Luttinger liquid fixed point for a 2D flat Fermi surface
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Luttinger liquid fixed point for a 2D flat Fermisurface
Vieri MastropietroOctober 27, 2018
Abstract
We consider a system of 2D interacting fermions with a flat Fermisurface. The apparent conflict between Luttinger and non Luttingerliquid behavior found through different approximations is resolved byshowing the existence of a line of non trivial fixed points, for theRG flow, corresponding to Luttinger liquid behavior; the presence ofmarginally relevant operators can cause flow away from the fixed point.The analysis is non-perturbative and based on the implementation,at each RG iteration, of Ward Identities obtained from local phasetransformations depending on the Fermi surface side, implying thepartial vanishing of the Beta function.
Pacs numbers : 71.10.Hf,71.10.Fd
The properties of the 2 D interacting fermions are still largely unknown, de-spite the tremendous effort devoted to their understanding in the last years.One of the most debated questions is on the possible existence of a Luttingerliquid phase , first suggested by Anderson [1] as an explanation of some prop-erties of high T c superconductors, as observed also in recent experiments, seee.g. [2] .It has been proved, in the case of symmetric, smooth and convex Fermisurfaces (like in the Jellium model [3] or in the Hubbard model in the nonhalf filled case [4]), that the wave function renormalization Z is essentiallytemperature independent up to exponentially small temperatures. As in aLuttinger liquid one expects instead a logarithmic behavior in this regime, i.e. Z ≃ O ( U log β ), such results rule out for sure the possibility ofLuttinger liquid behavior. 1n the contrary the presence in the Fermi surface of flat regions canproduce non Fermi liquid behavior. The simplest model with a flat Fermisurface is the 2D Hubbard model at half filling, in which the Fermi surfaceis a square. It was proved in [5] that the wave function renormalization is Z = 1 + O ( U log β ) up to exponentially small temperatures; the presenceof the log β is a consequence of the Van Hove singularities, related to thefact that the Fermi velocity is vanishing at the corners of the squared Fermisurface, and implies that also such a model does not show Luttinger liquidbehavior.It is important to stress that the results in [3],[4],[5] are rigorous as theyare based on expansions which are convergent provided that the temperatureis not too low, the finite temperature acting as an infrared cut-offs; how-ever such expansions cannot give any information on the zero temperatureproperties.A lot of attention has been devoted in recent years to the zero temper-ature properties of Fermi surfaces with flat regions and no corners, whichshare some features with the Fermi surfaces of some cuprates as seen inphotoemission experiments. Parquet methods results [6] and perturbativeRenormalization Group (RG) analysis [7] truncated at one loop indicate that,for repulsive interactions, there is no indication of a Luttinger liquid phaseat zero temperature; the effective couplings flow toward a strong couplingregime related to the onset of d-wave superconductivity. In a more recentRG analysis truncated at 2 loops [8] one still gets a flow to strong coupling,but in some intermediate region some indication of Luttinger liquid behavioris found.Apparently conflicting results are found by applying bosonization: in [9],[10] a model of electrons on a square Fermi surface was mapped in a collec-tion of fermions on coupled chains, and it is found that the correlations atzero temperature in momentum space are similar to the one of the Luttingermodel. A related but somewhat different strategy consists in proposing an ex-actly solvable 2D analogue of the Luttinger model; this approach was pursuedin [11] and [12] and again Luttinger liquid behavior up to zero temperaturewas found.A possible explanation of such conflicting results was suggested in [13],postulating the existence for the RG flow, in addition to the trivial fixedpoint associated to non interacting fermions, of a non trivial fixed pointassociated to Luttinger behavior, which could be made instable by the pres-ence of marginally relevant operators. In this paper we provide a quantitativeverification of such hypothesis showing explicitely the existence of a line nontrivial Luttinger fixed points for the RG flow of a system of 2D interactingfermions with a flat Fermi surface. It is would be not possible to derive such2esult directly from the perturbative expansions, as it is related to cancella-tions between graphs to all orders of the expansion which are too complexto be seen explicitly; it is indeed well known that even in 1D Ward Iden-tities (WI) are necessary to prove the existence of a Luttinger liquid fixedpoint [14]. Our analysis is based on the implementation, in an exact RGapproach, of WI with corrections due the the cut-offs introduced in the mul-tiscale analysis, extending a technique already used to establish Luttingerliquid behavior in a large class of 1D fermionic systems [15],[16] or 2D spinsystems [17]. Such methods are the only ones which can be applied to nonexactly solvable models, like the model analyzed in this paper. We consider a model with a square Fermi surface similar to the one consideredin [6], [8] or [9]; the Schwinger functions are given by functional derivativesof the generating functional e W ( φ ) = Z P ( dψ ) e V ( ψ )+ R d x [ ψ + x φ − x + ψ − x φ + x ] (1)with ψ ± k are Grassmann variables, k = ( k − , k + , k ), k ± = πL ± n ± , k = πβ ( n + ), n ± , n = 0 , ± , ± , ... and P ( dψ ) is the fermionic integrationwith propagator g k = X σ = ± X ω = ± H ( k − σ ) C − ( q a − ( k + v F ( | k σ | − p F ) )) − ik + v F ( k σ − ωp F ) ≡ X σ,ω = ± g σ,ω, k (2) H ( k − σ ) = χ ( a − k − σ ), χ ( t ) = 1 if t < C − ( t ) is a smoothcompact support function = 1 for t < t ≥ γ , γ >
1. Weassume, for definiteness, a ≤ p F , a ≤ p F so that the support of g k is over 4disconnected regions; the Fermi surface is defined as the set of in which g k for k = 0 is singular, in the limit β → ∞ .By using well known properties of Grassmann integrals, see [18], (6) allowsto write the Grassmann field as a sum of independent fields ψ ± k = X σ = ± X ω = ± ψ ± ω,σ, k (3)with ψ ± ω,σ, x independent Grassmann variables with propagator g ω,σ, k . Asin [6],[8] or [9] we can consider only interactions between parallel patches( V = L + L − ) V = X σ X ω V β ) X k ,... k U b v ( k − k ) ψ + ω ,σ, k ψ − ω ,σ, k ψ + ω ,σ, k ψ − ω ,σ, k δ ( k − k + k − k )(4)3ith v ( x ) a short range potential. The 2-point Schwinger function is givenby S ( x , y ) = ∂ W ( φ ) ∂φ x ∂φ y | φ =0 (5).Figure 1: The Fermi surface corresponding to the singularities of g k ; the foursides are labelled by ( σ, ω ) = ( ± , ± ). As the interaction does not couple different σ we can from now on fix σ = +for definiteness and forget the index σ . We analyze the functional integral(1) by performing a multiscale analysis, using the methods of constructiveQuantum Field Theory (for a general introduction to such methods, see [18]).The propagator (2) can be written as sum of ”single slice” propagators inthe following way g ω ( x − y ) = X h = −∞ e iωp F ( x + − y + ) g ( h ) ω ( x − y ) (6)where g ( h ) ω ( x − y ) = 1 V β X k e i k ( x − y ) H ( k − ) f h ( k ) − ik + ωv F k + (7)and f h ( k ) has support in a region O ( γ h ) around each flat side of the Fermisurface, at a distance O ( γ h ) from it, that is a γ h − ≤ k + v F k ≤ a γ h +1 ;note that in each term in (6) the change of variables k + → k + + ωp F hasbeen performed. The single scale propagator verify the following bound, forany integer M | g ( h ) ω ( x ) | ≤ C M | sin ax − x − | γ h γ h ( | x + | + | x | )] M (8)4he integration is done iteratively integrating out the fields with momentacloser and closer to the Fermi surface, renormalizing at each step the wavefunction. After the integration of the fields ψ (0) , ..., ψ ( h +1) we obtain Z P Z h ( dψ ( ≤ h ) ) e −V ( h ) ( √ Z h ψ ( ≤ h ) ) (9)where P Z h ( dψ ( ≤ h ) ) is the fermionic integration with propagator Z − h ( k ) g ( ≤ h ) ω, k ,with g ( ≤ h ) ω, k = P hk = −∞ g ( k ) ω, k and Z h is defined iteratively starting from Z = 1;moreover, if ~p F = (0 , p F , V ( h ) ( ψ ≤ h ) = ∞ X n =1 X ω βV ) n X k ,..., k n δ ( X i ε i ( k i + ω i ~p F )) " n Y i =1 b ψ ( ≤ h ) ε i ω i , k i W ( h )2 n ( k , ..., k n − ) (10)By using that R d k | g ( k ) ω,σ, k | ≤ Cγ k and | g ( k ) ω,σ, k | ≤ Cγ − k we see that the kernels c W ( k )2 n are O ( γ − k ( n − ); this means that the terms quadratic in the fields havepositive scaling dimension and the quartic terms have vanishing scaling di-mension, and all the other terms have negative dimension; we have then toproperly renormalize the terms with non-negative dimension.Calling ¯ k = ( k − , ,
0) we define an L operator acting linearly on thekernels of the effective potential:1) L c W ( h )2 n = 0 if n ≥ n = 1 L c W h ( k ) = c W h (¯ k ) + k ∂ k c W h (¯ k ) + k + ∂ + c W h (¯ k ) (11). 3) If n = 2 L c W h ( k , k , k ) = δ P i ε i ω i , c W h (¯ k , ¯ k , ¯ k ) . (12)Calling ∂ c W h (¯ k ) = − iz h ( k − ), ∂ + c W h (¯ k ) = ωz h ( k − ) (symmetry considera-tions are used) and l h ( k − , , k − , , k − , ) = c W h (¯ k , ¯ k , ¯ k ) we obtain LV h = 1 βV X k [ z h ( k − ) ωk + − ik z h ( k − )] b ψ +( ≤ h ) k ,ω b ψ − ( ≤ h ) k ,ω + (13) ∗ X ω,σ βV ) X k ,..., k l h ( k − , , k − , , k − , ) b ψ +( ≤ h ) k ,ω b ψ − ( ≤ h ) k ,ω b ψ +( ≤ h ) k ,ω b ψ − ( ≤ h ) k ,ω δ ( X i ε i k i ))5here P ∗ ω is constrained to the condition P i ε i ω i ~p F = 0 and we have usedthat, by symmetry, W h (¯ k ) = 0.We write (9) as Z P Z h ( dψ ( ≤ h ) ) e −LV ( h ) ( √ Z h ψ ( ≤ h ) ) −RV ( h ) ( √ Z h ψ ( ≤ h ) ) (14)with R = 1 − L . The non trivial action of R on the kernel with n = 2 canbe written as R c W h ( k , k , k ) = [ c W h ( k , k , k ) − c W h (¯ k , k , k )] (15)+[ c W h (¯ k , k , k ) − c W h (¯ k , ¯ k , k )] + [ c W h (¯ k , ¯ k , k ) − c W h (¯ k , ¯ k , ¯ k )]The first addend can be written as k , Z dt∂ k , c W h ( k − , , k + , , tk , ; k , k ) + k + , Z dt∂ k + , c W h ( k − , , tk + , , k , k ) (16)The factors k , and k + , are O ( γ h ′ ), for the compact support propertiesof the propagator associated to b ψ +( ≤ h ) ω , k , with h ′ ≤ h , while the derivativesare dimensionally O ( γ − h − ); hence the effect of R is to produce a factor γ h ′ − h − < R on the n = 1 terms. The effect of the L operation is to replace in W h ( k ) the momentum ~k with its projection on theclosest flat side of the Fermi surface. Hence the fact that the propagatoris singular over an extended region (the Fermi surface) and not simply ina point has the effect that the renormalization point cannot be fixed but itmust be left moving on the Fermi surface.In order to integrate the field ψ ( h ) we can write (14) as Z P Z h − ( dψ ( ≤ h ) ) e −L e V h ( √ Z h ψ ( ≤ h ) ) −RV ( h ) ( √ Z h ψ ( ≤ h ) ) (17)where P Z h − ( dψ ( ≤ h ) ) is the fermionic integration with propagator1 Z h − ( k ) H ( k − ) C − h ( k ) − ik + ωv F k + (18)with C − h ( k ) = P hk = −∞ f k and Z h − ( k ) = Z h ( k − )[1 + H ( k − ) C − h ( k ) z h ( k − )] (19)Moreover L e V h is the second term in (13).6e rescale the fields by rewriting the r.h.s. of (14) as Z P Z h − ( dψ ( ≤ h ) ) e −L b V h ( √ Z h − ψ ( ≤ h ) ) −RV ( h ) ( √ Z h − ψ ( ≤ h ) ) (20)where L b V h ( ψ ) = ∗ X ω βV ) X k ,.., k g h ( k − , , k − , , k − , ) b ψ + k ,ω b ψ − k ,ω b ψ + k ,ω b ψ − k ,ω δ ( X i ε i k i )(21)and the effective couplings g h ( k − , , k − , , k − , ) = [ Y i =1 vuut Z h ( k − ,i ) Z h − ( k − ,i ) ] l h ( k − , , k − , , k − , ) (22)After the integrations of the fields ψ (0) , ψ ( − , ..., ψ ( h ) we get an effective the-ory describing fermions with wave function renormalization Z h and effectiveinteraction (21). Note that Z h and g h are non trivial functions of the mo-mentum parallel to the Fermi surface.We write Z P Z h − ( dψ ( ≤ h − ) Z P Z h − ( dψ ( h ) ) e −L b V ( h ) ( √ Z h − ψ ( ≤ h ) ) −RV ( h ) ( √ Z h − ψ ( ≤ h ) ) (23)and the propagator of P Z h − ( dψ ) is b g hω,σ ( k ) = H ( k − ) 1 Z h − ( k − ) e f h ( k ) − ik + ωv F k + and e f h ( k ) = Z h − ( k − )[ C − h ( k ) Z h − ( k ) − C − h − ( k ) Z h − ( k − ) ] (24)with H ( k − ) e f h ( k ) having the same support that H ( k − ) f h ( k ). We integratethen the field ψ ( h ) and we get Z P Z h − ( dψ ( ≤ h − ) e −V ( h − ( √ Z h − ψ ( ≤ h − ) (25)and the procedure can be iterated.The above procedure allows us to write W ( h )2 n as a series in the effectivecouplings g k , k ≥ h , which is convergent , see [18], provided that L − is finiteand ε h = sup k ≥ h || g k || small enough; moreover || W n || = O ( γ − h ( n − ). Asimilar analysis can be repeated for the 2-point function.7owever even if the couplings g k starts with small values, they can possi-bly increase iterating the RG and at the end reach the boundary of the (es-timated) convergence domain; if this happen, all the above procedure loosesits consistency. A finite temperature acts as an infrared cut-off saying thatthe RG has to be iterated up to a maximum scale h β = O (log β ) and, up toexponentially small temperatures i.e. β ≤ O ( e κ | U | − ), then surely the effec-tive couplings are in the convergence domain; however, in order to get lowertemperatures, more information on the effective couplings are necessary. The RG analysis seen in the previous section implies that the effective cou-pling g h verify a flow equation of the form g h − = g h + β ( h ) g ( g h ; ... ; g ) (26)where the r.h.s. of the above equation is called Beta function , which isexpressed by a convergent expansion in the couplings if ε h is small enough.The first non trivial contribution to β ( h ) g , called β (2)( h ) g , is quadratic in thecouplings and it is given by β (2)( h ) g = β ( a ) h + β ( b ) h (27)where β ( a ) h = Z d p H ( k , − − p − ) H ( k , − + p − ) g h ( k , − , k , − − p − , k , − ) (28) g h ( k , − − p − , k , − , k , − + p − ) f h ( p ) C h ( p ) p + v F p β ( b ) h = − Z d p H ( k , − − p − ) H ( k , − − p − ) g h ( k , − , k , − − p − , k , − − p − ) g h ( k , − − p − , k , − , k , − ) f h ( p ) C h ( p ) p + v F p (29)The above expression essentially coincides with the one found in [6] or [8]; itis indeed well known that the lowest order contributions to the Beta functionare essentially independent by RG procedure one follows.The flow equation (26) encodes most of the physical properties of themodel, but its analysis is extremely complex. Some insights can be obtainedby truncating the beta function at second order, and by the numerical anal-ysis of the resulting flow by discretization of the Fermi surface; it is found,see [6] or [8],that g h ( k − , , k − , , k − , ) has a flow which, for certain values of8 − , , k − , , k − , increases and reach the estimated domain of convergence ofthe series for W ( k ) n . While this increasing can be interpreted as a sign of in-stability, mathematically speaking this means that the truncation procedurebecomes inconsistent.A basic question is about the fixed points of the flow equation (26); inparticular if there is,in addition to the trivial fixed point g h = 0, a non trivialfixed point corresponding to Luttinger liquid behavior. Note first that theset g h ( k − , , k − , , k − , ) = 1 L − δ ( k , − − k , − ) λ h (30)with λ h constant in k , is invariant under the RG flow, in the sense that if LV ( k ) has the form,for k ≥ h βV ) X k , k ′ , p ω,ω ′ λ h δ p − , b ψ + ω, k b ψ − ω, k + p b ψ + ω ′ , k ′ b ψ − ω ′ , k ′ − p (31)the same is true for LV ( h − . This can be checked by the graph expansion. Inthe graphs contributing to W ( h )4 , the external lines of the graphs contributingto W ( h )4 either comes out from a single point, or are connected by a chain ofpropagators with the same ω, k − . Moreover in each Feynman graph the onlydependence from the momenta of the external lines is through the function H ( k − ) which are 1 in the support of the external fields R d k H ( k − ) ψ ± k = R d k ψ ± k . For the same reasons also Z h is independent from k − .The crucial point is that, in the invariant set (30), some dramatic can-cellation are present implying the following asymptotic vanishing of the betafunction (which will be proved in the subsequent sections) β ( h ) g = O ( γ h ε h ) (32)saying that there is a cancellations between the graphs with four externallines and the graphs with two lines contributing to the square of Z h , see(22); such graphs are O (1) but there are cancellations making the size of thesum of them O ( γ h ). At the second order (32) can be verified from (29) and(30); at third order, it is compatible with (A16),(A15) and (4.8) of [8].The validity of (32) immediately implies the existence of a line of non-trivial fixed points for (26) of the form g −∞ ( k − , , k − , , k − , ) = 1 L − δ ( k , − − k , − ) λ −∞ (33)with λ −∞ k -independent and continuous function of U , λ −∞ = λ + O ( U ), λ = cU for a suitable constant c . 9ote also that to such fixed point is associated Luttinger liquid behavior,as, from (19), Z h ≃ γ ηh , with η = aλ + O ( U ) and ¯ p = (0 , ¯ p + , ¯ p ) a = lim h →−∞ X ω ′ λ L − h Z d k ′ d ¯ p + d ¯ p H ( k ′− ) C h ( k ′ ) − ik ′ + ω ′ v F k ′ + (34) H ( k ′− ) C h ( k ′ + ¯ p ) − i ( k ′ + ¯ p ) + ω ′ v F ( k ′ + + ¯ p + ) ∂∂k + H ( k − ) C − h ( k − ¯ p ) − i ( k − ¯ p ) + ωv F ( k + − ¯ p + ) | k = k + =0 Indeed the 2-point Schwinger function can be written as S ( x , y ) = X ω = ± e iωp F ( x + − y + ) V β X k e i k ( x − y ) g ( h ) ω ( k ) Z h [1 + A ( h ) ( k )]with A ( h ) ( k ) = O ( ε h ), so that S ( x , y ) = X ω = ± e iωp F ( x + − y + ) (35)1 V β X k e i k ( x − y ) H ( k − ) C − ( k ) − ik + ωv F k + A ( k ) | k + v F k | η with | A ( k ) | ≤ C | U | . This means that to the fixed point is associated Lut-tinger liquid behavior, as the wave function renormalization vanishes at theFermi surface as a power like with a non-universal critical index;the Luttingerliquid behavior is found only if L − is finite, as if L − → ∞ the critical indexis vanishing. Note also that the cancellation in (32) reduce to the one in 1Dif k − = k ′− in (31). There is essentially no hope of proving a property like (32) directly from thegraph expansion, as the algebra of the graphs is too cumbersome (exceptthan at one loop in which it is easy to check). We will follow instead thesame strategy for proving the asymptotic vanishing of the Beta function in1D followed in [15],[16], considering an auxiliary model with the same betafunction, up to irrelevant terms, but verifying extra symmetries, from whicha set of Ward Identities can be derived. In the present case, such identitiesare related to the invariance under local phase transformations dependingon the Fermi surface side, which in the model (1) is broken by the cut-offfunction C − ( k ) and by the lattice.We consider an auxiliary model with generating function is Z D ψ Y ω e R d k H − ( k − ) C h,N ( k )( − ik + ωk + v F ) b ψ + ω, k b ψ − ω, k + ¯ V ( ψ )+ P ε = ± R d x ψ εω, x φ − εω, x + R d x J x ρ ω, x (36)10ith C − h,N ( k ) = P Nk = h f k ( k + v F k ), h ≤ ρ ω, x = ψ + x ,ω ψ x ,ω ¯ V = UL − X ω,ω ′ Z d x dy − v ( x − y , x + − y + ) ψ + ω,x − ,x + ,x ψ − ω,x − ,x + ,x ψ + ω ′ ,y − ,x + ,x ψ − ω ′ ,y − ,x + ,x = UL − βV ) X k , k ′ βL + X p ,p + b v ( p , p + ) b ψ + ω, k b ψ − ω, k +(0 ,p + ,p ) b ψ + ω ′ , k ′ b ψ − ω ′ , k ′ − (0 ,p + ,p ) (37)with v ( x , x + ) a short range interaction. The above functional integral is verysimilar to the previous one, with the difference that there is an ultravioletcut–off γ N on the + variables , which will be removed at the end, and anultraviolet cut-off O (1) on the − variables; such features are present also inthe models introduced in [9] or [12].Again (36) can be analyzed by a multiscale integration based on a de-composition similar to (6), with the difference that the scale are from h to N . In the integration of the scales between N and 0, the ultraviolet scales ,there is no need of renormalization; apparently the terms with two or fourexternal lines have positive or vanishing dimension but one can use the nonlocality of the interaction to improve their scaling dimension. We integrate(with L = 0) the fields ψ ( N ) , ψ ( N − , .., ψ ( k ) and we call W ( k )2 n,m the kernels inthe effective potential multiplying 2 n fermionic fields and m J fields. Againthe dimension is γ − k ( n + m − , k ≥ W ( k )2 , ( x , y ) = Z d y U v ( x − y , , x + − y , + ) L − W ( k )0 , ( y ) g ( k,N ) ( x − y ) W ( k )2 , ( y ; y )+ U Z d y v ( x − y , , x + − y + , ) L − g ( k,N ) ( x − y ) W ( k )2 , ( y , y ; y ) + (38) U δ ( x − y ) Z d y v ( x − y , , x + − y , + ) L − W ( k )0 , ( y )The first and the third addend of Fig.2 are vanishing, by the symmetry g ( k , k + , k − ) = − g ( − k , − k + , k − ); hence, using that k g ( j ) k ≤ e Cγ − j and that W ( k )2 , is O ( U ) (by induction), we obtain the following bound k W ( k )2 , k ≤ C | U | L − k W ( k )2 , k · N X j = k k g ( j ) k ≤ C | U | γ k L − − γ − k (39)Note that we have a gain O ( L − − γ − k ) ), due to the fact that we are integratingover a fermionic instead than over a bosonic line.11= +Figure 2: Graphical representation of (38); the blobs represent W ( k ) n,m , thewiggly lines represent v , the lines g ( k,N ) + +Figure 3: Decomposition of W ( k )2 , Similar arguments can be repeated for W ( k )0 , , which can be decomposedas in Fig 3. The second term in the figure is bounded by O ( | U | L − − γ − k ). Asimilar bound is found for the third term in Fig.3; regarding the first term,we can rewrite it as Z d x d ¯ z [ g ( k,N ) ( z − x )] UL − v ( x − ¯ z , x + − ¯ z + ) W ( k )0 , (¯ z , y ) = Z d x d ¯ z U v (¯ z − z , ¯ z + − z + ) L − [ g ( k,N ) ( x − z )] W ( k )0 , (¯ z , y ) (40)+ Z d x d ¯ z UL − [ v (¯ z − x , ¯ z + − x + ) − v (¯ z − z , ¯ z + − z + )][ g ( k,N ) ( x − z )] W ( k )0 , (¯ z , y )and using that Z d x [ g ( k,N ) ( x − z )] = Z dk − H ( k − ) Z dk dk + C − k,N ( k )( − ik + k + ) = 0 (41)the first addend is vanishing; the second addend, by using the interpolationformula for v (¯ z − x , ¯ z + − x + ) − v (¯ z − z , ¯ z + − z + ), can be bounded by C | U | γ − k , as by induction || W ( k )0 , || ≤ C | U | . A similar analysis proves thebound for W k )4 , . 12fter the integration of the fields ψ ( N ) , ψ ( N − , ..., ψ ( − we get a Grass-man integral very similar to (1); the integration of the remaining fields ψ (0) , ψ ( − , .. is done following the same procedure as in section 3, with theeffective coupling of the form (30) and LV ( k ) of the form (31). The crucialpoint is that the beta function coincides with the beta function for the model(1) up to O ( γ h ) terms; hence it is enough to prove the validity of (32) in theauxiliary model. We derive now a set of Ward Identities relating the Schwinger functions ofthe auxiliary model (36); by performing the change of variables ψ ± ω, x → e ± iα ω, x ψ ± ω, x (42)and making a derivative with respect to α x ,ω and to the external fields weobtain Z d k ′ [ H − ( k ′− + p − ) C h,N ( k ′ + p )( − i ( k ′ + p ) + ωv F ( k ′ + + p + )) − H − ( k ′− ) C h,N ( k ′ )( − ik ′ + ωv F k ′ + )] h b ψ + ω, k ′ + p b ψ − ω, k ′ b ψ + ω ′ , k − p b ψ − ω ′ , k i = δ ω,ω ′ [ h b ψ + ω ′ , k − p b ψ − ω ′ , k − p i − h b ψ + ω ′ , k b ψ − ω ′ , k i ] (43)where h b ψ + ω, k ′ + p b ψ − ω, k ′ b ψ + ω ′ , k − p b ψ − ω ′ , k i is the derivative with respect to J p , φ + ω ′ , k − p , φ − ω ′ , k of (36). Computing (43) for p − = 0 we get,if ¯ p = (0 , ¯ p + , ¯ p )( − i ¯ p + ωv F ¯ p + ) h ρ ¯ p ,ω b ψ + k ,ω ′ b ψ − k − ¯ p ,ω ′ i + ∆( k , ¯ p ) = δ ω,ω ′ [ h b ψ + ω ′ , k − ¯ p b ψ − ω ′ , k − ¯ p i − h b ψ + ω ′ , k b ψ − ω ′ , k i ] (44)and ∆( k , ¯ p ) = Z d k ′ C ( k ′ , ¯ p ) h b ψ + ω, k ′ +¯ p b ψ − ω, k ′ b ψ + ω ′ , k − ¯ p b ψ − ω ′ , k i (45)with C ( k , ¯ p ) = ( − ik + ωv F k + ) (46)[ C h,N ( k + ¯ p ) − C h,N ( k )] + ( − i ¯ p + ωv F ¯ p + )[ C h,N ( k + ¯ p ) − Z d k ′ H − ( k ′− ) h b ψ + ω, k ′ +¯ p b ψ − ω, k ′ b ψ + ω ′ , k − ¯ p b ψ − ω, k i = Z d k ′ h b ψ + ω, k ′ +¯ p b ψ − ω, k ′ b ψ + ω ′ , k − ¯ p b ψ − ω ′ , k i (47)13or the compact support properties of the fields ψ k and H = H ; note alsothe crucial role of the condition p − = 0 in the above derivation.The presence of the term ∆( k , ¯ p ) in the Ward Identity (44) is related tothe presence of the ultraviolet cut-off; as in 1D, such a term is not vanishingeven in the limit N → ∞ and it is responsible of the anomalies, see [15],[16].The following correction identity holds, similar to the one in the 1D case∆( k , ¯ p ) = ν ( − i ¯ p − ωv F ¯ p + ) X ω ′′ = ± h ρ ¯ p ,ω ′′ b ψ + ω ′ , k − ¯ p b ψ − ω ′ , k i + R , ω ( k , ¯ p ) (48)with R , ω a small correction. Indeed R , ω can be written as functional deriva-tive, with respect to φ + , φ − , J , of e W ∆ ( J,φ ) = Z P ( dψ ) e − V ( ψ )+ P ω R d z [ ψ + ω, z φ − ω, z + φ + ω, z ψ − ω, z ]+ T ( J,ψ ) − T − ( J,ψ ) (49)with T ( ψ ) = Z d ¯ p + (2 π ) d ¯ p (2 π ) d k (2 π ) C ( k , ¯ p ) J ¯ p b ψ + k +¯ p ,ω b ψ − k ,ω (50) T − ( b ψ ) = X ω ′ Z d ¯ p + (2 π ) d ¯ p (2 π ) d k (2 π ) νJ ¯ p ( − i ¯ p − ωv F ¯ p + ) b ψ + k +¯ p ,ω ′ b ψ − k ,ω ′ (51)(49) can be evaluated by a multiscale integration similar to the previous one,the only difference being that R J ρ is replaced by T − T − . The terms withvanishing scaling dimension of the form J ψ + ψ − can be obtained from thecontraction of T and T − ; in the first case we can perform a decompositionsimilar to the one in Fig.3, see Fig. 4. Regarding the second and third term,+ +Figure 4: Terms obtained from the contraction of T ; the black dot represents C ( k , ¯ p )we can proceed exactly as in the previous section, the main difference beingthat at least one of the two fields have scale N so that they obey to the bound O ( γ − k γ − (1 / N − k ) ). This follows from the fact that when C is multiplied by14wo propagators we get C ( k , ¯ p ) g ( i ) ( k ) g ( j ) ( k + ¯ p ) = f i ( k ) − ik + ωv F k + [ f j ( k + ¯ p ) C − h,N ( k + ¯ p ) − f j ( k + ¯ p )] − f j ( k + ¯ p ) − i ( k + ¯ p ) + ωv F ( k + + ¯ p + ) [ f i ( k ) C − h,N ( k ) − f i ( k )] (52)which is non vanishing only if one among i or j are equal to h or N .The main difference with the analysis in the previous section is that inthe first term of Fig 4; the ”bubble” in Fig. 3 was vanishing, while here it isnot. We choose ν in (49) equal to the value of this bubble, in order to cancelit. The value of the bubble is given by ν = U v (¯ p , ¯ p + ) Z d k (2 π ) C ( k , ¯ p ) − i ¯ p − ωv F ¯ p + g ( ≤ N ) ω ( k ) g ( ≤ N ) ω ( k + ¯ p ) (53)and,in the limit N → ∞ , ν = U v (¯ p , ¯ p + ) a π . Hence for k , k + = O ( γ h ) | R , ω ( k , ¯ p ) | ≤ Cε h γ − h (54) An immediate consequence of the analysis in the previous section is that, formomenta computed at the infrared scale | k | = | k ′ | = | k + b p | = | k ′ − b p | = | b p | = γ h h b ψ − ω, k b ψ + ω, k + b p b ψ − ω ′ , k ′ b ψ + ω ′ , k ′ − b p i =1( Z h ) λ h L − g ( h ) ω, k g ( h ) ω ′ , k ′ g ( h ) ω, k ′ + b p g ( h ) ω ′ , k ′ − b p (1 + O ( ε h )) h b ψ − ω, k b ψ + ω, k i = g ( h ) ω, k Z h (1 + O ( ε h )) (55)This says that relations between the effective couplings at a certain scale h ≤ b p = (0 , b p + , b p ) and ¯ p = (0 , ¯ p + , ¯ p ) h b ψ − ω, k b ψ + ω, k − b p b ψ − ω ′ , k ′ b ψ + ω ′ , k ′ + b p i = X ω ′′ { UL − v ( b p , b p + ) g ω ′ , k ′ + b p h b ψ − ω ′ , k ′ b ψ + ω ′ , k ′ ih ρ b p ,ω ′′ b ψ − ω, k b ψ + ω, k − b p i + (56) UL − g ω ′ , k ′ + b p Z d ¯ p (2 π ) d ¯ p + (2 π ) v (¯ p , ¯ p + ) h ρ ¯ p ,ω ′′ b ψ − ω, k b ψ + ω, k − b p b ψ − ω ′ , k ′ b ψ + ω ′ , k ′ + b p − ¯ p i}
15 + p Figure 5: Graphical representation of (56); the dotted line represent the freepropagatorBy the WI (44),(48)( − i b p + ω ′ v F b p + )¯ h ρ b p ,ω ′ b ψ − k ,ω b ψ + k − b p ,ω i = A ω,ω ′ ( p )[ h b ψ − ω, k − b p b ψ + ω, k − b p i − h b ψ − ω, k b ψ + ω, k i ] + H (2 , ω,ω ′ ( k , p ) (57)with A ω,ω ( p ) = 1 + O ( ε h ), A ω, − ω ( p ) = O ( ε h ) and even in p ; moreover H ω,ω ′ ( k , p ) is a linear combination of the R , functions in (48) with boundedcoefficients. The WI for the 4 point function is given by( − i ¯ p + ωv F ¯ p + ) h ρ ¯ p ,ω b ψ − ω, k b ψ + ω, k − b p b ψ −− ω, k ′ b ψ + − ω, k ′ + b p − ¯ p i = (58) h b ψ − k − ¯ p ,ω b ψ + k − b p ,ω b ψ − k ′ , − ω b ψ + k ′ + b p − ¯ p , − ω i − h b ψ − k ,ω b ψ + k − b p +¯ p ,ω b ψ − k ′ , − ω b ψ + k ′ + b p − ¯ p , − ω i + ν X ω ′ ( − i ¯ p − ωv F ¯ p + ) h ρ ¯ p ,ω b ψ − k ,ω b ψ + k − b p ,ω b ψ − k ′ , − ω b ψ + k ′ + b p − ¯ p , − ω i + R , ω ( k , k ′ , p )and similar ones, so that the second addend of the l.h.s. is given by Z d ¯ p d ¯ p + χ ε (¯ p ) b v (¯ p , ¯ p + ) − i ¯ p + ω ′′ v F ¯ p + [ A (¯ p ) h b ψ − k − ¯ p ,ω b ψ + k − b p ,ω b ψ − k ′ ,ω ′ b ψ + k ′ + b p − ¯ p ,ω ′ i + A (¯ p ) h b ψ − k ,ω b ψ + k − b p +¯ p ,ω b ψ − k ′ ,ω ′ b ψ + k ′ + b p − ¯ p ,ω ′ i + A (¯ p ) h b ψ − k ,ω b ψ + k − b p ,ω b ψ − k ′ − ¯ p ,ω ′ b ψ + k ′ + b p − ¯ p ,ω ′ i + A (¯ p ) h b ψ − k ,ω b ψ + k − b p ,ω b ψ − k ′ ,ω ′ b ψ + k ′ + b p ,ω ′ i + H , ( k , k ′ , ¯ p )] (59)where χ ε (¯ p ) is a compact support function vanishing for ¯ p = 0, and suchthat it becomes the identity in the limit ε →
0; moreover the functions A i ( p )are bounded and even in p , and H , is a linear combination of the R , functions in (58) with bounded coefficients.16e have now to bound all the sums in the r.h.s. of (59). Note first that,by parity Z d ¯ p d ¯ p + χ ε (¯ p ) A (¯ p ) b v (¯ p , ¯ p + ) − i ¯ p + ω ′′ v F ¯ p + h b ψ − k ,ω b ψ + k − b p ,ω b ψ − k ′ ,ω ′ b ψ + k ′ + b p ,ω ′ i = 0 (60)Moreover the first term in the r.h.s. of (59) verifies | Z d ¯ p d ¯ p + b v (¯ p , ¯ p + ) − i ¯ p + v F ω ′′ ¯ p + A (¯ p ) χ ε (¯ p ) h b ψ − k − ¯ p ,ω b ψ + k − b p ,ω b ψ − k ′ ,ω ′ b ψ + k ′ + b p − ¯ p ,ω ′ i| ≤ Cε h γ − h Z h (61)and a similar bound is true for the second and third term. Finally, as in [16] | Z d ¯ p d ¯ p + b v (¯ p , ¯ p + ) − i ¯ p + v F ω ′′ ¯ p + H , ( k , k ′ , ¯ p ) | ≤ Cε h γ − h Z h (62)By inserting (55),(57),(59),(60),(61) in (56) we get λ h = λ + O ( U ), whichmeans that the effective interaction remain close to initial value for any RGiteration; a contradiction argument shows that this can be true only if thebeta function is asymptotically vanishing: as this beta function is the sameof the 2D model (1) with effective couplings (30), up to O ( γ h ) terms, then(32) follows. We have shown that the RG flow for a system of spinless fermions with flatFermi surface has, in addition to the trivial fixed point, a line of Luttingerliquid fixed points, corresponding to vanishing wave function renormalizationand anomalous exponents in the 2-point function; such fixed point is in theinvariant set (30). This makes quantitative the analysis in [10], in whichthe existence of a Luttinger fixed point in 2D was postulated on the basisof bosonization. With respect to previous perturbative RG analysis,the keynovelty is the implementation of WI at each RG iteration, in analogy to whatis done in 1D.Of course the other effective interactions should cause flows away fromthis fixed point. Indeed the situation is somewhat similar to the 1D (spin-ning) Hubbard model, in which there is a Luttinger liquid fixed point in theinvariant set obtained setting all but the backscattering and umklapp scatter-ing terms equal to zero (that is the set g ,h = g ,h = 0 in the g-ology notation,see [19]), and a flow to strong coupling regime driven by the backscatteringinteraction. We can in any case expect, as in 1D, that even if the Luttinger17xed point in 2D is not stable its presence has an important role in thephysical properties of the system.Fermi surfaces with flat or almost flat pieces and no van Hove singularitiesare found in the Hubbard model with next to nearest neighbor interactionsor in the Hubbard model close to half filling, and it is likely that our resultscan be extended, at least partially, to such models. Note however that insuch models the sides of the Fermi surface are not perfectly flat, so thatone expects a renormalization of the shape of the Fermi surface, as in [3],which is absent in the case of flat sides by symmetry. Another simplifyingproperty of the model considered here is that the modulation of the Fermivelocity is taken constant along the Fermi surface, contrary to what happensin more realistic models; a momentum dependent Fermi velocity producesextra terms in the WI, as it is evident from (43) ( v F should be replaced by v F ( k ′− ) in the first line and v F ( k ′− + p − ) in the second line), and their effectdeserves further analysis. Acknowledgments
I am grateful to A.Ferraz and E.Langmnann for veryinteresting discussions on the 2D Hubbard model, and to the Schroedingerinstitute in Wien where this paper was partly written.
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