Instabilities in Luttinger liquids

We discuss the appearance of magnetic and charge instabilities, named respectively metamagnetism (MM) and phase separation (PS), in systems which can be described by a perturbed Luttinger liquid. We argue that such instabilities can be associated with the vanishing of the effective Fermi velocity v, which in some cases coincides with a divergence of the effective Luttinger parameter K. We analyze in particular an XXZ chain with next-nearest-neighbor interactions in different limits where MM shows up and an extended Hubbard model where in turn, PS occurs. Qualitative agreement with previous studies is found.


I. INTRODUCTION
The study of instabilities in low dimensional strongly correlated electron systems has received much attention in the last few years. One of the main reasons is that a charge instability phenomenon (phase separation (PS)) often shows up in the vicinity of the superconducting transition in cuprates. In the case of double exchange models for manganese oxides that exhibit the "Colossal" magnetoresistance effect [1], this charge instability arises close to the transition to ferromagnetism and, interestingly, finite size studies of both the two-dimensional realistic model and its one-dimensional version display similar features in this respect. The case of magnetic instabilities (metamagnetism (MM)) has also received recent attention in connection to the one dimensional antiferromagnetic (AF) XXZ spin chain with next-nearest-neighbor interactions (NNN), where it was found that MM arises in a finite region of the phase space [2].
Generically, at the point where charge or magnetic instabilities occur, a divergent compressibility or magnetic susceptibility arises. This divergence is in turn associated respectively with the coexistence of two phases with different hole concentrations or magnetizations.
The aim of this paper is to discuss a general way to determine whether an instability could show up in a given one-dimensional model, using Abelian bosonization. Our discussion applies to any one-dimensional model describable as an integrable model plus perturbations whose effect is to renormalize the Luttinger liquid (LL) parameters, K and v. The effect of irrelevant operators is also discussed. Our discussion could be also relevant in the study of certain 2D systems which can realize the so called sliding Luttinger phase [3], since a divergence in the 1D susceptibility leads to a singularity in the 2D one [4]. Generically, our approach provides a quick tool to study the tendency of different perturbations to produce instabilities in the above mentioned systems.
As a sample case for magnetic systems, we analyze the XXZ chain with NNN exchange, treating first this last interaction perturbatively within a bosonization approach and find qualitative agreement with the results obtained in [2] in finite systems (This problem was also studied in [5]). We also study the opposite coupling limit, which we call hereafter "zig-zag" limit, where the system can be reinterpreted as a two-leg zig-zag ladder. In this case, we determine a region in the parameter space where MM occurs.
To analyze the case of charge instabilities we consider the charge sector of the Hubbard model at incommensurate fillings, perturbed by a nearest-neighbor density-density interaction V . In this case we find that for small Coulomb repulsion U there is a region where the system phase separates which corresponds to negative values of V , in agreement with previous studies [6].

II. GENERAL DISCUSSION
We consider a generic situation in which the low energy degrees of freedom (which could correspond to either charge or spin variables) are described by a Tomonaga-Luttinger liquid in the unperturbed case. The corresponding Hamiltonian once interactions are taken into account will generically be of the form (modulo irrelevant terms) where H 0 corresponds to the Tomonaga-Luttinger Hamiltonian The first two terms can be readily absorbed into a redefinition of the LL parameters K → K ef f and v → v ef f , while the third one changes the chemical potential in the case of charge variables and the magnetic field in the spin case.
Under the above mentioned conditions the compressibility for the charge modes, described by an effective LL with parameters K c and v c can be shown to be given by [7] κ and similarly, the magnetic susceptibility for the spin modes described by an effective LL with parameters K s and v s is given by One readily observes that a divergence in these quantities arises either when K −1 (c,s) or v (c,s) vanish. It should be pointed out that these two things could happen simultaneously, but this is not always the case and hence PS or MM instabilities are to be identified with the vanishing of the effective velocity. We argue that this is true, provided that the quantities Kv and K/v remain positive definite and irrelevant perturbations do not change the large scale behavior. In Ref. [8] an attempt to characterize these instabilities in one dimensional systems exhibiting a universal character described by a Tomonaga-Luttinger model was made. In this paper the authors identified the divergence of the above mentioned thermodynamic quantities with a divergence in the so called Luttinger parameter K. Although this interpretation led to a consistent analysis for the cases studied in [8], we argue that a more general criterion consists in identifying the instability regions with those where the velocity vanishes. This last statement can be understood as follows: by analogy with the unperturbed XXZ chain in a magnetic field, we see that when the Fermi velocity goes to zero we approach a ground state of FM nature (and K does not necessarily diverge, though this happens for the particular case of the XXZ chain for ∆ = −1 and zero field [8]). Then the magnetic susceptibility diverges as 1/v ef f and, if this happens before reaching saturation, the magnetization curve as a function of the applied magnetic field presents a jump. Once we have shown that there is a jump in the magnetization curve, for some critical value of the applied magnetic field, h c , we can conclude that two different values of the magnetization will coexist at this point [9].
In the next two Sections we study two sample cases where this general discussion applies, the XXZ chain with NNN interactions where MM has been shown to occur [2], [5] and the extended Hubbard model with nearest-neighbour interactions V , where PS appears for negative V [6].

III. THE XXZ AF CHAIN WITH NNN INTERACTIONS
The Hamiltonian of the XXZ AF with NNN interactions in a magnetic field is given by The large scale behaviour of the XXZ chain can be described by a U (1) free boson theory with Hamiltonian (2). The field φ i and its dualφ i are given by the sum and difference of the light-cone components, respectively. The constant K governs the conformal dimensions of the bosonic vertex operators and can be obtained exactly from the Bethe Ansatz solution of the XXZ chain (see e.g. [10] for a detailed summary). We have K = 1 for the SU (2) symmetric case (∆ = 1) and is related to the radius R of [10] by K −1 = 2πR 2 . In (2) v corresponds to the Fermi velocity of the fundamental excitations of the system.
In terms of these fields, the spin operators read where the colons denote normal ordering with respect to the groundstate with magnetization M . The Fermi momentum k F is related to the magnetization of the chain as k F = (1 − M )π/2. The effect of an XXZ anisotropy and/or the external magnetic field is then to modify the scaling dimensions of the physical fields through K and the commensurability properties of the spin operators, as can be seen from (6), (7). The constants a, b and c were numerically computed in the case of zero magnetic field [11] (see also [12]).
In what follows we study both the weak coupling (J ′ /J ≪ 1) and the zig-zag (J ′ /J ≫ 1) limits.
i) J ′ /J ≪ 1 limit Using (6) and (7) the NNN interaction term in the bosonic language reads where α = J ′ /J and g 1,2 , λ and λ ′ depend on ∆ and the non-universal constants a, b and c as and The first two terms in (8) have the effect of renormalizing both the compactification radius and the Fermi velocity in the following way Now we can make contact with the analysis of Ref. [2]: the MM region is identified within our approach as the set of phase space points in which the susceptibility diverges for 0 < M < M sat .
One immediately observes from (12) that, for zero magnetic field and without taking into account the effect of the λ ′ perturbation, both v ef f and K −1 ef f vanish simultaneously and that this happens when (v/K + 2αg 1 ) = 0 (see Fig. 1). However, this situation can change for two reasons: first, an extra term arises from the NNN perturbation, which renormalizes the external magnetic field h and hence changes both v ef f and K ef f through the Bethe Ansatz equations, generically in a way which removes the above mentioned simultaneity. Second, if one goes beyond the zero loop order, both K ef f and v ef f will renormalize in a different way due to the λ ′ term [13]. We will discuss this issue again in the context of charge instabilities in the next Section. In particular, the boundary between the MM and the FM phases is obtained as the set of points in which v ef f and/or K −1 ef f vanish/es for h = 0, and that between the MM and the AF phases as the set of points in which v ef f = 0 and K −1 ef f = 0 for all values of the magnetization M < M sat .
Let us focus on the MM-FM boundary, which is easily obtained using (4) and (12). In this case, we can use the numerical values for the non-universal constants a, b and c appearing in (6) and (7) obtained in [11]. The boundary obtained in this way agrees qualitatively with that obtained in [2] (see Fig. 1). The lack of quantitative agreement is presumably due to the perturbative treatment of the NNN interactions, which could be improved by considering higher loop contributions in a renormalization group analysis from the bosonization side and due to finite size effects from the numerical one. However, our main aim is to discuss the appearance of instabilities in a generic and simple way.
One should be careful also about the regime of validity of this approach, since due to the renormalization of the Luttinger parameter K ef f , the scaling dimensions of the many discarded irrelevant perturbations change, and nothing prevents one of these to become relevant. In fact, by analyzing the scaling dimension of the leading irrelevant λ ′ perturbation, we observe that it reaches the limiting value 2 at α c = v(K 2 −1) 2K(g1−g2) , and hence our approach ceases to be valid for α ≥ α c . This critical line separates the massless regime from a massive one, and our results compare qualitatively well with those obtained in [14] (dashed line in Fig. 1). The study of the AF-MM boundary is in this case more involved due to the appearance of the non-universal constants a, b and c in the bosonized operators which are not available for non-zero magnetic field. (The estimation of the field dependence of these constants for 0 ≤ ∆ < 1 has been done in [15] but its evaluation is more delicate on the ferromagnetic side ∆ < 0 [16].) ii) Zig-zag limit (J ′ /J ≫ 1) In this limit we reinterpret the Hamiltonian (5) as a two leg zig-zag ladder in the weak interchain coupling limit [17][18][19][20][21][22]. In this description one represents each of the chains by one free compactified U (1) boson. Thus, the whole ladder is represented by two bosons φ 1 , φ 2 , each governed by an action given by (2), plus the perturbative terms arising from the interchain zig-zag coupling.
The effective Hamiltonian governing the large scale behavior of φ diag is then given by where, calling β = 1/α, λ = β∆ M / √ 2π and v ef f and K ef f are given by There is however an extra term that mixes the fields φ rel and φ diag The effect of this term was studied in Ref. [20] where it was shown that it gives rise to a spin nematic phase close to ∆ = 0. The main point for our analysis is that the diagonal field is still described by a LL, and we expect the same picture to apply for −1 < ∆ ≤ 0. For this reason, we do not take into account this last term in what follows. One then observes from (14) that the non-universal constants do not appear in the perturbing terms, and hence we can obtain in this case both the FM-MM and MM-AF boundaries. The results are presented in Fig. 2.

IV. HUBBARD MODEL WITH NEAREST NEIGHBOR INTERACTIONS
We consider now the Hubbard model, defined as where In the absence of an external magnetic field it has been shown that this model presents charge-spin separation. It has also been shown (c.f. [23]) that the large scale behavior of the spin and charge degrees of freedom can be described by two decoupled boson field theories with dynamics governed by the Tomonaga-Luttinger Hamiltonian. The parameters K and v can in each case be exactly obtained for all values of µ and U via numerically solving the Bethe Ansatz equations in [24]. Approximate expressions for the velocity of the charge sector in the small and large U regimes are given in [23].
The addition of a density-density interaction between nearest neighbors leads to the so called Extended Hubbard model, whose Hamiltonian is given by (16) plus the term where n j = n j,↑ + n j,↓ .
It is a simple matter to show that, modulo irrelevant operators, the effect of the perturbation (18) is to renormalize the parameters µ and U as follows: Therefore, the low energy behavior of the charge sector of the Extended Hubbard model is that of a Luttinger liquid with parameters where K(U, µ) y v(U, µ), are given by the exact Bethe Ansatz solution of the non-Extended Hubbard model (16). This allows for the determination of the divergences in the compressibility κ, as given by (3), which we identify with the zeros of the effective velocity. We have only considered small values of V , since our approach is a perturbative one. Within this restriction, we have found no singularities, and therefore no PS, in the large U regime. On the other hand, regarding the small U ,V region, in which the effective velocity is given by we have found that the roots of the above expression present the same qualitative behavior as that described in [6]. It should be stressed that K ef f remains finite in the region where (22) vanishes. Following this approach, one could also study the appearance of instabilities in the magnetic sector.

V. CONCLUSIONS AND DISCUSSION
We have discussed the appearance of MM and PS in systems that can be described as perturbed Luttinger liquids. Specifically, we have studied the XXZ spin chain with NNN interactions and the extended Hubbard model with nearest-neighbor density-density interactions. We have found FM, AF and MM regions in the zig-zag limit of the XXZ-NNN spin chain. In the weak coupling limit, qualitative agreement with previous results was found for the MM-FM transition. We were not able to complete our search for MM due to lack of specific numerical data, an issue to be discussed elsewhere. The instabilities were in all cases identified as the roots of the inverse susceptibility (4).
Concerning the extended Hubbard model, according to our analysis, it does not present PS in the large U and small V limit. On the other hand, in the small U , V limit instabilities do show up, through the roots of the effective Fermi velocity, while the effective Luttinger parameter remains finite.
We argue that all the instabilities studied are associated in general with a vanishing effective velocity v ef f . As a matter of fact, it is the roots of v ef f that give a divergent compressibility of the extended Hubbard model, though in the other cases studied, this coincides with a divergence of the effective Luttinger parameter K ef f . This should not be regarded as the actual hallmark of the instabilities: although our treatment of the XXZ-NNN spin chain gives at the same time v ef f = K −1 ef f = 0, this is only true in the absence of an external magnetic field and moreover ceases to be valid beyond the zero loop order since both parameter renormalize in a different manner [13]. It should be stressed that our approach gives a simple way to qualitatively analyze instabilities in generic charge and magnetic systems provided they can be described as perturbed Luttinger Liquids. Henceforth, it provides a quick tool to study whether different perturbations could produce such instabilities.