Renormalization group domains of the scalar Hamiltonian

Using the local potential approximation of the exact renormalization group (RG) equation, we show the various domains of values of the parameters of the O(1)-symmetric scalar Hamiltonian. In three dimensions, in addition to the usual critical surface $S_{c}$ (attraction domain of the Wilson-Fisher fixed point), we explicitly show the existence of a first-order phase transition domain $S_{f}$ separated from $S_{c}$ by the tricritical surface $S_{t}$ (attraction domain of the Gaussian fixed point). $S_{f}$ and $S_{c}$ are two distinct domains of repulsion for the Gaussian fixed point, but $S_{f}$ is not the basin of attraction of a fixed point. $S_{f}$ is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the $\phi ^{4}$-coupling. This renormalized trajectory exists also in four dimensions making the Gaussian fixed point ultra-violet stable (and the $\phi_{4}^{4}$ renormalized field theory asymptotically free but with a wrong sign of the perfect action). We also show that very retarded classical-to-Ising crossover may exist in three dimensions (in fact below four dimensions). This could be an explanation of the unexpected classical critical behavior observed in some ionic systems.


Introduction
The object of this paper is to carry on studying the local potential approximation of the exact renormalization group (RG) equation for the scalar theory [1]. In a previous publication [2] (to be considered as the part I of the present work), we had already considered this approximation with a view to qualitatively discuss the connection between the standard perturbative renormalization of field theory (as it can be found in most textbooks on field theory, see for example [3]) and the modern view [4] in which the renormalized parameters of a field theory are introduced as the "relevant" directions of a fixed point (FP) of a RG transform. Actually the local potential approximation, which allows us to consider all the powers of the field ϕ on the same footing, is an excellent textbook example of the way infinitely many degrees of freedom are accounted for in (nonperturbative) RG theory. Almost all the characteristics of the RG theory are involved in this approximation. The only lacking features are related to phenomena highly correlated to the non local parts neglected in the approximation and when the critical exponent η is small (especially for d = 4 and d = 3), one expects the approximation to be qualitatively correct on all aspects of the RG theory [1].
In the following we look at the domains of attraction or of repulsion of fixed points in the O(1) scalar theory in three and four dimensions (d = 3 and d = 4).
At first sight, one could think that the issue considered is very simple since, with regard to criticality, the O(1)-symmetric systems in three dimensions are known to belong to the same class of universality (the Ising class). Now, because the Ising class is associated to the domain of attraction of the unique (non-trivial) Wilson-Fisher fixed point [5], then by adjusting one parameter (in order to reach the critical temperature 1 ) any O(1) scalar Hamiltonian should be driven to the Wilson-Fisher fixed point under the action of renormalization. Consequently there would have only two domains for the O(1) scalar theory: the critical subspace S c (of codimension 1) in the Wilson space (S) of infinite dimensions of the Hamiltonian parameters (in which the RG transforms generate flows) and the complement to S of S c (corresponding to noncritical Hamiltonians).
In fact, that is not correct because there is another fixed point in S: the Gaussian fixed point which, although trivial, controls tricritical behaviors in three dimensions. Now each FP has its own basin of attraction in S [5]. The attraction domain of the Gaussian FP is the tricritical subspace S t of codimension 2 (with no intersection with S c ). In addition, we show that there is a second subspace of codimension 1 in S, called S f , which is different from S c , and thus which is not a domain of attraction to the Wilson-Fisher fixed point. There is no FP to which a point of S f is attracted to. S f is characterized by a negative sign of the ϕ 4 -Hamiltonian parameter u 4 and is associated with systems undergoing a first-order phase transition [6]. We show that an endless attractive RG trajectory is associated to this domain of first-order transitions. It is a renormalized trajectory (denoted below by T " 1 ) that emanates from the Gaussian fixed point. The frontier between S f and S c corresponds to the tricritical subspace S t which is the domain of attraction of the Gaussian fixed point while S f and S c are two distinct domains of repulsion for the Gaussian fixed point.
Actually, the situation is conform to the usual view. Considering the famous ϕ 4 -model [Landau-Ginzburg-Wilson (LGW) Hamiltonian] in which the associated coupling u 4 is positive, then the Hamiltonian at criticality is attracted exclusively to the Wilson-Fisher fixed point, but if u 4 is negative, a ϕ 6 -term is required for stability, but then one may get either a tricritical phase transition or a second-or a first-order transition [7]. In the present study we do not truncate the Hamiltonian which involves all the powers of the field ϕ.
We explicitly show that a system which would correspond to an initial point lying very close to the frontier S t in the critical side (in S c ) would display a retarded classical-to-Ising crossover [8]. This result is interesting with regard to ionic systems (for example) in which a classical behavior has been observed while an Ising like critical behavior was expected. The eventuality of a retarded crossover from the classical to the Ising behavior has previously been mentioned but without theoretical explanation on how this kind of crossover could develop [9]. In [8] a calculation suggests that the RPM model for ionic systems would specifically correspond to a scalar Hamiltonian with a negative sign for the ϕ 4 -Hamiltonian parameter (but the order parameter chosen is not the bulk density [10]). That calculation has motivated the present study.
We also indicate that the renormalized trajectory T ′′ 1 still exists in four dimensions. This makes the Gaussian fixed point ultraviolet stable and the scalar field theory formally asymptotically free. However the associated "perfect" action [11] would have the wrong sign to provide us with an acceptable (well defined) field theory.
The organisation of the paper is as follows. In section 2 we briefly present the local potential approximation of the exact RG equation to be studied. We introduce the strategy we have chosen to solve the resulting nonlinear differential equation with a view to show the trajectories of interest in the space S of infinite dimension. Because the practical approach to the Gaussian fixed point is made difficult due to the logarithmic slowness characteristic of a marginally irrelevant direction (for d = 3), we found it useful to first test our numerical method with a close approach to the Wilson-Fisher fixed point. We present the characteristic results of this approach and the various kinds of domains corresponding to u 4 > 0 (somewhat a summary of [2]) In section 3 we describe the various kinds of attraction or repulsion domains of the Gaussian fixed point (for a negative value of the ϕ 4 -Hamiltonian parameter) corresponding to tricritical, critical and first-order subspaces. Then we discuss the consequences and especially explicitly show how a retarded crossover from the classical to the Ising behavior can be obtained.
We then shortly discuss the case d = 4 when u 4 < 0.
In two appendices we report on some technical aspects of the numerical treatment of the RG equation studied, in particular on the appearance of spurious nontrivial tricritical fixed points (Appendix A.1).

The RG equation studied
The local potential approximation has been first considered by Nicoll et al [12] from the sharp cutoff version of the exact RG equation of Wegner and Houghton [13], it has been rederived by Tokar [14] by using approximate functional integrations and rediscovered by Hasenfratz and Hasenfratz [15]. As in [2] we adopt the notation of the latter authors and consider the following nonlinear differential equation for the simple function f (l, ϕ): in which a prime refers to a derivative with respect to the constant dimensionless field ϕ (at constant l) and f (l, ϕ) = V ′ (l, ϕ) is the derivative of the dimensionless potential V (l, ϕ);ḟ stands for ∂f /∂l| ϕ in which l is the scale parameter defined by Λ Λ 0 =e −l and corresponding to the reduction to Λ of an arbitrary initial momentum scale of reference Λ 0 (the initial sharp momentum cutoff). Finally, K d is the surface of the d-dimensional unit sphere divided by (2π) d .
A fixed point is solution of the equationḟ = 0. The study of the resulting second order differential equation provides the following results: • d ≥ 4, no FP is found except the Gaussian fixed point.
If one represents the function f (l, ϕ) as a sum of monomials of the form: 27039 · · ·, u * 6 = 14.4005 · · ·, u * 8 = 32.31289 · · ·, etc. Once the FP is known, one may study its vicinity which is characterized by orthogonal directions corresponding to the infinite set of eigenvectors, solutions of the differential equation (1) linearized at f * . The eigenvectors associated to positive eigenvalues are said relevant; when the eigenvalues are negative they are said irrelevant and marginal otherwise [19].
The relevant eigenvectors correspond to directions along which the RG trajectories go away from the FP and the irrelevant eigenvectors correspond to directions along which the trajectories go into the FP. A marginal eigenvector may be relevant or irrelevant.
Our present FP f * has only one relevant direction and infinitely many irrelevant directions (no marginal direction, however see [1]). As already explained and shown in [2], in order to approach f * starting from an initial point in S, one must adjust one parameter of the initial function f (0, ϕ). This amounts to fixing the temperature of a system to its critical temperature.
Starting with a known initial function (at "time" l = 0) say: we adjust u 2 (0) to the critical value u c 2 = −0.29958691 · · · corresponding to u 4 (0) = 3 so that f (l, ϕ) [solution at time l of the differential equation (1)] approaches f * when l → ∞. The approach to f * is characterized by the least negative eigenvalue λ 2 = −1/ω 1 (ω 1 was noted ω in [2]). This means that, in the vicinity of f * any parameter u n (l) evolves as follows (l → ∞): u n (l) ≃ u * n + a n exp (−ω 1 l) Fig. 1 illustrates this feature for the first four u n (l)'s in the approach to f * . In [2] the two associated attractive trajectories (locally tangent to the least irrelevant eigenvector in the vicinity of f * ) was noted T 1 and T ′ 1 . One may also constrain the trajectory to approach f * along the second irrelevant direction (with the associated attractive trajectories noted T 2 or T ′ 2 in [2] and associated with the second least negative eigenvalue λ 3 = −1/ω 2 ). In this case a second parameter of the initial f must be adjusted, e.g., u 4 (0) must be adjusted to u c 4 and simultaneously u 2 (0) to the corresponding u c 2 , see [16,2]. Then, in the vicinity of f * , any parameter u n (l) will evolve as follows: Looking for this kind of approach to f * , we have found that 6.66151663 < u c 4 < 6.66151669 and u c 2 = −0.58328898880579 · · · This has allowed us to estimate ω 2 ∼ = 2.84. Although the shooting method is certainly not well adapted to the determination of the eigenvalues (see the huge number of digits required in the determination of u c 2 and u c 4 ), our estimate is close to ω 2 ∼ = 2.8384 found by Comellas and Travesset [20].
Because u c 4 cannot be perfectly determined, the trajectory leaves the trajectory T 2 before reaching f * to take one of the two directions T 1 or T Finally, far away from the fixed point, the RG trajectory approaches the trivial high temperature fixed point characterized by a classical eigenvalue (equal to 1 2 ). The global picture summarizing the evolution of ω eff (l) along a RG trajectory initialized in such a way as to approach f * first along T 2 , is drawn on fig. 3.
The values we have determined by this shooting method are (for eigenvalues other than ω 2 already mentioned): which are close to the values found, for example, in [15,20]: ω 1 ∼ = 0.5952 and ν ∼ = 0.6895.
3 Trajectories for u 4 < 0 In the preceding section, we have obtained a RG trajectory approaching the Wilson-Fisher fixed point f * along T 2 by adjusting two parameters of the initial Hamiltonian (u c 4 and u c 2 ). This is exactly the procedure one must follow to determine a tricritical RG trajectory approaching the Gaussian fixed point in three dimensions (because of its two relevant directions). The only difficulty is to discover initial points in S which are attracted to the Gaussian fixed point. To this end, we again use the shooting method.
From usual arguments on the LGW Hamiltonian and from the work done by Aharony on compressible ferromagnets [6], one expects to find the tricritical surface in the sector u 4 < 0 (and with u 2 > 0). Thus we have tried to approach the Gaussian fixed point starting with initial function f (0) of the form: with (not large) negative values of u 4 (0), for example u 4 (0) = −1.
Because the Gaussian fixed point is twice unstable, we must adjust two parameters to approach it starting with (3). We do that by successive tries (shooting method). For example, if we choose u 4 (0) = −1 and u 6 (0) = 3 and determine a value of u 2 (0) such as to get a trajectory which does not go immediately towards the trivial high temperature fixed point, the best we obtain is a trajectory which approaches the Wilson-Fisher fixed point (thus the corresponding initial point belongs to the attraction domain of f * although u 4 (0) < 0 [6]). But if u 6 (0) = 2, the adjustment of u 2 (0) with a view to counterbalance the effect of the most relevant direction of the Gaussian fixed point (which would drive the trajectory toward the high temperature FP) yields a runaway RG flow towards larger and larger negative values of u 4 (l). From now on, the target is bracketted: the tricritical trajectory corresponding to u 4 (0) = −1 can be obtained with a value of u 6 (0) in the range ]2, 3[ (we actually find a rather close approach to the Gaussian fixed point for 2.462280 > u 6 (0) > 2.4622788 and 6.4618440 · · · > u 2 (0) > 6.4618407 · · ·).
The upperscript "+" is just a reminder of the fact that the eigenfunctions are defined up to a global factor and thus the functions χ − k (ϕ) = −χ + k (ϕ) are also eigenfunctions with the same eigenvalue λ k .

Case d = 3
Similarly to χ + 2 , the direction provided by χ − 2 in S is a direction of instability of the Gaussian fixed point. Now χ + 2 is associated with the well known renormalized trajectory T 1 on which is defined the usual (massless) ϕ 4 3 -field theory [16,2], for the same reasons a renormalized trajectory T ′′ 1 locally tangent to χ − 2 in the vicinity of the origin of S exists with the same properties as T 1 (see [2]). The difference is that T ′′ 1 lies entirely in the sector u 4 < 0 and is endless (not ended by a fixed point).
This endless renormalized trajectory is associated with systems undergoing a first-order phase transition. This is due to the absence of fixed point [21], in which case the correlation length ξ cannot be made infinite although for some systems lying close to T ′′ 1 and attracted to it (i.e. at the transition temperature), ξ may be very large (because T ′′ 1 is endless), in which cases one may say that the transition is almost of second order [22]. Of course, a domain of first-order phase transition in S was expected from the usual arguments [7,6], we only specify better the conditions of realization, in S, of the first-order transition. Fig. 4 shows the attractive trajectory T ′′ 1 together with the attractive tricritical line approaching the Gaussian fixed point. The tricritical surface S t separates the first-order surface S f from the critical surface S c . Fig. 4 shows also that systems lying close to the tricritical surface may still be attracted to the Wilson-Fisher fixed point. In this case the effective exponents may undergo a very retarded crossover to the asymptotic Ising values compared to usual systems corresponding to initial points chosen in the sector u 4 > 0 of S. Fig. 5 illustrates how minus the inverse of (2) provides us with different evolutions [calculated from (1)] of the effective exponent ν eff (τ ) [with τ ∝ (T − T c )/T c ] according to the initial point chosen in S. It is worth to explain how we have defined ν eff (τ ).
We have seen at the end of section 2 that the quantity (2) undergoes a flat extremum (or a flat inflexion point) at an RG eigenvalue of f * each time the RG flow runs along an eigendirection in the vicinity of f * . Now it happens that this extremum is less and less flat as one chooses larger and larger values of (u 2 (0) − u c 2 ) (for the eigenvalue ν) but still exists. This provides us with a way to express the evolution of an effective exponent ν eff when the RG-substitute to τ , namely (u 2 (0) − u c 2 )/u c 2 , is varied. Fig. 6 shows such an evolution for some initial Hamiltonian (with u 4 (0) = 4). Notice that for such a Hamiltonian, the extremum disappears before ν eff reaches the trivial value 1 2 (associated with the approach to the trivial high temperature fixed point and to a regular -non critical-behavior) while in the case of a Hamiltonian initialized close to the tricritical surface, the classical-to-Ising crossover is complete (see fig. 4). This is because in the latter case the RG trajectory comes close to the Gaussian fixed point (and ν eff (τ ) has an extremum at 1 2 ) before approaching to f * . This reinforces the idea that the so-called classical-to-Ising crossover actually exists only between the Gaussian and Wilson-Fisher fixed points [23].
The same configuration displayed by fig. 4 has been obtained also by Tetradis and Litim [24] while studying analytical solutions of an exact RG equation in the local potential approximation for the O(N )-symmetric scalar theory in the large N limit. But they were not able to determine "the region in parameter space which results in first-order transitions" [24].

Case d = 4
To decide whether the marginal operator (associated with the eigenvalue equal to zero, i.e. λ 2 in four dimensions, or λ 3 in three dimensions) is relevant or irrelevant, one must go beyond the linear approximation. The analysis is presented in [15] for d = 4. If one considers a RG flow along χ + 2 such that g 2 (ϕ, l) = c(l)χ + 2 (ϕ), then one obtains, for small c: c(l) = c(0) [1 − Ac(0)l] with A > 0. Hence the marginal parameter decreases as l grows. As it is well known, in four dimensions the marginal parameter is irrelevant. However, if one considers the direction opposite to χ + 2 (i.e. χ − 2 ) then the evolution corresponds to changing c → −c. This gives, for small values of c: c(l) = c(0) [1 + A |c(0)| l] and the parameter becomes relevant. The parameter c is the renormalized φ 4 coupling constant u R and it is known that in four dimensions the Gaussian fixed point is IR stable for u R > 0 but IR unstable for u R < 0 [25].
We have verified that the trajectory T ′′ 1 survives when d = 4 (contrary to T 1 , see [2]). That trajectory T ′′ 1 is a renormalized trajectory on which we could define a continuum limit for the ϕ 4 4 -field theory and if the corresponding (perfect) action was positive for all ϕ, one could say that the φ 4 4 -field theory with a negative coupling is asymptotically free. Unfortunately, because the ϕ 4 -term is dominant for large ϕ in the vicinity of the origin of S (due to the relevant direction provided by χ − 2 ), the negative sign of the renormalized coupling prevents the path integral to be well defined.
However, because the action to which one refers in the continuum limit (the perfect action) is formal (because it involves an infinite number of parameters and cannot be written down, see [2]) we wonder whether the wrong sign of the action is actually a valid argument to reject the ϕ 4 4 -field theory with a negative renormalized coupling. It is worth to mention that the asymptotically free scalar field theory which has recently been considered on a lattice [26] could actually be the φ 4 4 -field theory with a negative coupling to which we refer here.

A The finite difference method used
For technical reasons, instead of studying Eq. (1), we consider the differential equation satisfied by g(ϕ) = f ′ (ϕ) (i.e. the second derivative of the potential with respect to the field): Starting with a known initial function (at "time" l = 0), we follow its evolution in S by approximating the differential equation (5) by finite differences and a two dimensional grid with the uniform spacings dy = 0.01 and dl = 0.000390625. The finite difference formulas for the derivatives g ′′ and g ′ have been chosen with the accuracy O(dy 4 ): g ′′ (y) = 16 The evolutionary function g(y, l) is known (calculated) at the discret set of points y i = i · dy with (i ≥ 0) and a maximum value i max = 82. (This value is large enough to study the approach to the Wilson-Fisher fixed point with a great accuracy but is too small for studying precisely the approach to the Gaussian fixed point.) At each time l k = k · dl, the derivatives are estimated from g(y i , l k ) = g(y i , l k−1 )+ġ(y i , l k−1 )·dl by using Eqs(6,7) which apply only for 1 < i < i max −1 . For the marginal points i = 0, 1 we use the parity of g(y) [by inserting g(−n · dy) = g(n · dy) for n = 1, 2 in Eqs (6,7)]. For the two other marginal points i = i max −1 , i max of the grid, there is no fixed solution and we shall used alternately the two following conditions [using the obvious abreviation g(i) instead of g(i · dy)]: Condition 2 is more accurate than condition 1 but leads sometimes to strong unstabilities which do not appear when we first use Condition 1 and then Condition 2 after some finite "time" l 0 . The validity of the procedure is tested by, for example, trying to approach a given fixed point (see the main part of the paper).
A.1 Appearance of spurious fixed points in the approach to the Gausian fixed point In trying to determine the attractive tricritical trajectory (approaching the Gaussian fixed point), we have encountered a spurious twice unstable fixed point lying at some finite and non negligible distance to the Gaussian fixed point. To understand the origin of this undesirable numerical effect, it is necessary to discuss a bit the solution of the fixed point equationḟ = 0. From (1) or (5), one sees that the fixed point equation is a second order non linear differential equation and a solution would be parametrized by two arbitrary constants. One of these two constants may easily be determined: since g * (ϕ) is expected to be an even function of ϕ [O(1) symmetry] then g * ′ (0) = 0 may be imposed. It remains one free parameter, thus a one-parameter family of (nontrivial) fixed points are solutions to the differential equation. But there is not an infinity of physically acceptable fixed points; all but a finite number of the solutions in the family are singular at some ϕ c [15,18,27]. Formally, by requiring the physical fixed point to be defined for all ϕ then the acceptable fixed points are limited to the Gaussian fixed point and (for d = 3) to the Wilson-Fisher fixed point.
However, in our study, because we numerically consider the function g(ϕ) in some finite range of values of ϕ (see above: i max = 82), it appears that in approaching the origin of S, infinitely many pseudo-fixed points exist which have there ϕ c -singularity located outside the finite range explicitly considered and there is at least one of them which looks like a tricritical fixed point. When we enlarge the range of ϕ to i max = 200, the previously observed nontrivial tricritical fixed point disappeared to the benefit of another one located closer to the origin. In conclusion, a larger and larger number of grid-points must be considered as one tries to come closer and closer to the Gaussian fixed point. This particularity together with the slowness of the approach along a marginal direction makes it excessively difficult to come very close to the Gaussian fixed point. not real (it is accidental, due to a projection onto a plane of trajectories lying in a space of infinite dimension). The points A or B could correspond to the restricted primitive model of ionic systems (see [8]). 5. Evolutions of an effective exponent ν eff (τ ) [with τ ∝ (T − T c )/T c ] along three different families of RG trajectories (see text for additional details). The full squares indicate the evolution of ν eff (τ ) for a family of trajectories initialized in the sector u 4 > 0 with u 4 (0) = 3 and for various values of u 2 (0) (the same system at criticality corresponds to the white circle "Simple fluid" of fig. 4). When u 2 (0) → u c 2 the effective exponent approaches the critical exponent value ν ∼ = 0.69 compatible with the present study. One observes that the crossover towards the classical value 1 2 is not complete because ν eff (τ ) ceases to make sense before τ becomes large. This is not the case of the evolution represented by the full circles which corresponds to trajectories initialized close to the Gaussian fixed point. In this case the complete crossover reproduces the interpolation between the Gaussian and the Wilson-Fisher fixed points and typically corresponds to the usual answer given by field theory [28]. The third evolution (full triangles) corresponds to a family of Hamiltonian initialized close to the tricritical surface but still attracted to the Wilson-Fisher fixed point. One sees that the classical-to-Ising crossover is complete but highly retarded compared to the two other cases. This is because at criticality, the RG flow is first attracted to the Gaussian fixed point (showing then an apparent classical value of ν) before interpolating between the Gaussian and the Wilson-Fisher fixed point.