{'fraction_non_alphanumeric': 0.04846157837389094, 'fraction_numerical': 0.013957349660146319, 'mean_word_length': 4.108401802279353, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Luttinger liquid fixed point for a 2D flat Fermi surface

We consider a system of 2D interacting fermions with a flat Fermi surface. The apparent conflict between Luttinger and non Luttinger liquid behavior found through different approximations is resolved by showing the existence of a line of non trivial fixed points, for the RG flow, corresponding to Luttinger liquid behavior; the presence of marginally 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 phase transformations depending on the Fermi surface side, implying the partial vanishing of the Beta function.


Introduction
The properties of the 2D interacting fermions are still largely unknown, despite the tremendous effort devoted to their understanding in the last years. One of the most debated questions is on the possible existence of a Luttinger liquid phase, first suggested by Anderson [1] as an explanation of some properties of high T c superconductors, as observed also in recent experiments, see e.g. [2] .
It has been proved, in the case of symmetric, smooth and convex Fermi surfaces (like in the Jellium model [3] or in the Hubbard model in the non half filled case [4]), that the wave function renormalization Z is essentially temperature independent up to exponentially small temperatures. As in a Luttinger liquid one expects instead a logarithmic behavior in this regime, i.e. Z ≃ 1 + O(U 2 log β), such results rule out for sure the possibility of Luttinger liquid behavior.
On the contrary the presence in the Fermi surface of flat regions can produce non Fermi liquid behavior. The simplest model with a flat Fermi surface is the 2D Hubbard model at half filling, in which the Fermi surface is a square. It was proved in [5] that the wave function renormalization is Z = 1 + O(U 2 log 2 β) up to exponentially small temperatures; the presence of the log 2 β is a consequence of the Van Hove singularities, related to the fact that the Fermi velocity is vanishing at the corners of the squared Fermi surface, and implies that also such a model does not show Luttinger liquid behavior.
It is important to stress that the results in [3], [4], [5] are rigorous as they are based on expansions which are convergent provided that the temperature is not too low, the finite temperature acting as an infrared cut-offs; however such expansions cannot give any information on the zero temperature properties.
A lot of attention has been devoted in recent years to the zero temperature properties of Fermi surfaces with flat regions and no corners, which share some features with the Fermi surfaces of some cuprates as seen in photoemission experiments. Parquet methods results [6] and perturbative Renormalization Group (RG) analysis [7] truncated at one loop indicate that, for repulsive interactions, there is no indication of a Luttinger liquid phase at zero temperature; the effective couplings flow toward a strong coupling regime related to the onset of d-wave superconductivity. In a more recent RG analysis truncated at 2 loops [8] one still gets a flow to strong coupling, but in some intermediate region some indication of Luttinger liquid behavior is 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 collection of fermions on coupled chains, and it is found that the correlations at zero temperature in momentum space are similar to the one of the Luttinger model. A related but somewhat different strategy consists in proposing an exactly solvable 2D analogue of the Luttinger model; this approach was pursued in [11] and [12] and again Luttinger liquid behavior up to zero temperature was found.
A possible explanation of such conflicting results was suggested in [13], postulating the existence for the RG flow, in addition to the trivial fixed point associated to non interacting fermions, of a non trivial fixed point associated to Luttinger behavior, which could be made instable by the presence of marginally relevant operators. In this paper we provide a quantitative verification of such hypothesis showing explicitely the existence of a line non trivial Luttinger fixed points for the RG flow of a system of 2D interacting fermions with a flat Fermi surface. It is would be not possible to derive such result directly from the perturbative expansions, as it is related to cancellations between graphs to all orders of the expansion which are too complex to be seen explicitly; it is indeed well known that even in 1D Ward Identities (WI) are necessary to prove the existence of a Luttinger liquid fixed point [14]. Our analysis is based on the implementation, in an exact RG approach, of WI with corrections due the the cut-offs introduced in the multiscale analysis, extending a technique already used to establish Luttinger liquid behavior in a large class of 1D fermionic systems [15], [16] or 2D spin systems [17]. Such methods are the only ones which can be applied to non exactly solvable models, like the model analyzed in this paper.

The model
We consider a model with a square Fermi surface similar to the one considered in [6], [8] or [9]; the Schwinger functions are given by functional derivatives of the generating functional with ψ ± k are Grassmann variables, k = (k − , k + , k 0 ), k ± = 2π L ± n ± , k 0 = 2π β (n 0 + 1 2 ), n ± , n 0 = 0, ±1, ±2, ... and P (dψ) is the fermionic integration with propagator , χ(t) = 1 if t < 1 and 0 otherwise, C −1 0 (t) is a smooth compact support function = 1 for t < 1 and = 0 for t ≥ γ, γ > 1. We assume, for definiteness, a ≤ p F 4 , a 0 ≤ p F 20 so that the support of g k is over 4 disconnected regions; the Fermi surface is defined as the set of in which g k for k 0 = 0 is singular, in the limit β → ∞.
By using well known properties of Grassmann integrals, see [18], (6) allows to write the Grassmann field as a sum of independent fields with ψ ± ω,σ,x independent Grassmann variables with propagator g ω,σ,k . As in [6], [8] or [9] we can consider only interactions between parallel patches with v(x) a short range potential. The 2-point Schwinger function is given by . Figure 1: The Fermi surface corresponding to the singularities of g k ; the four sides are labelled by (σ, ω) = (±, ±).

Renormalization Group analyis
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 constructive Quantum Field Theory (for a general introduction to such methods, see [18]). The propagator (2) can be written as sum of "single slice" propagators in the following way where and ; note that in each term in (6) the change of variables k + → k + + ωp F has been performed. The single scale propagator verify the following bound, for any integer M The integration is done iteratively integrating out the fields with momenta closer and closer to the Fermi surface, renormalizing at each step the wave function. After the integration of the fields ψ (0) , ..., ψ (h+1) we obtain where By using that dk|g ) ); this means that the terms quadratic in the fields have positive scaling dimension and the quartic terms have vanishing scaling dimension, and all the other terms have negative dimension; we have then to properly renormalize the terms with non-negative dimension. Callingk = (k − , 0, 0) we define an L operator acting linearly on the kernels of the effective potential: .
where * ω is constrained to the condition i ε i ω i p F = 0 and we have used that, by symmetry, W h 2 (k) = 0. We write (9) as with R = 1 − L. The non trivial action of R on the kernel with n = 2 can be written as The first addend can be written as The factors k 0,1 and k +,1 are O(γ h ′ ), for the compact support properties of the propagator associated to ψ ; hence the effect of R is to produce a factor γ h ′ −h−1 < 1 making its scaling dimension negative. Similar considerations can be done for the action of R on the n = 1 terms. The effect of the L operation is to replace in W h 2 (k) the momentum k with its projection on the closest flat side of the Fermi surface. Hence the fact that the propagator is singular over an extended region (the Fermi surface) and not simply in a point has the effect that the renormalization point cannot be fixed but it must be left moving on the Fermi surface.
In order to integrate the field ψ (h) we can write (14) as where Moreover L V h is the second term in (13).
We rescale the fields by rewriting the r.h.s. of (14) as where and the effective couplings After the integrations of the fields ψ (0) , ψ (−1) , ..., ψ (h) we get an effective theory describing fermions with wave function renormalization Z h and effective interaction (21). Note that Z h and g h are non trivial functions of the momentum parallel to the Fermi surface. We write and the propagator of with H(k − ) f h (k) having the same support that H(k − )f h (k). We integrate then the field ψ (h) and we get and the procedure can be iterated.
The above procedure allows us to write W (h) 2n as a series in the effective couplings g k , k ≥ h, which is convergent, see [18], provided that L − is finite and ε h = sup k≥h ||g k || small enough; moreover ||W 2n || = O(γ −h(n−2) ). A similar analysis can be repeated for the 2-point function.
However even if the couplings g k starts with small values, they can possibly increase iterating the RG and at the end reach the boundary of the (estimated) convergence domain; if this happen, all the above procedure looses its consistency. A finite temperature acts as an infrared cut-off saying that the RG has to be iterated up to a maximum scale h β = O(log β) and, up to exponentially small temperatures i.e. β ≤ O(e κ|U | −1 ), then surely the effective couplings are in the convergence domain; however, in order to get lower temperatures, more information on the effective couplings are necessary.

RG flow and Luttinger liquid fixed point
The RG analysis seen in the previous section implies that the effective coupling g h verify a flow equation of the form where the r.h.s. of the above equation is called Beta function, which is expressed 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 the couplings and it is given by The above expression essentially coincides with the one found in [6] or [8]; it is indeed well known that the lowest order contributions to the Beta function are essentially independent by RG procedure one follows. The flow equation (26) encodes most of the physical properties of the model, but its analysis is extremely complex. Some insights can be obtained by truncating the beta function at second order, and by the numerical analysis of the resulting flow by discretization of the Fermi surface; it is found, see [6] or [8],that g h (k −,1 , k −,2 , k −,3 ) has a flow which, for certain values of k −,1 , k −,2 , k −,3 increases and reach the estimated domain of convergence of the series for W (k) n . While this increasing can be interpreted as a sign of instability, mathematically speaking this means that the truncation procedure becomes inconsistent.
A basic question is about the fixed points of the flow equation (26); in particular if there is,in addition to the trivial fixed point g h = 0, a non trivial fixed point corresponding to Luttinger liquid behavior. Note first that the set 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 either comes out from a single point, or are connected by a chain of propagators with the same ω, k − . Moreover in each Feynman graph the only dependence from the momenta of the external lines is through the function H(k − ) which are 1 in the support of the external fields dkH(k − )ψ ± k = dkψ ± k . For the same reasons also Z h is independent from k − . The crucial point is that, in the invariant set (30), some dramatic cancellation are present implying the following asymptotic vanishing of the beta function (which will be proved in the subsequent sections) saying that there is a cancellations between the graphs with four external lines 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 the sum 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 nontrivial fixed points for (26) of the form with λ −∞ k-independent and continuous function of U, λ −∞ = λ 0 + O(U 2 ), λ 0 = cU for a suitable constant c.
Note also that to such fixed point is associated Luttinger liquid behavior, as, from (19), Indeed the 2-point Schwinger function can be written as This means that to the fixed point is associated Luttinger liquid behavior, as the wave function renormalization vanishes at the Fermi surface as a power like with a non-universal critical index;the Luttinger liquid behavior is found only if L − is finite, as if L − → ∞ the critical index is vanishing. Note also that the cancellation in (32) reduce to the one in 1D if k − = k ′ − in (31).

The auxiliary model
There is essentially no hope of proving a property like (32) directly from the graph expansion, as the algebra of the graphs is too cumbersome (except than at one loop in which it is easy to check). We will follow instead the same strategy for proving the asymptotic vanishing of the Beta function in 1D followed in [15], [16], considering an auxiliary model with the same beta function, up to irrelevant terms, but verifying extra symmetries, from which a set of Ward Identities can be derived. In the present case, such identities are related to the invariance under local phase transformations depending on the Fermi surface side, which in the model (1) is broken by the cut-off function C −1 0 (k) and by the lattice. We consider an auxiliary model with generating function is with v(x 0 , x + ) a short range interaction. The above functional integral is very similar to the previous one, with the difference that there is an ultraviolet cut-off γ N on the + variables , which will be removed at the end, and an ultraviolet cut-off O(1) on the − variables; such features are present also in the models introduced in [9] or [12]. Again (36) can be analyzed by a multiscale integration based on a decomposition 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 four external lines have positive or vanishing dimension but one can use the non locality of the interaction to improve their scaling dimension. We integrate (with L = 0) the fields ψ (N ) , ψ (N −1) , .., ψ (k) and we call W (k) 2n,m the kernels in the effective potential multiplying 2n fermionic fields and m J fields. Again the dimension is γ −k(n+m−2) , k ≥ 0 and we have to improve the bounds using the non-locality of the interaction. We can write 2,1 (y, y 2 ; y 1 ) + (38) The first and the third addend of Fig.2 are vanishing, by the symmetry g(k 0 , k + , k − ) = −g(−k 0 , −k + , k − ); hence, using that g (j) 1 ≤ Cγ −j and that W (k) 2,1 is O(U) (by induction), we obtain the following bound Note that we have a gain O(L −1 − γ −2k) ), due to the fact that we are integrating over a fermionic instead than over a bosonic line.
. A similar bound is found for the third term in Fig.3; regarding the first term, we can rewrite it as 0,2 (z, y 0 ) and using that the first addend is vanishing; the second addend, by using the interpolation After the integration of the fields ψ (N ) , ψ (N −1) , ..., ψ (−1) we get a Grassman integral very similar to (1); the integration of the remaining fields ψ (0) , ψ (−1) , .. is done following the same procedure as in section 3, with the effective coupling of the form (30) and LV (k) of the form (31). The crucial point 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 the auxiliary model.

Ward Identities
We derive now a set of Ward Identities relating the Schwinger functions of the auxiliary model (36); by performing the change of variables and making a derivative with respect to α x,ω and to the external fields we obtain where ψ + ω,k ′ +p ψ − ω,k ′ ψ + ω ′ ,k−p ψ − ω ′ ,k is the derivative with respect to J p , φ + ω ′ ,k−p , φ − ω ′ ,k of (36). Computing (43) for p − = 0 we get,ifp = (0,p + ,p 0 ) and with In deriving the above equation we have used that for the compact support properties of the fields ψ k and H 2 = H; note also the 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 to the presence of the ultraviolet cut-off; as in 1D, such a term is not vanishing even 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 with R 2,1 ω a small correction. Indeed R 2,1 ω can be written as functional derivative, with respect to φ + , φ − , J, of e W ∆ (J,φ) = P (dψ)e −V (ψ)+ ω dz[ψ + ω,z φ − ω,z +φ + ω,z ψ − ω,z ]+T 0 (J,ψ)−T − (J,ψ) with T 0 (ψ) = dp + (2π) dp 0 (2π) dk (2π) 3 C(k,p)Jp ψ + k+p,ω ψ − k,ω (49) can be evaluated by a multiscale integration similar to the previous one, the only difference being that Jρ is replaced by T 0 − T − . The terms with vanishing scaling dimension of the form Jψ + ψ − can be obtained from the contraction of T 0 and T − ; in the first case we can perform a decomposition similar to the one in Fig.3 we can proceed exactly as in the previous section, the main difference being that at least one of the two fields have scale N so that they obey to the bound O(γ −2k γ −(1/2)(N −k) ). This follows from the fact that when C is multiplied by fixed point in 2D is not stable its presence has an important role in the physical properties of the system. Fermi surfaces with flat or almost flat pieces and no van Hove singularities are found in the Hubbard model with next to nearest neighbor interactions or in the Hubbard model close to half filling, and it is likely that our results can be extended, at least partially, to such models. Note however that in such models the sides of the Fermi surface are not perfectly flat, so that one 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 simplifying property of the model considered here is that the modulation of the Fermi velocity is taken constant along the Fermi surface, contrary to what happens in more realistic models; a momentum dependent Fermi velocity produces extra 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 effect deserves further analysis.