The 2d gross-neveu model at finite temperature and density with finite N corrections

Abstract

We use the linear delta expansion, or optimized perturbation theory, to evaluate the effective potential for the two dimensional Gross-Neveu model at finite temperature and density obtaining analytical equations for the critical temperature, chemical potential and fermionic mass which include finite N corrections. Our results seem to improve over the traditional large-N predictions.

Non perturbative methods; Gross-Neveu model; Finite temperature; Finite density

The 2d gross-neveu model at finite temperature and density with finite N corrections

Jean-Loïc Kneur^I; Marcus Benghi Pinto^II; Rudnei O. Ramos^III

^ILaboratoire de Physique Mathématique et Théorique - CNRS - UMR 5825 Université Montpellier II, France

^IIDepartamento de Física, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, SC, Brazil

^IIIDepartamento de Física Teórica, Universidade do Estado do Rio de Janeiro, 20550-013 Rio de Janeiro, RJ, Brazil

ABSTRACT

We use the linear d expansion, or optimized perturbation theory, to evaluate the effective potential for the two dimensional Gross-Neveu model at finite temperature and density obtaining analytical equations for the critical temperature, chemical potential and fermionic mass which include finite N corrections. Our results seem to improve over the traditional large-N predictions.

Keywords: Non perturbative methods; Gross-Neveu model; Finite temperature; Finite density

I. INTRODUCTION

The development of reliable analytical non-perturbative techniques to treat problems related to phase transitions in quantum chromodynamics (QCD) represents an important domain of research within quantum field theories. The appearance of large infrared divergences, happening for example in massless field theories, like in QCD [1], close to critical temperatures (in field theories displaying a second order phase transition or a weakly first order transition [2]) can only be dealt with in a non-perturbative fashion. Among the analytical non-perturbative techniques one of the most used is the 1/N approximation [3]. Though a powerful resummation method, this approximation can quickly become cumbersome after the resummation of the first leading contributions, like for the N = 3 case which regards QCD. This is due to technical difficulties such as the formal resummation of infinite subsets of Feynman graphs and their subsequent renormalization. In this work we employ an alternative non-perturbative method known as the linear d expansion (LDE) [4] to investigate the breaking and restoration of chiral symmetry within the two dimensional Gross-Neveu model [5] at finite temperature (T) and chemical potential (µ). As we shall see, the LDE great advantage is that the actual selection and evaluation,including renormalization, of the relevant contributions are carried out in a completely perturbative way. Non-perturbative results are generated through the use of a variational optimization procedure known as the principle of minimal sensitivity (PMS) [6]. The two dimensional Gross-Neveu model offers a perfect testing ground for the LDE-PMS because, apart from sharing common features with QCD, it is exactly solvable in the large-N limit. The large-N result for the critical temperature (at zero chemical potential) of the Gross-Neveu model is T_c 0.567 m_F(0) where m_F(0) is the fermionic mass at T = 0. However, due to the appearance of kink-anti-kink configurations, the exact critical temperature for this model should be zero [7]. Because kink configurations are unsuppressed the system is segmented into regions of alternating signs of the order parameter, at low temperatures. Then, the net average value of the order parameter is zero. At leading order, the 1/N approximation misses this effect because the energy per kink goes to infinity as N ® ¥ while the contribution from the kinks has the form e^-N. Our strategy will be twofold. First, we show that the LDE-PMS exactly reproduces, within the N ® ¥ limit, the ''exact" large-N result. Next we show explicitely that already at the first non trivial order the LDE takes into account finite N corrections which induce a lowering of T_c as predicted by Landau's theorem. Here, the calculations are performed for three cases which are: (a) T = 0 and µ = 0, (b) T ¹ 0 and µ = 0 and (c) T = 0 and µ 0. Our main results include analytical relations for the fermionic mass at T = 0 and µ = 0, T_c (at m = 0) and µ_c (at T = 0) which include finite N corrections. The case T ¹ 0 and µ 0, which allows for the determination of the tricritical points and phase diagram is more complex, due to the numerics. This situation is currently being treated by the present authors [8]. In the next section we review the Gross-Neveu effective potential at finite temperature and chemical potential in the large-N approximation. The LDE evaluations are presented in section III. The results are discussed in section IV while section V contains our conclusions.

II. THE GROSS-NEVEU EFFECTIVE POTENTIAL AT FINITE TEMPERATURE AND CHEMICAL POTENTIAL IN THE LARGE-N APPROXIMATION

The Gross-Neveu model is described by the Lagrangian density for a fermion field y_k (k = 1,...,N) given by [5]

When m_F = 0 the theory is invariant under the discrete transformation

displaying a discrete chiral symmetry (CS). In addition, Eq. (1) has a global SU(N) flavor symmetry.

For the studies of the Gross-Neveu model in the large-N limit it is convenient to define the four-fermion interaction as g²N = l. Since g² vanishes like 1/N, we then study the theory in the large-N limit with fixed l [5]. As usual, it is useful to rewrite Eq. (1) expressing it in terms of an auxiliary (composite) field s, so that [9]

As it is well known, using the 1/N approximation, the large-N expression for the effective potential is [5,9]

The above equation can be extended at finite temperature and chemical potential applying the usual associations and replacements. E.g., momentum integrals of functions f(p₀,p) are replaced by

where w_n = (2 n +1) pT, n = 0, ±1, ±2, ¼, are the Matsubara frequencies for fermions [10]. For the divergent, zero temperature contributions, we choose dimensional regularization in arbitrary dimensions 2w = 1-e and carry the renormalization in the scheme, in which case the momentum integrals are written as

where M is an arbitrary mass scale and g_E 0.5772 is the Euler-Mascheroni constant. The integrals are then evaluated by using standard methods.

In this case, Eq. (4) can be written as

where . The sum over the Matsubara's frequencies in Eq. (5) is also standard [10] and gives for the effective potential, in the large-N approximation, the result

After integrating and renormalizing the above equation one obtains

where

with a = s_c/T and b = µ/T. Taking the T = 0 and µ = 0 limit one may look for the effective potential minimum (_c) which, when different from zero signals dynamical chiral symmetry breaking (CSB). This minimization produces [5,9]

One may proceed by numerically investigating m_F as a function of T and µ as shown in Figure 1 which shows a smooth phase (second order) transition at µ = 0. At this point, the exact value for the critical temperature (T_c) at which chiral symmetry restoration (CSR) occurs can be evaluated analytically producing [11]

while, according to Landau's theorem, the exact result should be T_c = 0. By looking at Figure 1 one notices an abrupt (first order) transition when T = 0. The analytical value at which this transition occurs has also been evaluated, in the large-N limit, yielding [12]

In the T-µ plane there is a (tricritical) point where the lines describing the first and second order transition meet. This can be seen more clearly by analyzing the top views of figure 1. Figure 2 shows these top views in a way which uses shades (LHS figure) and contour lines (RHS figure). The tricritical point (P_tc) values can be numerically determined producing P_tc = (T_tc,µ_tc) = [0.318 m_F(0), 0.608 m_F(0)] [13].

III. THE LINEAR d EXPANSION AND FINITE N CORRECTIONS TO THE EFFECTIVE POTENTIAL

According to the usual LDE interpolation prescription [4] the deformed original four fermion theory displaying CS reads

So, that at d = 0 we have a theory of free fermions. Now, the introduction of an auxiliary scalar field s can be achieved by adding the quadratic term,

to

_d(y, ). This leads to the interpolated model

where h_* = h-(h- s_c)d. The counterterm Lagrangian density, _ct,d, has the same polynomial form as in the original theory while the coefficients are allowed to be d and h dependent. Details about renormalization within the LDE can be found in Ref. [14].

From the Lagrangian density in the interpolated form, Eq. (14), we can immediately read the corresponding new Feynman rules in Minkowski space. Each Yukawa vertex carries a factor -id while the (free) s propagator is now -il/(N d). The LDE dressed fermion propagator is

where h_* = h-(h- s_c)d.

Finally, by summing up the contributions shown in figure 3 one obtains the complete LDE expression to order-d

where I₁(a,b) is defined by Eq. (8), with a = h/T. Also,

and

Notice once more, from Eq. (16), that our first order already takes into account finite N corrections. Now, one must fix the two non original parameters, d and h, which appear in Eq. ( 16). Recalling that at d = 1 one retrieves the original Gross-Neveu Lagrangian allows us to choose the unity as the value for the dummy parameter d. The infra red regulator h can be fixed by demanding to be evaluated at the point where it is less sensitive to variations with respect to h. This criterion, known as Principle of the Minimal Sensitivity (PMS) [6] can be written as

In the next section the PMS will be used to generate the non-perturbative optimized LDE results.

IV. OPTIMIZED RESULTS

From the PMS procedure we then obtain from Eq. ( 16), at h = , the general result

where we have defined the function

Let us first consider the case N ® ¥. Then, Eq. (20) gives two solutions where the first one is = s_c which, when plugged in Eq. ( 16), exactly reproduces the large-N effective potential, Eq. ( 7). This result was shown to rigorously hold at any order in d provided that one stays within the large-N limit [15]. The other possible solution, which depends only upon the scales M,T and µ, is considered unphysical [15].

A. The case T = 0 and µ = 0

Taking Eq. ( 20) at T = µ = 0 one gets

As discussed previously, the first factor leads to the model independent result, = M/e, which we shall neglect. At the same time the second factor in (22) leads to a self-consistent gap equation for , given by

The solution for obtained from Eq. (23) is

where W(x) is the Lambert W function, which satisfies W(x)exp[W(x)] = x.

To analyze CS breaking we then replace h by Eq. (24) in Eq. ( 16), which is taken at T = 0 and µ = 0. As usual, CS breaking appears when the effective potential displays minima at some particular value _c¹ 0. Then, one has to solve

Since m_F = _c, after some algebraic manipulation of Eq. (25) and using the properties of the W(x) function, one finds

where we have defined the quantity (l,N) as

Eq. (26) is our result for the fermionic mass at first order in d which goes beyond the large-N result, Eq. (9). Note that in the N ® ¥ limit, F(l,N ® ¥) = exp( -p/l). Therefore, Eq. (26) correctly reproduces, within the LDE non perturbative resummation, the large-N result, as already discussed. In Fig. 4 we compare the order-d LDE-PMS results for _c with the one provided by the large-N approximation. One can now obtain an analytical result for evaluated at _c = s_c. Eqs. (24) and (26) yield

Fig. 5 shows that (_c) is an increasing function of both N and l kickly saturating for N 3. The same figure shows the results obtained numerically with the PMS.

B. The case T ¹ 0 and µ = 0

Let us now investigate the case T ¹ 0 and µ = 0. In principle, this could be done numerically by a direct application of the PMS the LDE effective potential, Eq. (16). However, as we shall see, neat analytical results can be obtained if one uses the high temperature expansion by taking h/T = a 1 and µ/T = b 1. The validity of such action could be questioned, at first, since h is arbitrary. However, we have cross checked the PMS results obtained analytically using the high T expansion with the ones obtained numerically without using this approximation. This cross check shows a good agreement between both results. Expanding Eq. (8) in powers of a and b, the result is finite and given by [16]

and

where z(3) 1.202. If we then expand Eq. (16) at high temperatures, up to order h²/T², we obtain

Now, one sets d = 1 and applies the PMS to Eq. ( 31) to obtain the optimum LDE mass

The above result is plugged back into Eq. ( 31) which, for consistency, should be re expanded to the order h²/T². This generates a nice analytical result for the thermal fermionic mass

Figure 6 shows _c(T)/M given by Eq. ( 33) as a function of T/M, again showing a continuous (second order) phase transition for CS breaking/restoration. The numerical results illustrated by Fig. 6 show that the transition is of the second kind and an analytical equation for the critical temperature can be obtained by requiring that the minima vanish at T_c. From Eq. ( 33) one sees that _c(T = T_c) = 0 can lead to two possible solutions for T_c.

The one coming from

can easily be seen as not been able to reproduce the known large-N result, when N ® ¥, T_c= M exp(g_E-p/l)/p. However, the other possible solution coming from

gives for the critical temperature, evaluated at first order in d, the result

with

_l (N) as given before, by Eq. (27). Therefore, Eq. (36) also exactly reproduces the large-N result for N ® ¥. The results given by this equation are plotted in Fig. 7 in terms of l for different values of N. The (non-perturbative) LDE results show that T_c is always smaller (for the realistic finite N case) than the value predicted by the large-N approximation. According to Landau's theorem for phase transitions in one space dimensions, our LDE results, including the first 1/N correction, seem to converge to the right direction.

C. The case T = 0 and µ 0

One can now study the case T = 0,µ 0 by taking the limit T ® 0 in the integrals I₁, I₂ and J₂ which appear in the LDE effective potential, Eq. ( 16). In this limit, both functions are given by

Then, one has to analyze two situations. In the first, h > µ, the optimized is given by

while for the second, h < µ, is found from the solution of

Note that the results given by Eqs. (37-39) vanish for µ < h. Fig. 8 shows µ_c, obtained numerically, as a function of l for different values of N. Our result is contrasted with the ones furnished by the 1/N approximation. The analytical expressions for (_c), Eq. (28), and T_c, Eq. (36), suggest that an approximate solution for µ_c at first order in d is given by

It is interesting to note that both results, for T_c, Eq. (7), and µ_c, Eq. (42), follow exactly the same trend as the corresponding results obtained from the large-N expansion, Eqs. (10) and (11), respectively, which have a common scale given by the zero temperature and density fermion mass m_F(0). Here, the common scale is given by evaluated at s_c = _c and T = µ = 0, (_c) = M (l,N).

V. CONCLUSIONS

We have used the non-perturbative linear d expansion method (LDE) to evaluate the effective potential of the two dimensional Gross-Neveu model at finite temperature and chemical potential. Our results show that when one stays within the large-N limit the LDE correctly reproduces the 1/N approximation leading order results for the fermionic mass, T_c and µ_c. However, as far as T_c is concerned the large-N predicts T_c 0.567 m_F(0) while Landau's theorem for phase transitions in one space dimensions predicts T_c = 0. Having this in mind we have considered the first finite N correction to the LDE effective potential. The whole calculation was performed with the easiness allowed by perturbation theory. Then, the effective potential was optimized in order to produce the desired non-perturbative results. This procedure has generated analytical relations for the relevant quantities (fermionic mass, T_c and µ_c) which explicitely display finite N corrections. The relation for T_c, for instance, predicts smaller values than the ones predicted by the large-N approximation which hints on the good convergence properties of the LDE in this case. The LDE convergence properties in critical temperatures has received support by recent investigations concerned with the evaluation of the critical temperature for weakly interacting homogeneous Bose gases [17]. In order to produce the complete phase diagram, including the tricritical points, we are currently investigating the case T ¹ 0 and µ 0 [8].

Acknowledgments

M.B.P. and R.O.R. are partially supported by CNPq. R.O.R. acknowledges partial support from FAPERJ and M.B.P. thanks the organizers of IRQCD06 for the invitation.

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Received on 10 September, 2006

Publication Dates

• Publication in this collection
  11 May 2007
• Date of issue
  Mar 2007

History

• Received
  10 Sept 2006