv F ˜ θ δ n · B c ⊥ (¯ q ⊥ ) + 2i˜ θ ∗ B c (¯ q ⊥ ) o tr [Ψ ˜ n ( − Q, ˜ κ )]+ 12 ν Z d n d˜ n d κ d˜ κ X KQ u × tr h Ψ n ( − K, κ ) σ k Ψ n + q ⊥ pF (cid:0) K + Q, κ (cid:1)i × n ˜ uνv F ˜ θ δ n · B s ⊥ (¯ q ⊥ ) + 2i˜ θ ∗ B s (¯ q ⊥ ) o tr (cid:2) σ k Ψ ˜ n ( − Q, ˜ κ ) (cid:3) ,and the quadratic action as S = − ν Z d n d˜ n d κ d˜ κ X Q (4.3) × tr [Ψ n ( Q, κ )] tr [ − Ψ ˜ n ( − Q, ˜ κ )] × n (2i θ ∗ )(2i˜ θ ∗ ) ∆ c (¯ q ⊥ ) + 2(2i θ ∗ )(˜ uνv F ˜ θ δ n ) ∆ c ⊥ (¯ q ⊥ )+ ( uνv F θ δ n )(˜ uνv F ˜ θ δ n ) ∆ c ⊥⊥ (¯ q ⊥ ) o − ν Z d n d˜ n d κ d˜ κ X Q × tr (cid:2) σ k Ψ n ( Q, κ ) (cid:3) tr (cid:2) − σ k Ψ ˜ n ( − Q, ˜ κ ) (cid:3) × n (2i θ ∗ )(2i˜ θ ∗ ) ∆ s (¯ q ⊥ ) + 2(2i θ ∗ )(˜ uνv F ˜ θ δ n ) ∆ s ⊥ (¯ q ⊥ )+ ( uνv F θ δ n )(˜ uνv F ˜ θ δ n ) ∆ s ⊥⊥ (¯ q ⊥ ) o .Equations (4.1)–(4.3) reduce to the original formulation,Eqs. (2.55)–(2.58), if we insert the bare values for thecoupling constants, which are given byΓ c/s (¯ q ⊥ ) (cid:12)(cid:12)(cid:12) = B c/s (¯ q ⊥ ) (cid:12)(cid:12)(cid:12) = ∆ c/s (¯ q ⊥ ) (cid:12)(cid:12)(cid:12) = γ c/s c n ˜ n f (¯ q ⊥ ) ,B c/s ⊥ (¯ q ⊥ ) (cid:12)(cid:12)(cid:12) = ∆ c/s ⊥ (¯ q ⊥ ) (cid:12)(cid:12)(cid:12) = − ¯ q ⊥ γ c/s c n ˜ n f (¯ q ⊥ ) ,∆ c/s ⊥⊥ (¯ q ⊥ ) (cid:12)(cid:12)(cid:12) = − ¯ q ⊥ γ c/s c n ˜ n f (¯ q ⊥ ) .(4.4)The notations for the (one-dimensional) angular vari-able δ n and the transverse momentum ¯ q ⊥ are taken fromEqs. (3.10) and (3.11). Since the coupling constants even-tually enter only at backscattering, c n ˜ n = π or δ n → q k in boththe cutoff functions and the prefactors have been omit-ted. The former is justified according to the discussionpreceding Eq. (3.17). In the prefactors ( n · q ) of the origi-nal actions S and S , Eqs. (2.57) and (2.58), the parallelmomenta ¯ q k are irrelevant for the non-analyticities fol-lowing the discussions after Eq. (A5) and in Appendix B.This justifies the latter. According to Eq. (4.4), the cutoff functions f (¯ q ⊥ ) areabsorbed into the coupling constants. This is a conve-nient definition since we have seen in Eq. (3.19) that thequantities flowing with the RG are γ c/s c n ˜ n f (¯ q ⊥ ) rather thanthe interaction constants γ c/s c n ˜ n themselves. The couplingconstants with index “ ⊥ ” also contain the transverse mo-mentum ¯ q ⊥ as prefactors. As the flowing coupling con-stants in Eq. (4.4) are functions of ¯ q ⊥ , we are formallyapplying a functional RG procedure. B. Renormalization group equations
We develop an RG scheme using the momentum shellintegration. In one RG step, the large energy cutoff Λfor the one-dimensional parallel spectrum v F ¯ q k (and theMatsubara frequencies) is reduced to a still large butmuch smaller cutoff Λ ′ ≪ Λ by integrating out thefields with parallel momenta of orders between Λ ′ /v F and Λ /v F . This yields an action at the energy scale Λ ′ with renormalized coupling constants. Repeatedly ap-plied RG steps make the coupling constants flow. ThisRG flow stops at the latest as soon as the cutoff ap-proaches the order of the temperature T .In this work, we study the RG flow of the couplingconstants in Eq. (4.4), which comprise the physics ofthe anomalous low energy behavior of a two-dimensionalFermi liquid, at the leading one-loop order. This corre-sponds to a summation of all orders of ln(Λ /T ) at leadingorder in γ c/sπ . The relevant one-loop diagrams have beencompletely identified in Sec. III B.
1. RG equations for S Equations (3.8), (3.15), and (3.19) determine how thecoupling constant Γ c (¯ q ⊥ ) in the quartic interaction S ,Eq. (4.1), or its Fourier transform Γ c ( r ⊥ ), cf. Eq. (3.18),is renormalized if the energy cutoff Λ is reduced to Λ /b ≪ Λ. Defining d ξ = 4 u ˜ uν ∗ ν ln b , (4.5)the correction dΓ c ( r ⊥ ) to the coupling constant Γ c ( r ⊥ )in one RG step would bedΓ c ( r ⊥ ) = − (cid:2) Γ c ( r ⊥ ) (cid:3) d ξ (4.6)if we could neglect the spin channel Γ s ( r ⊥ ).Including the spin channel, we have to examine thespin structure of the one-loop diagram Fig. 4(a). Equa-tion (4.6) is the result of attributing both S -blocks to Γ c .Replacing in one of these blocks Γ c by Γ s , reproducesthe spin structure of the spin Γ s -vertex. If both blocksbelong to the spin channel, the algebra of the Pauli ma-trices allocates renormalizations to both the Γ c and the8Γ s vertices. Then, Eq. (4.6) should be replaced by theRG equations which are written asdΓ c ( r ⊥ )d ξ = − (cid:8)(cid:2) Γ c ( r ⊥ ) (cid:3) + 3 (cid:2) Γ s ( r ⊥ ) (cid:3) (cid:9) , (4.7)dΓ s ( r ⊥ )d ξ = − (cid:8) c ( r ⊥ ) Γ s ( r ⊥ ) − (cid:2) Γ s ( r ⊥ ) (cid:3) (cid:9) .This system of two differential equations is decoupled forthe linear combinationsΓ I ( r ⊥ ) = Γ c ( r ⊥ ) − s ( r ⊥ ) , (4.8)Γ II ( r ⊥ ) = Γ c ( r ⊥ ) + Γ s ( r ⊥ ) ,whose bare values are given byΓ I ( r ⊥ ) f ( r ⊥ ) (cid:12)(cid:12)(cid:12) ξ =0 = γ I c n ˜ n = γ c c n ˜ n − γ s c n ˜ n , (4.9)Γ II ( r ⊥ ) f ( r ⊥ ) (cid:12)(cid:12)(cid:12) ξ =0 = γ II c n ˜ n = γ c c n ˜ n + γ s c n ˜ n .In terms of Γ I / II ( r ⊥ ), the RG equations (4.7) take theform dΓ I / II ( r ⊥ )d ξ = − (cid:2) Γ I / II ( r ⊥ ) (cid:3) . (4.10)Equation (4.10) with Eq. (4.9) as boundary condition iseasily solved, yieldingΓ I ( r ⊥ ; ξ ) = γ I c n ˜ n f ( r ⊥ )1 + γ I c n ˜ n f ( r ⊥ ) ξ , (4.11)Γ II ( r ⊥ ; ξ ) = γ II c n ˜ n f ( r ⊥ )1 + γ II c n ˜ n f ( r ⊥ ) ξ ,where in the relevant backscattering limit δ n →
0, thequantity ξ varies between 0 and 4 u ˜ u ( ν ∗ /ν ) ln(Λ /T ).The renormalized coupling constants Γ I / II , Eq. (4.11),can also be obtained summing the relevant ladder dia-grams built of the quartic vertices. Such a ladder is con-structed adding rungs of S -blocks one by one in the waycorresponding to the diagram in Fig. 4(a). Consideringthe correspondence of the boson diagrams to the conven-tional fermion ones as discussed in Appendix C, one cansee that these ladders are related to the usual particle-particle Cooper ladders. The evaluation of the arisinggeometric series is depicted in form of diagrammaticalBethe-Salpether equations in Fig. 7. The two equationsdecouple in a complete analogy with Eq. (4.8) and even-tually yield the same result Eq. (4.11) as obtained usingthe RG equations.The correspondence between the ladder form of therenormalized quartic vertex and conventional Cooperladders suggests that the logarithmic renormalizations ofthe coupling constants can be attributed to the supercon-ducting correlations which in case of an attractive inter-action cause a phase transition toward superconductivity at a critical temperature T c . Re-expressing the bare cou-pling constants γ I / II c n ˜ n , Eq. (4.11), in terms of the originalinteraction potential, Eq. (2.56), we come at backscatter-ing c n ˜ n = π to the relations γ I π = ν n ˜ V (0) + ˜ V (2 p F ) o , (4.12) γ II π = ν n ˜ V (0) − ˜ V (2 p F ) o .In final results, contributions containing γ I π can be inter-preted in terms of a spin singlet while contributions dueto γ II π , which enter final results with a prefactor of three,can be attributed to spin triplets. Note that the lattercoupling constant vanishes in models with a contact in-teraction.
2. RG equations for S The relevant one-loop diagrams for the cubic interac-tion S , Eq. (4.2), are shown in Fig. 6(a) and (a’). Incontrast to the quartic interaction S , the renormalizedcubic vertex cannot be obtained from simple ladder sum-mations and an RG procedure seems unavoidable. Onecan understand this fact from Fig. 6, which shows thatone has always two different choices in attaching anotherquartic block when building higher-order logarithmic di-agrams. As a result, a diagram for a general leadingvertex correction acquires a rather complicated topologyand momentum structure.Similarly to the RG equations for S , Eq. (4.7), theRG equations for the cubic action take the formdB c / ⊥ ( r ⊥ )d ξ = − (cid:8) Γ c ( r ⊥ ) B c / ⊥ ( r ⊥ ) + 3Γ s ( r ⊥ ) B s / ⊥ ( r ⊥ ) (cid:9) ,dB s / ⊥ ( r ⊥ )d ξ = − (cid:8) Γ c ( r ⊥ ) B s / ⊥ ( r ⊥ ) + Γ s ( r ⊥ ) B c / ⊥ ( r ⊥ ) − s ( r ⊥ ) B s / ⊥ ( r ⊥ ) (cid:9) . (4.13)Inserting in analogy with Eq. (4.8) the linear combina-tions B I0 / ⊥ ( r ⊥ ) = B c / ⊥ ( r ⊥ ) − s / ⊥ ( r ⊥ ) , (4.14)B II0 / ⊥ ( r ⊥ ) = B c / ⊥ ( r ⊥ ) + B s / ⊥ ( r ⊥ )into the the RG equations in Eq. (4.13) reduces theseequations to the formdB I / II0 / ⊥ ( r ⊥ )d ξ = − Γ I / II ( r ⊥ ) B I / II0 / ⊥ ( r ⊥ ) . (4.15)With the knowledge of the renormalized coupling con-stants Γ I / II ( r ⊥ ) of the quartic action, Eq. (4.11), the RGequation Eq. (4.15) is nothing but a homogeneous lineardifferential equation for the coupling constant B I / II0 / ⊥ ( r ⊥ ).9 FIG. 7: Diagrammatical Bethe-Salpether equations for the S -ladders. Dark gray blocks stand for the renormalized verticeswhile light gray blocks stand for the bare quartic vertices. P c and P s are defined to project a general quartic vertex onto itsvertex component of the form [ΨΨ][ΨΨ] or [Ψ σ k Ψ][Ψ σ k Ψ], respectively.
Using the boundary conditionsB I / II0 ( r ⊥ ) (cid:12)(cid:12)(cid:12) ξ =0 = γ I / II c n ˜ n f ( r ⊥ ) , (4.16)B I / II ⊥ ( r ⊥ ) (cid:12)(cid:12)(cid:12) ξ =0 = γ I / II c n ˜ n (cid:2) i q ∂ r ⊥ f ( r ⊥ ) (cid:3) ,cf. Eq. (4.4), the solutions of the RG equations inEq. (4.15) are found to beB I / II0 ( r ⊥ ; ξ ) = γ I / II c n ˜ n f ( r ⊥ ) (cid:2) γ I / II c n ˜ n f ( r ⊥ ) ξ (cid:3) , (4.17)B I / II ⊥ ( r ⊥ ; ξ ) = γ I / II c n ˜ n (cid:2) i q ∂ r ⊥ f ( r ⊥ ) (cid:3)(cid:2) γ I / II c n ˜ n f ( r ⊥ ) ξ (cid:3) .Equipped with the knowledge about quartic and cubiccoupling constants at any energy scale, we are now readyto finally determine the renormalized coupling constantsof the quadratic interaction S .
3. RG equations for S The quadratic action S , Eq. (4.3), is renormalizedaccording to the one-loop diagrams in Fig. 5(a), (b),and (b’). The spin structure of these diagrams is com-pletely analogous to the one-loop corrections to the quar-tic and cubic terms. Introducing corresponding combi-nations of the coupling constants in Eq. (4.4) as∆ I0 / ⊥ / ⊥⊥ ( r ⊥ ) = ∆ c / ⊥ / ⊥⊥ ( r ⊥ ) − s / ⊥ / ⊥⊥ ( r ⊥ ) ,(4.18)∆ II0 / ⊥ / ⊥⊥ ( r ⊥ ) = ∆ c / ⊥ / ⊥⊥ ( r ⊥ ) + ∆ s / ⊥ / ⊥⊥ ( r ⊥ ) ,we immediately write the decoupled RG equations.These ared∆ I / II0 ( r ⊥ )d ξ = − Γ I / II ( r ⊥ ) ∆ I / II0 ( r ⊥ ) − (cid:2) B I / II0 ( r ⊥ ) (cid:3) ,d∆ I / II ⊥ ( r ⊥ )d ξ = − Γ I / II ( r ⊥ ) ∆ I / II ⊥ ( r ⊥ ) − B I / II0 ( r ⊥ ) B I / II ⊥ ( r ⊥ ) ,d∆ I / II ⊥⊥ ( r ⊥ )d ξ = − Γ I / II ( r ⊥ ) ∆ I / II ⊥⊥ ( r ⊥ ) + (cid:2) B I / II ⊥ ( r ⊥ ) (cid:3) .(4.19) Since Γ I / II ( r ⊥ ), Eq. (4.11), and B I / II0 / ⊥ ( r ⊥ ), Eq. (4.17), areknown functions of ξ , we are once more to solve lineardifferential equations, which are now — in contrast to theRG equations for the cubic action, Eq. (4.15) — non-homogeneous. The boundary conditions for Eq. (4.19)take the form∆ I / II0 ( r ⊥ ) (cid:12)(cid:12)(cid:12) ξ =0 = γ I / II c n ˜ n f ( r ⊥ ) ,∆ I / II ⊥ ( r ⊥ ) (cid:12)(cid:12)(cid:12) ξ =0 = γ I / II c n ˜ n (cid:2) i q ∂ r ⊥ f ( r ⊥ ) (cid:3) , (4.20)∆ I / II ⊥⊥ ( r ⊥ ) (cid:12)(cid:12)(cid:12) ξ =0 = γ I / II c n ˜ n (cid:2) q ∂ r ⊥ f ( r ⊥ ) (cid:3) ,cf. Eq. (4.4). As a result, the solutions of the RG equa-tions (4.19) are∆ I / II0 ( r ⊥ ; ξ ) = γ I / II c n ˜ n f ( r ⊥ ) 1 − γ I / II c n ˜ n f ( r ⊥ ) ξ (cid:2) γ I / II c n ˜ n f ( r ⊥ ) ξ (cid:3) , (4.21)∆ I / II ⊥ ( r ⊥ ; ξ ) = i q ∂∂ r ⊥ γ I / II c n ˜ n f ( r ⊥ ) (cid:2) γ I / II c n ˜ n f ( r ⊥ ) ξ (cid:3) , (4.22)∆ I / II ⊥⊥ ( r ⊥ ; ξ ) = q ∂ ∂ r ⊥ γ I / II c n ˜ n f ( r ⊥ )1 + γ I / II c n ˜ n f ( r ⊥ ) ξ . (4.23)Equations (4.21)–(4.23) constitute the final result forthe renormalized quadratic vertex and complete theRG study of our low energy field theory, Eqs. (4.1)–(4.4). Stopping the RG flow at ξ = 4 u ˜ u ( ν ∗ /ν ) ln(Λ /T )yields the renormalized coupling constants in the relevantbackscattering limit δ n →
0, which enter the effective ac-tion after integrating out all superfields at energy scaleslarger than T . Therefore in low temperature calculationsof thermodynamic quantities based on the renormalizedaction, the summation of all orders in the large loga-rithm ln(Λ /T ) is at leading order in γ I / II π automaticallyincluded. V. SPECIFIC HEAT
As a result of the renormalization group proceduredeveloped in the previous section, we have the fullknowledge about the physics of our low energy theory,0Eqs. (4.1)–(4.4), at any energy scale Λ ′ ≪ Λ. The rele-vant energy scale for the calculation of the specific heatis according to Eqs. (3.1) and (3.2) of the order of thetemperature T ≪ Λ. So, inserting the quadratic vertex with the renormalized coupling constants, Eqs. (4.21)–(4.23), into the formula for the thermodynamic poten-tial − ( T / hS i , we obtain instead of Eq. (3.4) the fol-lowing expression: δ Ω( T ) = T X ω − Z d ω π ! Z d q (2 π ) d n d˜ n v F δ n ) [i ω − v F ( n · q )][ − i ω + v F (˜ n · q )] × Z Z d u d˜ u u ˜ u n(cid:0)(cid:2) ∆ I ⊥ (¯ q ⊥ ; ξ ) (cid:3) − ∆ I0 (¯ q ⊥ ; ξ )∆ I ⊥⊥ (¯ q ⊥ ; ξ ) (cid:1) + 3 (cid:0)(cid:2) ∆ II ⊥ (¯ q ⊥ ; ξ ) (cid:3) − ∆ II0 (¯ q ⊥ ; ξ )∆ II ⊥⊥ (¯ q ⊥ ; ξ ) (cid:1)o . (5.1)For ξ = 4 u ˜ u ( ν ∗ /ν ) ln(Λ /T ), Eq. (5.1) yields the anoma-lous contribution to the thermodynamic potential in allorders in the logarithm ln(Λ /T ) ∼ ln( ε F /T ) at leadingorder in the bare couplings γ I / II π . This means that inthe limit of low temperatures T ≪ ε F considered here,Eq. (5.1) gives the full physical result. Explicit results forthe specific heat can be extracted by expansion or speci-fying the cutoff function f (¯ q ⊥ ) that enters the renormal-ized coupling constants, cf. Eqs. (4.21)–(4.23).An important observation in Eq. (5.1) is that the en-tire contribution is expressed as a sum of two contribu-tions. The first one contains only the bare coupling con-stant γ I π while the second contribution contains only thecoupling constant γ II π . In other words, the fluctuationsattributed to the spin singlet [ γ I π ] are in leading ordercompletely separated from the fluctuations due to thespin triplet [ γ II π ], cf. the discussion in Secs. V C and V D. A. Low order perturbation theory
Before evaluating the specific heat in all orders fora certain cutoff function f ( r ⊥ ), let us explicitly calcu-late the non-analytic contribution to the specific heatin the third order in the interaction. This approxi-mation is valid for not very low temperatures T , suchthat the quantity γ I / II π ln(Λ /T ) is still small, i.e. if T ≫ Λ exp( − [ γ I / II π ] − ) is fulfilled. Results in this limitare known from conventional perturbation theory andtherefore, we can check our bosonization approach andsee how it works. In Appendix B, we recalculate thethird order starting from the bosonic diagrams ratherthan from Eq. (5.1).In the second order in γ I / II π , the renormalized verticesin Eqs. (4.21)–(4.23) are reduced to∆ I / II0 ( r ⊥ ) ≃ γ I / II c n ˜ n f ( r ⊥ ) − (cid:2) γ I / II c n ˜ n f ( r ⊥ ) (cid:3) ξ , (5.2)∆ I / II ⊥ ( r ⊥ ) ≃ i q ∂∂ r ⊥ n γ I / II c n ˜ n f ( r ⊥ ) − (cid:2) γ I / II c n ˜ n f ( r ⊥ ) (cid:3) ξ o ,∆ I / II ⊥⊥ ( r ⊥ ) ≃ q ∂ ∂ r ⊥ n γ I / II c n ˜ n f ( r ⊥ ) − (cid:2) γ I / II c n ˜ n f ( r ⊥ ) (cid:3) ξ o . Returning to the momentum representation by Fouriertransformation, we find∆ I / II0 (¯ q ⊥ ) ≃ γ I / II c n ˜ n f (¯ q ⊥ ) − (cid:2) γ I / II c n ˜ n (cid:3) [ f ∗ f ](¯ q ⊥ ) ξ , (5.3)∆ I / II ⊥ (¯ q ⊥ ) ≃ − ¯ q ⊥ n γ I / II c n ˜ n f (¯ q ⊥ ) − (cid:2) γ I / II c n ˜ n (cid:3) [ f ∗ f ](¯ q ⊥ ) ξ o ,∆ I / II ⊥⊥ (¯ q ⊥ ) ≃ − ¯ q ⊥ n γ I / II c n ˜ n f (¯ q ⊥ ) − (cid:2) γ I / II c n ˜ n (cid:3) [ f ∗ f ](¯ q ⊥ ) ξ o with [ f ∗ f ] denoting a convolution as in Eq. (3.17).Inserting Eq. (5.3) into Eq. (5.1), using the relation ξ =4 u ˜ u ( ν ∗ /ν ) ln(Λ /T ), and taking the result from the cal-culation in Appendix A, we find the non-analytic ther-modynamic potential correction in the third order in theinteraction as δ Ω ≃ ζ (3) T πv F n [ γ I π ] + 3[ γ II π ] − ν ∗ ν [ f ∗ f ](0) (cid:16) [ γ I π ] + 3[ γ II π ] (cid:17) ln (cid:16) Λ T (cid:17)o . (5.4)Choosing the cutoff function as f (¯ q ⊥ ) = Θ( q − | ¯ q ⊥ | ),the prefactor [ f ∗ f ](0) just gives unity. In this case,we can also estimate ν ∗ /ν ∼ /
2, following the linesafter Eq. (3.16). Finally applying the thermodynamicrelation δc = − T ∂ δ Ω /∂T yields the third order non-analytic specific heat contribution at low temperatures δc ≃ − ζ (3) T πv F n [ γ I π ] + 3[ γ II π ] − (cid:16) [ γ I π ] + 3[ γ II π ] (cid:17) ln (cid:16) Λ T (cid:17)o . (5.5)This perturbative result is applicable in the limit of notvery low temperatures such that on one hand ln(Λ /T ) ≫ γ I / II π ln(Λ /T ) ≪ δc , Eq. (5.5), fully agrees with the result of the laterpublication by Chubukov and Maslov .To be more specific, the result for the third order non-analytic contribution to the specific heat at low temper-ature by Chubukov and Maslov can be recasted into theform δc (3) = 3 ζ (3)2 πv F (cid:8) ( u + u π ) h ( u ϑ + u π − ϑ ) i + 3( u − u π ) h ( u ϑ − u π − ϑ ) i (cid:9) T ln(Λ /T ) (5.6)Here, ϑ is the scattering angle, u ϑ = ν ˜ V (2 p F sin( ϑ/ h g ϑ i is the angular average for an arbitrary function g ϑ . Following the decoupling into soft modes, Eq. (2.7),angular averages h g ϑ i are to be replaced by ( g + g π ) / δc (3) at third order as in Eq. (5.5).Thus, on one hand Eq. (5.5) serves as a good check ofour low energy model, on the other hand we confirm theestimate ν ∗ ∼ ν/ γ I / II π termsof all orders in the logarithm ln(Λ /T ). This calculationshall complete the picture of the non-analytic correctionsto the specific heat at low temperatures T . B. Full low temperature result
In this section, we will extract from Eq. (5.1) theanomalous contribution to the specific heat in all ordersin ln(Λ /T ). As a result, we obtain the full picture of thenon-analyticities in the Fermi liquid thermodynamics atlow temperatures T .In order to accomplish this task, we should choose amodel for the cutoff function f (¯ q ⊥ ), which controls thetwo soft modes represented by Figs. 1(a) and (b). Fol-lowing Ref. 15, a suitable candidate is a Lorentzian ofthe form f (¯ q ⊥ ) = 11 + | ¯ q ⊥ | /q . (5.7)This choice implies f (0) = 1 so that the result from sec-ond order perturbation theory, Eq. (3.6), remains thesame. We recall that within the low energy theory, it isimplied that q ≪ p F . Fourier transforming Eq. (5.7)yields f ( r ⊥ ) = π −| r ⊥ | (5.8)according to the definition of the Fourier transform inEq. (3.18).Inserting the model function f ( r ⊥ ), Eq. (5.8), intothe renormalized quadratic vertices, Eqs. (4.21)–(4.23),we are in a position to Fourier transform them to themomentum representation, which is needed for formula Eq. (5.1). Since only the leading quadratic in ¯ q ⊥ termof the expression [∆ I / II ⊥ ] − ∆ I / II0 ∆ I / II ⊥⊥ is relevant, we mayneglect higher orders in ¯ q ⊥ from the beginning. Thus,the evaluation of the Fourier integrals yields∆ I / II0 (¯ q ⊥ ) = γ I / II c n ˜ n (cid:0) π γ I / II c n ˜ n ξ (cid:1) , (5.9)∆ I / II ⊥ (¯ q ⊥ ) = − ¯ q ⊥ γ I / II c n ˜ n π γ I / II c n ˜ n ξ , (5.10)∆ I / II ⊥⊥ (¯ q ⊥ ) = − ¯ q ⊥ πξ ln (cid:16) π γ I / II c n ˜ n ξ (cid:17) . (5.11)Applying ξ = 4 u ˜ u ( ν ∗ /ν ) ln(Λ /T ) to the renormal-ized couplings, Eqs. (5.9)–(5.11), inserting them intoEq. (5.1), using the integral Z Z d u d˜ u u ˜ u + x − ln(1 + xu ˜ u )(1 + xu ˜ u ) = ln (1 + x )2 x ,and adopting the result of Appendix A for the remain-ing integrations, we obtain for the low temperature non-analytic part of the thermodynamic potential δ Ω the for-mula δ Ω = ζ (3) T πv F ( ln (1 + γ I π L ) L + 3 ln (1 + γ II π L ) L ) .(5.12)Herein, the quantity L is defined as L = πν ∗ ν ln (cid:16) Λ T (cid:17) (5.13)with ν ∗ given by Eq. (3.16). The bare coupling con-stants γ I / II π are expressed in terms of the original fermioninteraction potential ˜ V as γ I / II π = ν n ˜ V (0) ± ˜ V (2 p F ) o , (5.14)cf. Eq. (4.12).Equation (5.12) constitutes our final result for the non-analyticities of a two-dimensional Fermi gas with repul-sive interaction. In the following, we discuss the correc-tions to the specific heat of the Fermi liquid and possibleinstabilities. C. Corrections to the Fermi liquid
For the Fermi liquid model, the thermodynamic poten-tial correction δ Ω, Eq. (5.12), is a regular function for allrelevant γ I / II π . By means of the formula δc = − T ∂ δ Ω ∂T (5.15)2 FIG. 8: The non-analytic correction to the specific heat of atwo-dimensional Fermi liquid for a repulsive contact interac-tion as a function of γ I π L . The logarithmic-linear plot in thebackground illustrates the behavior at large γ I π L , i.e. in theasymptotic limit of very low temperatures T . we find the anomalous contribution to the specific heatat low temperatures T in the form δc = − ζ (3) T πv F ( ln (1 + γ I π L ) L + 3 ln (1 + γ II π L ) L ) .(5.16)This formula has been obtained from Eqs. (5.12)and (5.15) by differentiating only the T prefactor butnot the quantity L , Eq. (5.13). Derivatives of L wouldyield subleading terms in the most interesting low tem-perature limit.Equation (5.16) constitutes our main result for theanomalous correction δc to the specific heat of the two-dimensional Fermi liquid at low temperature in the weakcoupling regime. Figure 8 shows the plot of δc dividedby the second order result (3.7) as a function of γ I π L .For simplicity, only the case γ II π = 0 (contact interaction)is plotted. For a general γ II π , a contribution of the sameshape as in Fig. 8 but dependent on γ II π L has to be added.With a further decrease of the temperature, the quan-tity L , Eq. (5.13), grows such that the non-analyticpart δc of the specific heat of the Fermi liquid asymp-totically reaches zero as − (ln L ) /L . This asymptoticbehavior fully agrees with the estimate by Chubukov andMaslov in the limit of small ν ∗ /ν . This estimate wasbased on the conjectured relation δc = − ζ (3) T πv F (cid:2) f c ( π ) + 3 f s ( π ) (cid:3) , (5.17)where f c ( π ) and f s ( π ) denote the charge and spincomponents of the fully renormalized backscatteringamplitude. The asymptotic agreement between ourbosonization approach and the conjecture (5.17) sup-ports its validity. Moreover, full agreement between ourresult Eq. (5.16) and Eq. (5.17) can be shown in the limit ν ∗ /ν ≪ L for a long-range inter-action, considering the potential V q to be non-zero onlyclose to q = 0. We also note that in the supersymmet-ric approach of Ref. 15, where certain effects of the Fermisurface curvature were neglected, the dependence on L ofthe spin contribution to the anomalous specific heat wasfound to have the same analytic shape as the result ofthe present work for the contribution due to both chargeand spin excitations, Eq. (5.16).At not very low temperatures, γ I / II π ln (Λ /T ) ≪
1, thefull result for the anomalous specific heat, Eq. (5.16), re-duces to the perturbative result of Eq. (5.5) [except fora prefactor of π/ s - and p -waveCooper pairs). This is supported by the form of thecoupling constants γ I π and γ II π , see Eq. (5.14). The com-bination of the interaction amplitudes ˜ V (0) and ˜ V (2 p F )precisely corresponds to what one obtains when summingup the singlet [for γ I π ] and triplet [for γ II π ] Cooper ladders.Following this line of reasoning, we interpret the firstand the second terms in Eq. (5.16) as contributionscoming from singlet and triplet superconducting fluctu-ations. Of course, for models with a contact interaction,˜ V = const, the coupling constant γ II π vanishes and onlythe singlet Cooper pairs contribute.It appears that finding the anomalous correction tothe specific heat δc , Eq. (5.16), from conventional dia-grammatic expansions is rather difficult. Identifying theCooper “wheels” as the relevant diagrams and especiallyidentifying the soft modes in these diagrams for an ar-bitrary order in the perturbation theory can be rathertricky, which is why earlier works restricted themselvesto low orders while estimating infinite order results byplausibility. The bosonization approach presented in thispaper allows for a field theory study based on simple el-ementary diagrams that can be used as building blocksfor a subsequent renormalization group analysis. As aresult, an explicit derivation of the low temperature non-analytic specific heat of higher-dimensional Fermi liq-uids becomes possible in all orders in the large loga-rithm ln(Λ /T ).Formula (5.16) for the anomalous specific heat con-tribution δc has to replace the well-known second ordercontribution δc (2) , Eq. (3.7), as soon as the logarithmin L , Eq. (5.13), becomes of the order of [ γ I / II π ] − . Theexperimental study on He fluid monolayers presented inRef. 14 confirmed the validity of the second order per-turbation theory down to temperatures of order 1 mK.In this experimental setting, however, the coupling con-stant γ π is of order unity, which is beyond the applica-bility of our theory as it is based on the assumption ofweak interaction. Nevertheless, provided the logarith-mic renormalization remains valid at least qualitativelyalso for strong interactions, the logarithms should be de-tectable in measurements, although one might need to3investigate a broader temperature interval including tem-peratures considerably below the mK scale. D. Instabilities
According to the main result, Eq. (5.16), discussed inthe last section, the non-analytic corrections to the Fermiliquid thermodynamics at low temperatures are smallas the function δc ( T ) /T decays logarithmically in thelimit of T →
0. This statement remains valid as long asboth singlet and triplet constants, γ I π ∝ ˜ V (0) + ˜ V (2 p F )and γ II π ∝ ˜ V (0) − ˜ V (2 p F ), are positive. The assumptionof their positivity guarantees that the system is in theFermi liquid regime.If one of these constants is negative, the function ln(1+ γ I / II π L ) in the thermodynamic potential δ Ω, Eq. (5.12),approaches a pole at | γ I / II π | L = 1 when lowering the tem-perature T , i.e. when boosting the quantity L , Eq. (5.13).This pole corresponds to the divergency of the geomet-ric series in the summation of the ladders in Fig. 7 for γ I / II π L = − u = ˜ u = 1 and r ⊥ = 0. The emergence ofthis singularity means the breakdown of the perturbativeapproach close to and beyond | γ I / II π | L = 1.For a repulsive interaction, the coupling constant γ I π always remains positive. It would be negative if the ef-fective interaction ˜ V were negative, corresponding to at-tractive interaction. We can identify the resulting sin-gularity in the thermodynamic potential with the well-known Cooper instability. As a result, one obtains theconventional s -wave superconductivity below a certaintransition temperature T c obtained from the condition γ I π L = − γ I π is strictly positive for a repulsive interac-tion, the triplet coupling constant γ II π becomes negativeas soon as ˜ V (2 p F ) > ˜ V (0). This may actually happenfor a repulsive interaction. For instance, for contact in-teraction, ˜ V (2 p F ) = ˜ V (0) and already a small increase of˜ V (2 p F ) — for instance due to the closeness to a quantumcritical point — may make the constant γ II π negative. Ifthis happens, one comes again to an instability in thethermodynamic potential δ Ω, Eq. (5.12), at γ II π L = − p -wavesuperconducting pairing. One can come to this conclu-sion recalling once more the superconducting particle-particle ladders in the spin triplet representation. It iseasy to see that for a small angle and backward scatteringthe combination ˜ V (0) − ˜ V (2 p F ) enters the pre-factor infront of the logarithms, which confirms our assumption.The condition for criticality γ II π L = − T c , T c = Λ exp n − νπν ∗ | γ II π | o , (5.18)corresponding to the transition temperature into thetriplet superconducting state. The nature of this transi-tion is similar to the Kohn-Luttinger transition towards p -wave pairing although in the Kohn-Luttinger sce-nario, the inverse coupling constant 1 / | γ II π | enters the ex-ponent of the critical temperature T c , Eq. (5.18), with adifferent power. Everywhere in this paper, we considered the limit ofweak interactions. A more interesting situation may oc-cur near a quantum critical point (QCP) of a transitioninto a magnetic or charge density wave state. In thevicinity of such a point, the collective mode propagator χ ( q , Ω) dressed by the electron-hole bubble can be writ-ten as χ ( q , Ω) ∼ χ ξ − + q + ¯ γ ( | Ω | / | q | ) , (5.19)where χ is the susceptibility, ξ − determines the close-ness to QCP [ ξ − = 0 at QCP], and ¯ γ is proportionalto the interaction. The propagator χ ( q , Ω), Eq. (5.19),is written in the limit of small bosonic Matsubara fre-quencies Ω. Higher-order corrections to the propaga-tor χ ( q , Ω) can be considered but these are highly non-trivial [see, e.g. Refs. 35–39], which invalidates the earlyconjecture that one can describe a QCP by a conven-tional φ -field theory with the bare propagator χ ( q , Ω),Eq. (5.19).Using the conventional fermionic diagrammatic tech-nique for studying the critical behavior near a QCP isvery difficult. At the same time, the bare bosonic prop-agator g n ( K ), Eq. (2.53), in our bosonization approachdescribes directly the electron-hole excitations. Near aQCP, it could be modified and take a form similar to theone of Eq. (5.19). Then, we would be able to derive a su-perfield theory with a modified bare action. Within sucha theory, diagrams to be disregarded for the Fermi liquidlike, e.g., diagram Fig. 4(b) are not necessarily small andrequire a special care. This adds to the complexity of thetheory with very intriguing consequences. We hope thatour bosonization method can help in studying the QCPproblem. VI. CONCLUSION
Singling out the low energy spin and charge excitationsof a higher-dimensional clean Fermi gas with a repulsiveinteraction, we have modified our general bosonizationscheme to derive an effective low energy superfieldtheory. This representation allows to conveniently cal-culate thermodynamic quantities in the low temperaturelimit. During the derivation, special care has been givento the role of Fermi surface geometry in higher dimen-sions. As a result, all curvature effects are preserved inthe final action, correcting the earlier supersymmetricapproach from Ref. 15.The superfields in our low energy theory are anticom-muting but periodic in imaginary time. Consequently,the described excitations obey Bose statistics. Thesebosonic excitations include both spin and charge exci-tations, interacting in a non-trivial way as described by4quartic, cubic, and quadratic terms in the action of thesuperfield theory.A perturbative study of the low energy superfield the-ory in the backscattering limit yields the well-knownleading non-analyticities in the thermodynamics and alsothe logarithmic corrections in any dimension d . In di-mensions d > d > δc , Eq. (5.16). This result is of infinite orderin the large logarithm ln( ε F /T ) and leading order in thecoupling constants of the weak interaction. As such, itis valid for an arbitrary low temperature T ≪ ε F . Thedependence of δc on the logarithm ln( ε F /T ) – plotted inFig. 8 – indicates that the function δc ( T ) /T decays as1 / ln ( ε F /T ) for T →
0. As discussed in Sec. V C, our re-sult is in full agreement with asymptotic results of earlierworks based on conventional diagrammatic expansions.Remarkably, the thermodynamic potential and thespecific heat correction, Eqs. (5.12) and (5.16), consist oftwo separate terms of an identical analytical form con-trolled by two different coupling constants γ I π and γ II π .The contribution of the term containing only the cou-pling constant γ II π comes with a factor of three as com-pared to the term with the coupling γ I π . In the discussionin Sec. V C, we interpret these two contributions as com-ing from superconducting fluctuations of spin singlet andspin triplet types. This statement is supported by com-parison of the conventional Cooper ladders with the lad-ders of the quartic bosonic action and by the analyticalform of the effective coupling constants γ I / II π , Eq. (5.14).For a contact interaction, γ II π = 0 and only the singletsuperconducting fluctuations contribute. We note thatwithin our approximation of small fluctuations at back-ward scattering, we cannot distinguish between angularharmonics of the same parity, e.g. between s - and d -pairing, merely distinguishing between the singlet andtriplet excitations.If one of the coupling constants — γ I π or γ II π — becomesnegative, the Fermi liquid picture breaks down at a crit-ical temperature T c and a superconducting phase transi-tion takes place. The constant γ I π can be negative onlyfor an attractive interaction, leading to the conventionalspin singlet superconducting transition. In contrast, γ II π can become negative also for certain models of repulsiveinteraction. This scenario of triplet superconductivity, which is similar to the Kohn-Luttinger one, is discussedin Sec. V D. Since our perturbative approach breaks downclose above the critical temperature T c , Eq. (5.18), thecritical behavior itself should be studied introducing thesuperconducting order parameter.A very interesting situation may arise near a quantumphase transition into, e.g., a ferromagnetic state. Nearthis point, both superconducting and paramagnetic ex-citations are important. In the language of the su-perfield theory developed here, the non-interacting para-magnetic excitations should be described by the bare ac-tion, Eq. (2.51), while the superconducting fluctuationsappear as a result of the interaction between these ex-citations. Due to the special form of these excitations,cf. Eq. (5.19), the perturbation theory is more compli-cated than the one considered here for the Fermi liquidand logarithmic contributions may arise in more classesof one-loop diagrams. Since calculations in this inter-esting situation are not simple within the conventionaldiagrammatic approaches , we hope that the presentbosonization technique will become a helpful analyticaltool for future studies on this topic.
Acknowledgements
We are grateful to A.V. Chubukov and D.L. Maslovfor invaluable discussions. H.M. and K.E. acknowledgefinancial support from the SFB/Transregio 12 of theDeutsche Forschungsgemeinschaft. H.M, C.P., and K.E.acknowledge the hospitality of the International Instituteof Physics in Natal where parts of this work were done.
Appendix A: Evaluation of the integral in Eq. (3.4)
In this appendix, we explicitly evaluate the second or-der contribution δ Ω (2) ( T ), Eq. (3.4), δ Ω (2) ( T ) = T X ω − Z d ω π ! Z d q (2 π ) d n d˜ n f ( q ) × γ c n ˜ n v F ( n · q ) v F (˜ n · q )[i ω − v F ( n · q )][ − i ω + v F (˜ n · q )] . (A1)The explicit form of the cutoff function f ( q ) is not im-portant for the second order loop. It is sufficient to knowthat the estimate | q | . q ≪ p F holds. Eventually, thecalculation shows that effectively only those q enter f ( q )which satisfy | q | ≪ q . Consequently, we can effectivelyput q = 0 in f ( q ) and assume f (0) = 1.First, we perform the sum and integral over thefrequency ω . The coth-functions resulting from thefinite temperature Matsubara summation are conve-niently combined with the sign functions from the zero-temperature frequency integration using the relationcoth x = sgn x [1 + 2 P ∞ l =1 exp( − n | x | )]. As a result, we5obtain δ Ω (2) ( T ) = v F Z d n d˜ n d q (2 π ) γ c n ˜ n ( n · q )(˜ n · q )( n · q ) − (˜ n · q ) × ∞ X l =1 n sgn( n · q ) exp (cid:16) − lv F | ( n · q ) | T (cid:17) − sgn(˜ n · q ) exp (cid:16) − lv F | (˜ n · q ) | T (cid:17)o . (A2)From Eq. (A2), we understand that relevant “parallel”momenta ( n · q ) or (˜ n · q ) are of order T /v F . It it thisobservation which makes our entire low energy approachuseful.In order to carry out the remaining integrations, weshould transform the variables into a frame that wouldbetter reflect the physics of the scattering processes un-der consideration. For this purpose, we introduce newangular variables¯ n = 12 ( n − ˜ n ) , δ n = 12 ( n + ˜ n ) (A3)and the corresponding projections of the momentum vec-tor q , ¯ q k = (¯ n · q ) , ¯ q ⊥ = q − ¯ q k ¯ n . (A4)The choice of the angular coordinates and the notation inEq. (A3) are motivated by the observation that the mostimportant contributions to δ Ω (2) ( T ), Eq. (A1), will comefrom small δ n . The phase space region around δ n = 0corresponds to backscattering, c n ˜ n ∼ π , and is also ex-actly the region where the logarithmic renormalizationsof the scattering amplitude γ c n ˜ n take place.Reexpressing Eq. (A2) with the help of the variablesfrom Eqs. (A3) and (A4), we obtain δ Ω (2) ( T ) = v F Z d n d˜ n d¯ q ⊥ π d¯ q k π γ c n ˜ n ∞ X l =1 × ( | ¯ q k | ¯ q k − ( δ n · ¯ q ⊥ ) − | ¯ q k | ) exp (cid:16) − lv F | ¯ q k | T (cid:17) . (A5)The second term − | ¯ q k | in the curly brackets produces aninsensitive to small | δ n | and thus purely analytic contri-bution to the thermodynamic potential. Focussing on thenonanalytic contributions, we neglect such terms. Onecan observe at this point that the nonanalytic contribu-tions arise completely from the term ( δ n · ¯ q ⊥ ) in the nu-merator ( n · q )(˜ n · q ) = ( δ n · ¯ q ⊥ ) − ¯ q k of Eq. (A1). In otherwords, the momentum components ¯ q k of the terms with( n · q ) θ and (˜ n · q )˜ θ in the actions S and S are effectivelyirrelevant for the analysis of the nonanalyticities.We note that the integrand in Eq. (A5) does not de-pend on the angular variable ¯ n but only on δ n . There-fore, we should transform the integration variables { n , ˜ n } to { ¯ n , δ n } and while we do so, we should already have inmind that the integral will be dominated by small δ n . Let us thus parameterize n = (cos φ, sin φ ) and ˜ n =(cos ˜ φ, sin ˜ φ ) with both φ and ˜ φ varying between 0 and 2 π .Then the normalized integration measure is given byd n d˜ n = (d φ/ π )(d ˜ φ/ π ). For a suitable parametrizationof the variables ¯ n and δ n , Eq. (A3), we introduce the an-gles ¯ φ and δφ in such a way that φ = ¯ φ − ( δφ + π/
2) and˜ φ = ¯ φ + ( δφ + π/
2) with δφ being small in the backscat-tering limit. Then, to linear order in δφ , we find ¯ n ≃ (sin ¯ φ, − cos ¯ φ ) and δ n ≃ − δφ (cos ¯ φ, sin ¯ φ ), which alsogives | δ n | ≃ | δφ | in the integrand of Eq. (A5). The in-tegration measure transforms as d n d˜ n = (d ¯ φ/ π )d δφ/π .The integration over ¯ φ is immediately performed over(0 , π ) and consequently yields unity. After that, thenonanalytic correction δ Ω (2) ( T ) reads δ Ω (2) ( T ) = v F γ π π Z d δ n d¯ q ⊥ π d¯ q k π × ∞ X l =1 | ¯ q k | ¯ q k − ( δ n · ¯ q ⊥ ) exp (cid:16) − lv F | ¯ q k | T (cid:17) . (A6)Here and in the following, δ n is identified with the one-dimensional variable δφ and typically takes small values | δ n | ≪ q k as ¯ q k i κ k , Eq. (A6) is reduced toEq. (7.17b) of Ref. 15. Explicitly, we recast Eq. (A6)into the form δ Ω (2) ( T ) = − v F γ π π Z d δ n d¯ q ⊥ π ∞ X l =1 × Re "Z ∞ d κ k π κ k κ k + ( δ n · ¯ q ⊥ ) exp (cid:16) − i lv F κ k T (cid:17) .(A7)For the integration over δ n , we indeed notice that themost important contributions come from | ( δ n · ¯ q ⊥ ) | /κ k . κ k ∼ T /v F and ¯ q ⊥ ∼ q , therefore fromthe backscattering region of small δ n where the esti-mate | δ n | . T / ( v F q ) ≪ γ π become indeed active. Before we come tothe δ n -integral, we integrate over the “imaginary” mo-mentum κ k and in order to facilitate that, we recast therational integrand as a Fourier integral,1 κ k + ( δ n · ¯ q ⊥ ) = 12 q | δ n | κ k Z e − κ k | r ⊥ | / ( q | δ n | ) e − i r ⊥ ¯ q ⊥ /q d r ⊥ . (A8)Because of the cutoff q for the momentum ¯ q ⊥ , typ-ical | r ⊥ | are of order 1. Inserting Eq. (A8) into theexpression for δ Ω (2) ( T ), Eq. (A7), we see that the pre-exponential is just a power of κ k and the corresponding6 FIG. 9: Diagrams for the third order backscattering contri-bution to the thermodynamic potential. integration is easily performed, δ Ω (2) ( T ) = − γ π T π v F Z e − i r ⊥ ¯ q ⊥ /q (cid:18) v F q | δ n | T | r ⊥ | (cid:19) × ∞ X l =1 Re "(cid:26) l (cid:18) v F q | δ n | T | r ⊥ | (cid:19)(cid:27) − v F q d δ n T | r ⊥ | d r ⊥ d¯ q ⊥ πq .(A9)According to the estimates discussed in the pre-ceding text, essential values of the quantity Φ = v F q | δ n | / ( T | r ⊥ | ) are of order 1. Transforming from theintegration variable δ n to Φ, the integration limits willbe of order v F q /T ≫
1, allowing to extend the domainof integration to ±∞ . Using the integral Z ∞ Re (cid:20) Φ (1 + iΦ) (cid:21) dΦ = − π δ Ω (2) ( T ) = γ π T πv F ∞ X l =1 l Z e − i r ⊥ ¯ q ⊥ /q d r ⊥ d¯ q ⊥ πq . (A10)The sum over l just gives Ap´ery’s constant ζ (3), the re-maining integrals are trivial, and we obtain for δ Ω (2) ( T )the result presented in Eq. (3.6). Appendix B: Third order correction from bosonicdiagrams
In this appendix, we calculate in the leading logarith-mic order the anomalous specific heat in third order inthe interaction by the explicit evaluation of the bosonicdiagrams.As a result of the analysis in Sec. III, the relevant di-agrams are those shown in Fig. 9. Other diagrams areeither insensitive to the backscattering region or theirlogarithmic divergency is prohibited by the effects of thecurvature of the Fermi surface. Since for the diagrams inFigs. 9(a) and (b) the curvature terms are not relevant,we will omit them in the following formulas. In analytical terms, diagram Fig. 9(a) yields the con-tribution∆Ω (3a) ( T ) = − ν Z d n d˜ n X QQ ′ × (cid:16) [ γ I c n ˜ n ] + 3[ γ II c n ˜ n ] (cid:17) f ( q ) f ( q ′ ) f ( q + q ′ ) × (cid:8) v F ( n · q )(˜ n · q ) − v F ( n · q ′ )(˜ n · q ) (cid:9) × g n ( Q ) g ˜ n ( − Q ) g n ( Q ′ ) g ˜ n ( − Q ′ ) (B1)and the second diagram Fig. 9(b) corresponds to∆Ω (3b) ( T ) = − ν Z d n d˜ n X QQ ′ × (cid:16) [ γ I c n ˜ n ] + 3[ γ II c n ˜ n ] (cid:17) f ( q ) f ( q ′ ) f ( q + q ′ ) × (cid:8) − v F ( n · q )(˜ n · q ) − v F ( n · q )(˜ n · q ′ ) + 2 v F ( n · q ′ )(˜ n · q ′ ) (cid:9) × g ˜ n ( Q + Q ′ ) g n ( − Q ) g n ( Q ′ ) g ˜ n ( − Q ′ ) . (B2)The coupling constants γ I c n ˜ n and γ II c n ˜ n have been intro-duced in Eq. (4.9).The third order diagrams for ∆Ω (3a) ( T ) and ∆Ω (3b) ( T )contain two loops with correspondingly two running four-momenta Q and Q ′ . Following the idea of Eq. (3.1), weshould subtract the contribution at T = 0 and deal withthe quantity δ Ω (3) ( T ) = ∆Ω (3) ( T ) − ∆Ω (3) (0) rather thanwith ∆Ω (3) ( T ) itself. Therefore, one four-momentum ef-fectively varies on the scale . T while the other oneconversely needs to vary on large scales ≫ T in order toproduce the leading logarithmic correction.Let us begin the explicit evaluation with the expres-sion for diagram Fig. 9(a), Eq. (B1). The first term in thecurly brackets behaves in a considerably different way forthe two cases of small or large Q — corresponding to largeor small Q ′ , respectively. In the case of small Q , the inte-gral over Q is calculated analogously to the second orderintegral Eq. (3.4) while the integral over Q ′ is essentiallythe logarithmic one-loop integral from Eq. (3.9). As a re-sult, we obtain a correction of relative order γ I / II π ln(Λ /T )to the second order result, Eq. (3.6).The opposite case of large Q constitutes an unpleasantdivergency at large v F ¯ q k , which is only formally cut bythe cutoff functions f ( q ). Fortunately, this ultraviolet di-vergency is exactly compensated by an ultraviolet diver-gency appearing with the opposite sign in the first term inthe curly brackets in the expression for diagram Fig. 9(b),Eq. (B2), such that the result is eventually regular. Wenote once more in this context that the parallel momen-tum ¯ q k of the ( n · q )-terms in the actions S and S isirrelevant for thermodynamic quantities, cf. the discus-sion after Eq. (A5). Furthermore, since the seeming ul-traviolet divergency in v F ¯ q k is compensated — an obser-vation that is easily generalized to diagrams of arbitraryorder —, it is completely safe to neglect the v F ¯ q k part ofthe ( n · q )-terms in the cubic and quadratic parts of theinteraction as done in Eqs. (4.1)–(4.3).7The second term in the curly brackets of Eq. (B1) isodd in both Q and Q ′ and for this reason, one might betempted to disregard that term. However, because of thepresence of the cutoff functions, the overall integrand isnot odd in the perpendicular momenta ¯ q ⊥ and ¯ q ′⊥ , cf.Eq. (3.11) for the notation. Explicitly, since Z ¯ q ′⊥ f (¯ q ′⊥ ) f (¯ q ⊥ + ¯ q ′⊥ ) d¯ q ′⊥ q = − ¯ q ⊥ Z f (¯ q ′⊥ ) f (¯ q ⊥ + ¯ q ′⊥ ) d¯ q ′⊥ q , (B3)we observe that the second term of Eq. (B1) gives thesame contribution as the regular part of the first term —with one half coming from small Q and one half comingfrom small Q ′ .Now, let us turn our attention to the expressionfor ∆Ω (3b) , Eq. (B2). The presence of Q ′ in three de-nominators implies that Q is necessarily the large four-momentum. The first term in the curly brackets ofEq. (B2) consequently does nothing more than neutralizethe ultraviolet divergency in ∆Ω (3a) as discussed above.The second term is treated in complete analogy with thesecond term in Eq. (B1) while, finally, the third termis effectively of the same form as the first term of thediagram Fig. 9(a) in the limit of small Q .As to the cutoff functions f ( q ), they have played animportant role in understanding the seemingly odd termsin Eqs. (B1) and (B2). After applying Eq. (B3), all rele-vant terms have the form of the first term in Eq. (B1) atsmall Q . The remaining integration of Q is completelyequivalent to the second order integral presented in Ap-pendix A. There, we learned that the transverse momen-tum of the small four-momentum could be safely put tozero in the cutoff functions. Thus, the integral of the“large” transverse momentum over the cutoff functionsyields a prefactor of f (0) R f (¯ q ⊥ )[d¯ q ⊥ / q ]. For thechoice f (¯ q ⊥ ) = Θ( q − | ¯ q ⊥ | ), this prefactor is just unity.Collecting all the terms, we find that diagram Fig. 9(a)gives 4 / / δ Ω (3) ( T ) to the thermodynamic potential, which canbe written as δ Ω (3) ( T ) = − ν ∗ ν ζ (3) πv F (cid:0) [ γ I π ] + 3[ γ II π ] (cid:1) T ln (cid:18) Λ T (cid:19) .This result clearly agrees with the one obtained from thelow order expansion of the renormalized coupling con-stants, Eq. (5.4). Appendix C: Boson model versus fermion picture
In Sec. III of Ref. 24, it was checked in the second orderin the interaction that our method of bosonization allowsfor, in principle, an exact reformulation of the originalfermion model in terms of bosonic excitations and repro-duces exactly each single contribution from the fermionicdiagrammatics. The choice of a proper diagrammatical
TABLE I: Vertices of the bosonic excitations and related di-agrammatical structures of the Hartree and Fock soft modesin conventional fermion language.FIG. 10: Correspondence between the bosonic low energyrepresentation and conventional diagrammatics: (a) the barebackscattering diagram from Fig. 3(a), (b) its representationin Hartree and Fock soft modes according to Table I, andfinally (c) the conventional fermionic diagrams. representation allowed us to identify the bosonic contri-butions with the ones of the fermionic picture already onthe level of diagrams — before explicitly evaluating theanalytical expressions.Following the decoupling into soft modes in Sec. II,the vertices in the bosonic theory collect at the sametime the fermionic Hartree vertices with a momentumtransfer close to zero and Fock vertices transferring mo-menta of order 2 p F . As discussed in Ref. 24, where anexact Hartree-like decoupling scheme has been applied,the bosonic propagator corresponds to the propagation ofa particle-hole pair in the fermion picture, which will bereflected diagrammatically by opposite oriented double-lines. Following the derivation of the exact supersymmet-ric representation in Ref. 24 for eachwise the Hartree andthe Fock decoupling schemes, we obtain the diagrammat-ical representation in Table I. Table I can be understoodas a dictionary translating diagrams in the boson pictureinto corresponding standard fermionic diagrams.As an example, let us consider the diagram for thebare anomalous contribution, Fig. 10(a). Its evaluationin Sec. III A returns Eq. (3.6) for the correction to Ω,8 FIG. 11: The bosonic one-loop diagram Fig. 4(a) and its re-lation to conventional diagrams. The latter form particle-particle ladders, giving logarithms independently from thedimension d of the system.FIG. 12: The bosonic one-loop diagram Fig. 4(b) and its re-lated conventional diagrams. The latter are of particle-holeladder and polarization bubble type, reflecting the absence ofa logarithmic divergency in d > which yields the leading anomalous T -term in the spe-cific heat c .Redrawing the bosonic quadratic vertices with the helpof Table I in all possible ways that the soft Hartree andFock vertices may enter, we obtain the diagrams shownin Fig. 10(b). Finally, “literally” interpreting the bosonicpropagators as pairs of opposite directed fermion ones,we identify the corresponding conventional diagrams,Fig. 10(c), which share the same low energy physical con-tent with Fig. 10(a). Indeed, standard fermion perturba-tion theory yields exactly Eq. (3.6) as the anomalouscontribution to Ω, which exclusively comes from the con-ventional second order diagrams in Fig. 10(c). One-loop diagrams in the fermion picture
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