Semiclassical partition function for the double-well potential
D. Kroff, A. Bessa, C. A. A. de Carvalho, E. S. Fraga, S. E. Jorás
aa r X i v : . [ h e p - ph ] N ov Semiclassical partition function for the double-well potential D. Kroff , ∗ , A. Bessa † , C. A. A. de Carvalho ‡ , E. S. Fraga , , § and S. E. Jor´as ¶ Instituto de F´ısica, Universidade Federal do Rio de Janeiro,Caixa Postal 68528, 21941-972, Rio de Janeiro, RJ , Brazil Institute de Physique Th´eorique CEA/DSM/Saclay,Orme des Merisiers 91191 Gif-sur-Yvette cedex, France Escola de Ciˆencias e Tecnologia, Universidade Federal do Rio Grande do Norte,Caixa Postal 1524, 59072-970, Natal, RN , Brazil Institute for Theoretical Physics, J. W. Goethe-University, D-60438 Frankfurt am Main, Germany Frankfurt Institute for Advanced Studies, J. W. Goethe University, D-60438 Frankfurt am Main, Germany (Dated: August 6, 2018)We compute the partition function and specific heat for a quantum mechanical particle underthe influence of a quartic double-well potential non-perturbatively, using the semiclassical method.Near the region of bounded motion in the inverted potential, the usual quadratic approximationfails due to the existence of multiple classical solutions and caustics. Using the tools of catastrophetheory, we identify the relevant classical solutions, showing that at most two have to be considered.This corresponds to the first step towards the study of spontaneous symmetry breaking and thermalphase transitions in the non-perturbative framework of the boundary effective theory.
I. INTRODUCTION AND MOTIVATION
In the analytic description of phase transitions in par-ticle physics and nuclear theory, one usually relies onthe effective model approach, given the complexity of thefundamental theories involved. If we consider strong in-teractions, the phase diagram related to chiral symmetryrestoration and deconfinement is a particularly interest-ing example, since they are within experimental reachand currently being investigated by different experimentsat RHIC-BNL and LHC-CERN [1]. Usually, one gener-ally adopts low-energy effective models such as the linearsigma model [2, 3] and the Nambu–Jona-Lasinio model[4], which can be combined with different versions of thePolyakov loop model [5]. The standard approach, then,corresponds to the computation of a thermal effective po-tential from which one can extract information on the dif-ferent phases and all thermodynamic quantities, so thatone can build a phase diagram.In most cases, the computation is performed in themean-field approximation with one-loop thermal correc-tions assuming homogeneous and static background fields[6]. Frequently, vacuum loop contributions are ignored,even in a theory with spontaneous symmetry breaking,where the presence of a condensate always modifies themasses, which then become medium-dependent quanti-ties, affecting significantly the phase structure [7–16].So, the highly non-linear behavior of the effective po-tential for large fields is completely missed, as well asnon-perturbative effects (with the exception of the treat-ment within the functional renormalization group [17]). ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] Those aspects can, in principle, dramatically modify thephase structure provided by a given effective model.The boundary effective theory formalism [18, 19] fur-nishes a non-perturbative method to calculate the parti-tion function of quantum systems in thermal equilibriumin which configurations that are not strictly periodic playthe main role. In such approach, one can compute thethermal one-loop effective potential for a system of mass-less scalar fields with quartic interaction [20]. The cal-culation relies on the solution of the classical equation ofmotion for the field, and Gaussian fluctuations aroundit. The result is non-perturbative and differs from thestandard one-loop effective potential [21] for field valueslarger than
T / √ λ , T being the temperature and λ thecoupling [20].The natural extension would be the calculation of theeffective potential in the case with spontaneous symme-try breaking. That would allow for the description ofphase transitions in effective models incorporating non-linear and non-perturbative effects, as well as controllingthe infrared divergences of thermal field theory in a well-defined and relatively simple way [18, 19, 22]. However,to develop the method to be applied in this case, it isnecessary to deal with multiple classical solutions, sincemore than one solution may satisfy the boundary condi-tions in euclidean time.As a first step towards the study of spontaneous sym-metry breaking and thermal phase transitions using theboundary effective theory, in this paper we focus onthe simpler case of computing the semiclassical partitionfunction for a quartic double-well potential in quantumstatistical mechanics. Although apparently trivial andstraightforward, the inverted potential in this case hasa region of bounded motion. Therefore, one also hasto deal with multiple solutions and their coalescence asthe temperature changes. The usual quadratic approx-imation may yield good results when such solutions arefar away from each other in functional space, but, as weshall see later on, this is not so in the opposite scenario.Among the numerous solutions, we use the framework ofcatastrophe theory to identify the only two relevant ones,following Refs. [23, 24]. We then compute the partitionfunction and specific heat, obtaining the correct limits atboth high and low temperatures, and a regular behaviorwhere the usual quadratic approximation diverges.The paper is organized as follows. In Section II wereview general characteristics of the semiclassical path-integral representation of the partition function and dis-cuss the case of multiple solutions in the double well. InSection III we use the tools from catastrophe theory todeal with the coalescence of solutions and identificationof relevant minima. In Section IV we present results forthe partition function and the specific heat, discussingtheir controlled behavior and the domain of validity ofour approximation. Section V contains our summary. El-ements and some technical details of catastrophe theoryare presented in appendices. II. SEMICLASSICAL PATH-INTEGRALREPRESENTATION OF THE PARTITIONFUNCTIONA. General Features
In statistical mechanics, the partition function for asystem in contact with a thermal reservoir at tempera-ture T is given by the sum of a probabilistic weight, thediagonal elements of the density matrix, over a stochasticvariable that labels the state of the system. This objectis of fundamental importance, as it encodes all the ther-modynamic information.For a one-dimensional quantum-mechanical systemconsisting of a single particle, the stochastic variable canbe chosen as the Schr¨odinger-picture position operatoreigenvalue. Therefore, if the dynamics is dictated by theHamiltonian operator ˆ H , the partition function is writtenas (1 /β ≡ k B T ): Z = Z ∞−∞ dx h x | exp( − β ˆ H ) | x i . (1)The matrix element in the previous equation can beunderstood as the analytic continuation of the transi-tion amplitude h x | exp[ − i ( ˆ H/ ~ )( t − t ′ )] | x i to imaginarytime, allowing for a formal expression for the diagonal el-ements of the density matrix in terms of path integrals[25]. If we restrict our analysis to systems subject tovelocity-independent potentials, the desired expressionhas the well-known form: h x | exp( − β ˆ H ) | x i = Z x (0)= x ( β ~ )= x [ D x ( τ )] e − S E / ~ , (2)where S E [ x ] = Z β dτ " m (cid:18) dxdτ (cid:19) + V ( x ) . (3) In other words, the diagonal elements of the density ma-trix are obtained integrating the exponential of the Eu-cliden action S E over the paths x ( τ ) in imaginary timesatisfying the conditions x (0) = x ( β ~ ) = x .For convenience we define the dimensionless quan-tities q ≡ x/x N , θ ≡ ω N τ , Θ ≡ β ~ ω N , U ( q ) ≡ V ( x N q ) /mω N x N and g ≡ ~ /mω N x N where ω − N and x N are the natural time and length scales of the problemunder consideration, respectively. In terms of these, thepartition function can be written as follows: Z (Θ) = Z ∞−∞ dq Z q (0)= q (Θ)= q [ D q ( θ )] e − I/g , (4)where I [ q ] = Z Θ0 dθ " (cid:18) dqdθ (cid:19) + U ( q ) . (5)In general, it is not possible to solve exactly the pathintegral above, but we can still resort to approximationprocedures in order to evaluate it. A very natural ap-proach is the JWKB [26] asymptotic expansion in ~ (or g ) — also known as semiclassical approximation — thatwe briefly discuss below.The trajectories that extremize the euclidean action I [ q ] are those satisfying the Euler-Lagrange equation( U ′ ≡ dU/dq ), d q c dθ − U ′ ( q c ) = 0 , (6)subject to the boundary conditions q (0) = q (Θ) = q .In other words, these are the classical solutions describ-ing the motion of a particle under the influence of theinverted potential − U ( q ).Due to its euclidean nature, the path integral in equa-tion (2) is dominated by the functions in the vicinity ofthose that minimize I [ q ]. So, one has to determine amongthe solutions of (6) those representing minima, which wedenote by ¯ q ic . Expanding the action around the minima,we have I [¯ q ic + η ] = I [¯ q ic ] + I [¯ q ic , η ] + δI [¯ q ic , η ], where I [¯ q ic , η ] = 12 Z Θ0 dθ η ( θ ) (cid:20) − d dθ + U ′′ (¯ q ic ) (cid:21) η ( θ ) , (7) δI [¯ q ic , η ] = ∞ X k =3 k ! Z Θ0 dθ U ( k ) (¯ q ic ) η k ( θ ) . (8)Keeping only terms up to quadratic order in the fluctu-ations, one obtains the so-called standard semiclassicalapproximation for the partition function: Z ≈ Z ∞−∞ dq X i exp( − I [¯ q ic ] /g )∆ − / , (9)where ∆ is the determinant of the quadratic fluctuationoperator∆ = det ˆ F [¯ q ic ] = det (cid:20) − d dθ + U ′′ (¯ q ic ) (cid:21) . (10)As an example, let us consider a single-well potential U ( q ), whose global minimum is located at the point q m ,as depicted in Fig. 1. Following the method describedabove, one has to obtain the solutions describing the clas-sical motion of the particle under the influence of the in-verted potential that leaves the point q at θ = 0 andreturns after a time interval Θ. q m q - U FIG. 1: Single-well inverted potential.
As the potential − U ( q ) is unbounded from below, ifthe particle departs from a point such that q < q m ( q > q m ), it will only return to the initial position ifits initial velocity points to the right (left), otherwise theparticle will move directly towards −∞ (+ ∞ ). However,the initial velocity can not be arbitrarily large, for if theparticle energy is greater than the height of the potentialbarrier, it will not return to its initial position either, asit will move directly towards + ∞ ( −∞ ). Thus, the max-imum possible value for the particle energy is exactly thebarrier height.For a fixed value of q , the time the particle spendsgoing from the initial position up to the turning point q t is a function of q t only, given by the following expression:Θ2 = sign( q t − q ) Z q t q dq p U ( q ) − U ( q t )] . (11)Clearly, the previous expression vanishes when q t = q .But, as the turning point moves further up the barrier,the time of flight increases continuously, diverging when q t is exactly at the top. Therefore, for any value of Θ,it is possible to determine the one solution satisfying q (0) = q (Θ) = q by choosing the appropriate turningpoint. It is, then, a straightforward task to implementthe semiclassical method, as was demonstrated in detailin Ref. [27]. In fact, even the D -dimensional case can betreated for central potentials [28]. B. Double-well potential: multiple solutions
The problem becomes more intricate in the case ofdouble-well potentials. Suppose now that U ( q ) representsa double-well potential with degenerate minima locatedat q = a and q = b , with a < b , e.g. like the one sketchedin Fig. 2. As we shall see, it is now necessary to dealwith multiple classical solutions . a b q - U FIG. 2: Double-well inverted potential. If q < a or q > b , the particle lies in a region ofunbounded motion under the potential − U , resemblingthe single-well case. It is trivial to extend the argumentsgiven in the previous section and conclude that, as before,each pair ( q , Θ) defines a unique solution to equation (6).Let us now analyze what happens when a < q < b , i.e.when the particle starts in the well of − U ( q ). Once again,its energy has to be smaller than the barrier height, oth-erwise the particle will leave the well towards ±∞ with-out ever returning to its initial position. In other words,there is a maximum allowed speed for such particles andall the solutions departing from a point in the well mustalways remain therein. In this region of bounded mo-tion, we see a much richer structure, with the possibilityof multiple classical solutions for a given q , dependingon Θ.If the temperature is high enough, the available timeof flight is still very restrictive. Accordingly, since thespeed is limited, the particle will be able to move onlytowards the nearest peak ( q t and q will have the samesignal — see Fig. 3, lower panel), but it will not be ableto reach points too far from its initial position. Thus,in this limit, we still have a single solution for every q .Lowering the temperature (increasing Θ), the particlewill be able to go further away and eventually it will be The quartic double-well potential at finite temperature has beeninvestigated previously using semiclassical, variational and per-turbative methods, e.g. in Refs. [29–32]. It is essential for this piece of the argument that the maxima ofthe inverted potential are degenerate. The method discussed inthe present work can be generalized and applied to the case ofnon degenerate minima. able to reach also the opposite side of the potential welland return to its initial position. From then on (i.e, forlower temperatures), a fixed q will no longer define aunique classical solution [23].In order to apply the semiclassical method with mul-tiple solutions, one has to be able to identify and keeponly those representing minima of the euclidean action infunctional space, discarding maxima and saddle points,which both correspond to unstable solutions with at leastone negative eigenvalue of the quadratic fluctuation op-erator ˆ F [ q c ], defined in equation (10).In the next section, we restrict ourselves to a quarticdouble-well potential and, using the language of catas-trophe theory, we not only identify how the number ofsolutions changes as we vary the parameters ( q , Θ), butalso find a straightforward criterium to determine whichclassical trajectories must be taken into account.
III. COALESCENCE OF SOLUTIONSA. Caustics and catastrophes for the quarticdouble-well potential
From now on, we consider the specific case of a quar-tic double-well potential, V ( x ) = − mω x / λx / q ≡ x/x N , x N ≡ p mω /λ : U ( q ) = λm ω V ( x N q ) = − q + 14 q . (12)As discussed previously, if the temperature is suffi-ciently high, there is only one closed path, with a singleturning point, for every q . Lowering the temperature,we go from a single-solution to a three-solution regime.Lowering it even further, we reach a five-solution regimeand so on [23].Fig. 3 is a clear illustration of the feature of solutionbifurcation. It shows the plots of the time of flight Θ vs. the first turning point q t of the classical path for two dif-ferent values of q . One can read directly from the plotsthe values of q t that allow the particle to return to q in atime interval Θ. As fixing the initial position and the firstturning point defines univocally the classical trajectory,the plot shows the number of classical solutions relatedwith each value of the time of flight.Thus, the plane ( q , Θ) is divided into several regionswith different numbers of solutions, as shown in Fig. 4.Moving from a certain region to a neighbouring one, twosolutions are either created or annihilated. Exactly at thefrontier between those regions, two classical trajectoriescoalesce. The curves defining the frontiers between twosuch regions are named caustics , for they are analogous The solution q c ( θ ) is a minimum when all the eigenvalues arepositive, a maximum when they are all negative and a saddle-point otherwise. - - q t Q - - q t Q FIG. 3: Plots of the time of flight Θ vs the turning point q t for q = 0 (upper) and q = 0 . q = q t = 0 = q ( t ) ∀ t , although valid forall Θ, is not shown. to the optical phenomenon. In our case, the classicalsolutions play the role of the light rays and the actionreplaces the optical distance [23]. - - q Q FIG. 4: The plane ( q , Θ) is divided into regions with dif-ferent number of classical solutions. The frontiers betweenthose regions, named caustics, are shown above. The smoothcurves indicate the coalescence of strictly periodic solutions,i.e, those that begin and end at the same position and at thesame velocity.
The information depicted in Figures 3 and 4 is com-bined into a single 3D plot in Figure 5.To apply the semiclassical method to compute the par-tition function for the double-well potential we must de-
FIG. 5: Three dimensional plot of q t × q × Θ (upper panel)and its projections onto the planes q t × Θ (bottom left) and q × Θ (bottom right). termine which of the solutions are actual minima of theaction and track them down. To do so, we use the frame-work of catastrophe theory [33], which classifies and stud-ies how the extrema of certain functions coalesce andemerge. A brief summary of a few ingredients of catas-trophe theory is presented in Appendix A.
B. Finding the minima
Following Appendix A, new real solutions of (6), i.e.new extrema of the euclidean action, emerge wheneverthe fluctuation determinant ∆ vanishes. This provides acriterium to determine the location of the caustics.It is clear that, for the problem under consideration,there are only two variables controlling the pattern of ac-tion extrema, which we choose to be q and Θ. There-fore, we are dealing with catastrophes whose codimen-sion is not greater than two. The only two catastrophessatisfying this condition are the fold and the cusp, bothhaving only one essential variable or coordinate [33]. Inother words, we know that, in our case, only one eigen-value of the fluctuation operator vanishes when a causticis crossed.Therefore, we can focus on one direction of the func- The third available quantity, q t , can be written in terms of thechosen ones. tional space: the one defined by the eigenfunction whoselowest eigenvalue vanishes. If we project the action ontothat direction and perform a change of variables (see Ap-pendix B), we will reach the so-called normal form (seeTable I): I N ( z ) = 14 z + u z + vz + s , (13)where z is the coordinate associated to the aforemen-tioned direction in functional space and u and v are thecontrol parameters. The bifurcation set is then given by dI N dz = d I N dz = 0 ⇒ v + 4 u = 0 . (14)The previous equation defines a cusp in the control pa-rameter space ( u, v ), dividing it in two parts — see Fig.6. To the right of the curve the action has one minimum,to its left there are one maximum and two minima. FIG. 6: Cuspid 27 v +4 u = 0 in the control parameter space.See text and Figs. 7 and 8 for explanation of the arrows. If the cusp is crossed at its vertex, as in arrow 1 of Fig.6, the original minimum becomes a maximum and twosymmetric minima appear — see Fig. 7. On the otherhand, if the crossing happens at any other point, as inarrow 2 in Fig. 6, the original minimum remains and twonew solutions appear — a maximum and a new (local)minimum — see Fig. 8. In the former case, one solutionsplits into three; in the later, two new solutions emerge(out of the coalescence of their complex counterparts —see next paragraph) while the previously existing mini-mum is unaffected. This picture agrees with our previousstatement that the number of solutions of (6) increasesby two.It is useful to think of exactly the same merging of ex-trema that happens in the algebraic equation (13). Beinga fourth-order polynomial with real coefficients, there arealways three extrema which may be either real or imag-inary, depending on the values taken by the control pa-rameters { u, v } . Then, one usually speaks of the coales-cence of complex solutions (which always come in pairsand are conjugate to each other) and their subsequentseparation along the real axis, as opposed to their plain creation out of nothing. FIG. 7: Behavior of the action in functional space when thecusp is crossed at the vertex. See arrow 1 in Fig. 6.FIG. 8: Behavior of the action in functional space when thecusp is crossed at a point that is not the vertex. See arrow 2in Fig. 6.
As one lowers the temperature, at the next catastrophethe classical trajectory with highest action is the one thatgives rise to two new solutions. So, the solutions emerg-ing after the second caustic are still maxima along thedirection (in functional space) of the first catastrophe.On all the forthcoming catastrophes, the same happens:new solutions originate from the one with the highestaction. Thus, in the multiple-solution regime, the onlysolutions that are actual minima of the action are thosethat are minima along the direction of the first catastro-phe. So, to apply the semiclassical method, we just haveto be concerned about at most two classical solutions.Moreover, as the first catastrophe happens before theemergence of strictly periodic solutions (in fact, thesesolutions appear only in the second catastrophe), we canguarantee that the solutions we have to keep have a singleturning point. Hence, there is a criterium that allowsto determine which solutions of (6) we must use whenapplying the semiclassical method: at the end we needonly the single-turning-point trajectories . IV. PARTITION FUNCTION AND SPECIFICHEAT FOR THE DOUBLE-WELL POTENTIAL
The solutions of the classical equation of motion forthe potential given in equation (12) can be expressed interms of Jacobi Elliptic Functions [34, 35]. In particular,the solutions we are interested in, the ones with a singleturning point, can be written as q ( θ ) = q t cd (cid:20)q − q t / θ − Θ / , k (cid:21) (15)where we define k ≡ q t / p − q t . Θ q H Θ L - Θ q H Θ L FIG. 9: Plots of the classical solutions q ( θ ) for q = 0 . For those trajectories, the fluctuation determinant ∆defined in (10) can be expressed as [23, 27]:∆ = 4 πg [ U ( q t ) − U ( q )] U ′ ( q t ) (cid:18) ∂ Θ ∂q t (cid:19) q (16)That being so, we can now use the previous equation in(9) to obtain the semiclassical partition function.The standard semiclassical method yields a very goodapproximation for the density matrix before and after thefirst caustic. However, as discussed in Ref. [24] (see alsobelow), the method breaks down at the caustic, since,there, by construction, the determinant ∆ vanishes —see equation (9). This can be easily understood in thefunctional space (see Figs. 7 and 8): the second deriva-tive of the action vanishes whenever two (or three) solu-tions coalesce. Therefore, any approximation that stopsat the quadratic term is bound to diverge at this point.This singularity, however, is integrable (as also notedin [24]). This statement can be proved if we perform achange of variables in (9) from q to q t . Using (11) and(16) we can write, following Ref. [27]: (cid:18) ∂q ∂q t (cid:19) Θ = − U ′ ( q t )∆4 πgv ( q , q t ) . (17)Thus, the standard semiclassical partition function iswritten as Z = − πg X i Z q +Θ q − Θ dq t U ′ ( q t )∆ / v ( q , q t ) exp( − I [¯ q ic ] /g ) , (18)where v ( q , qt ) ≡ sign( q t − q ) p U ( q ) − U ( q t )] and q ± Θ ≡ lim q →±∞ q t ( q , Θ). Therefore, the change of vari-ables removes the singularity and this procedure, sum-ming over the two minima of the euclidean action, shouldgive a reasonable approximation to the partition func-tion.However, thermodynamic quantities are obtained tak-ing derivatives of the partition function and thus theyare affected by the singularity. Therefore, as we are in-terested in computing the specific heat, we shall take ourcalculation up to the fourth order in the fluctuations.Notice that this is still a semiclassical expansion, for weassume that the main contribution comes from the clas-sical solution. The calculation is depicted in AppendixB, where one can also promptly recognise the standardsemiclassical expansion if one stops at the second termon the right-hand side of Eq. (B8). Nevertheless, eventhe full expression is not useful for practical purposes,for its calculation requires the knowledge of the eigen-function y ( θ ) and its eigenvalue c . There is, however,a shortcut [24]: just as in a plain 4th-order polynomialof the form (see Eq. (13) and Table I) f ( x ) = 14 x + a x + bx + c , (19)the coefficients { a, b, c } are completely determined by thevalues of the function f ( x ) in 3 points. In the presentcase, all we need are the values of the action at the 3extrema, easily calculated from the classical trajectories(15).The following plots present the results obtained for thespecific heat for g = 0 . π ). On the right-hand side (higher-temperature, lower-Θ) of the plot, the action has only1 minimum; on the left-hand side (lower-temperature,higher-Θ), the action has 2 minima. Accordingly, weplot (dotted/magenta line) the standard semiclassical ap-proximation around the global minimum of the action,(dashed/blue line) the standard semiclassical approxima-tion around both minima of the action, considered inde-pendent and far apart from each other, and (solid/redline) the current approach. The former two calculations It is also obvious then when this approximation fails: by neglect-ing terms of order c and higher, Eq. (B8) yields the usual term∆ − / , which diverges at the caustic. are supposed to diverge at the catastrophe due to thecoalescence of the classical trajectories. Our results arealso to compared the classical one in Fig. 11. -0.2 0 0.2 0.4 0.6 0.8 0.2 0.25 0.3 0.35 0.4 0.45 s pe c i f i c hea t Θ FIG. 10: Specific heat vs / Θ for g = 0 .
1. Dotted/magentaline: only the global minimum is taken into account.Dashed/blue line: both minima are taken into account.Solid/red line: current approach. s pe c i f i c hea t Θ FIG. 11: Specific heat vs / Θ for g = 0 .
1. Green/dashed:classical result; Red/solid: current approach.
Here, we show the corrections that should be takeninto account when the standard semiclassical approxima-tion fails, going beyond second order in the perturbationwhenever this approximation yielded divergent results.On the other hand, we still rely on the assumption thatthe classical solution is responsible for the main contri-bution to the partition function (and, consequently, torelevant thermodynamic quantities, such as the specificheat). In other words, we assume throughout the pa-per that the first term of the JWKB expansion of thepartition function is a good approximation.Just as in any standard quantum mechanics calcula-tion, one does not expect the JWKB approximation tohold when the thermal energy is close to the height of thebarrier, where the potential changes quickly and the clas-sical turning points are too close to each other. There-fore, one must require here that E b /E T ≡ Θ / (4 g ) ≫ E b ≡ V (0) − V ( ± p mw /λ ) is the height of thebarrier and E T ≡ /β ≡ k B T corresponds to the ther-mal energy. In other words, unless g ≪ Θ c / ∼ .
8, theJWKB approximation itself will break down before thefirst catastrophe sets in at Θ c = π . V. SUMMARY
Semiclassical approximations usually uncover impor-tant non-perturbative information about quantum sys-tems. They are specially suited to the construction of ef-fective theories at finite temperature, since the perturba-tive approach suffers from serious infrared problems andneeds involved resummation techniques to provide sen-sible results. The boundary effective theory has provedto be very adequate to describe the thermodynamics ofa thermal massless scalar theory, providing an excellentresult for the pressure at leading order [19], as well as aconsistent description of the thermal effective potentialin the symmetric sector [20]. Its extension to the casewhere spontaneous symmetry breaking is present is, nev-ertheless, subtle, the main obstacle being the existenceof multiple extrema of the action for sufficiently low tem-peratures and the associated bifurcations of classical so-lutions.In this paper we have considered, as a toy model (how-ever not an academic one, since the double well has,of course, applications in statistical mechanics and con-densed matter physics), the analogous case in quantumstatistical mechanics. We have shown how to use thetools of catastrophe theory to deal with caustics and pro-vide finite and well-behaved results for the partition func-tion and the specific heat. In particular, we have provedthat one needs at most two relevant classical solutionsin the procedure, which renders the method of practicaluse. As mentioned previously, this corresponds to a firststep towards the study of spontaneous symmetry break-ing and thermal phase transitions using the boundaryeffective theory, on which we plan to report soon.
Acknowledgments
This work was partially supported by CAPES-COFECUB project 663/10, CNPq, FAPERJ, FAPERN,FUJB/UFRJ and ICTP. The work of ESF was finan-cially supported by the Helmholtz International Cen-ter for FAIR within the framework of the LOEWE pro-gram (Landesoffensive zur Entwicklung Wissenschaftlich-¨Okonomischer Exzellenz) launched by the State of Hesse.
Appendix A: Elements of Catastrophe Theory
Consider a function S ( x ; ν ) that depends on a set ofcoordinates x = { x , x , ... } and certain control param-eters ν = { ν , ν , ... } . The number of coordinates is the dimension of the catastrophe, while the number of con-trol parameters defines the so-called codimension of thecatastrophe.In two dimensions, S can be seen, for instance, as de-scribing the terrain height of a certain landscape. Itsmaxima, minima and saddle points represent the peaks,valleys and throats. In this picture, the role of the pa-rameters ν is to deform the topography of the landscape,changing the position of the extrema and eventually split-ting or merging some of them.The aim of catastrophe theory is to study how thepattern of the so-called generating function S is qualita-tively altered when the control parameters are changed.Within this framework, one is able to understand howthe extrema coalesce and separate as the parameters ν k are varied, in a systematic and quite general approach.Catastrophe theory [33] characterizes the stable sin-gularities under changes in the generating functional S :those are the so-called elementary catastrophes. Thesplitting lemma [33] guarantees that it is always pos-sible to write such stable generating functions in theirnormal forms, according to Table I. They can also be ar-ranged hierarchically : whenever a given catastrophe isidentified, all of its subordinated ones — those with thesame dimension and smaller codimension — will also bepresent.Let us consider the phase space ( x, ν ) defined by boththe coordinates and control parameters of the function S . Obviously, the locus of the extrema, the so-calledequilibrium surface, of S is given by ∂S∂x i ( x e , ν ) = 0 . (A1)i.e., if for certain values of the control parameters ν , thepoint x e represents an extremum of S , and the point( x e , ν ) is said to lie on the equilibrium surface.Note, however, that no information about the natureof the extrema is given by (A1). In order to determinewhether a given extremum is a minimum, maximum ora saddle-point, one has to study the eigenvalues of theHessian matrix H calculated at the equilibrium points x e , whose elements are defined as H ij = ∂ S∂x i ∂x j (cid:12)(cid:12)(cid:12)(cid:12) x e . (A2)When all the eigenvalues of H are positive, we have aminimum; when all are negative, a maximum and when k are negative and the others positive, the extremumunder consideration is a k -saddle-point.It is clear from the previous considerations that thenature of the extrema changes when some of the relatedeigenvalues of H change sign. Therefore, if none of theeigenvalues is zero, a small change of the parameters willnot affect the nature of the extrema.At the bifurcation set, when one or more of the eigen-values vanish, the situation changes drastically, as anysmall change of the control parameters will make the catastrophe codim dim normal formfold 1 1 x / ux cusp 2 1 x / ux / vx swallowtail 3 1 x / ux / vx / wx elliptic umbilic 3 2 x − xy − u ( x y − vx − wx hyperbolic umbilic 3 2 x y uxy − vx − wy TABLE I: The five simplest elementary catastrophes, theircodimensions (number of control parameters), dimensions(number of coordinates) and the normal forms of their gener-ating functions. eigenvalue(s) positive or negative, changing the natureof the extremum. In other words, the qualitative aspectof the function is changed whenever the determinant of H vanishes.One can see such behavior clearly present in Fig. 6,which represents the bifurcation set in the control pa-rameter space, with codimension 2: crossing at the ver-tex correponds to the coalescence of 3 extrema (Fig. 7):this is the cusp catastrophe. Along the bifurcation set,however, there is only one free control parameter (codi-mension 1) — since Eq. (14) introduces a constraintbetween the two of them. On this curve, only 2 trajec-tories coalesce (Fig. 8): this is the fold, subordinated tothe cusp.In the next Appendix, we show how one can write theaction in the normal form corresponding to the cusp. Appendix B: The normal form of the action
In this section, we show how the action can be writtenin normal form, as in Eq. (13).In the first place, we write q ( θ ) = q cl ( θ ) + η ( θ ), so thatthe euclidean action is cast in the form: I [ q ( θ ) + η ( θ )] = I [ q cl ( θ )]+ 12 Z Θ0 η ( θ ) (cid:20) − d dθ − q cl ( θ ) (cid:21) η ( θ ) dθ + Z Θ0 (cid:20) q cl ( θ ) η ( θ ) + 14 η ( θ ) (cid:21) dθ . (B1)Notice that the classical solution was not specified.There are two interesting cases: the identically null func-tion ( q cl ≡ η ( θ ) in terms of theeigenfunctions of the fluctuation operator, i.e. in termsof the functions y j ( θ ) satisfying the following equation: (cid:20) − d dθ − q cl ( θ ) (cid:21) y j ( θ ) = α j y j ( θ ) . (B2)The eigenfunctions can be taken as orthonormal in theinterval [0 , Θ]: Z Θ0 y i ( θ ) y j ( θ ) dθ = δ ij . (B3) Furthermore, they must satisfy the following boundaryconditions y j (0) = y j (Θ) = 0 ∀ j . (B4)Expanding η ( θ ) in terms of y j ( θ ), we have η ( θ ) = ∞ X j =0 c j y j ( θ ) . (B5)Thus, using the expansion of the fluctuations and theorthonormalization conditions, we can write the actionas: I = I cl + 12 X j c j α j + X ijk c i c j c k Z Θ0 q cl y i y j y k dθ + 14 X i c i . (B6)We have to impose the fact that the classical solution q cl is an extremum of the action, therefore the fluctua-tions vanish, i.e. c j = 0, at q cl . Equivalently: ∂I∂c i (cid:12)(cid:12)(cid:12)(cid:12) c i =0 = 0 . (B7)This leads to the following expression for the action: I ≈ I cl + α c + c Z Θ0 q cl y dθ + 14 c + X j =0 α j c j . (B8)In the previous equation, j = 0 denotes the eigenfunctionwhose eigenvalue is about to vanish. Besides, we haveneglected terms of the order c j for j = 0.The difference between this expression and the usualsaddle-point approximation is the inclusion of higher-order terms in the variable c , the one related with thevanishing eigenvalue, while only terms up to second orderin the other variables, related with the other directionsin functional space.Now we perform the following change of variables: z ≡ c + Υ (B9a) u ≡ α − (B9b) v ≡ Υ(2Υ − α ) (B9c) s ≡ I cl + Υ (cid:18) α −
32 Υ (cid:19) (B9d)Υ ≡ Z Θ0 q cl y dθ (B9e)allowing us to write the action in the so called normalform, as in Eq. (13).0 [1] T. Ullrich, B. Wyslouch and J. W. Harris, Nucl. Phys. A , 1c (2013).[2] M. Gell-Mann and M. Levy, Nuovo Cim. , 705 (1960).[3] B. Lee, Chiral Dynamics (Gordon and Breach, 1972).[4] S. P. Klevansky, Rev. Mod. Phys. , 649 (1992).[5] R. D. Pisarski, Phys. Rev. D , 111501 (2000).[6] M. Le Bellac, Thermal Field Theory (Cambridge Uni-versity Press, 2000); J. I. Kapusta and C. Gale,
Finite-Temperature Field Theory: Principles and Applications (Cambridge University Press, 2006).[7] A. Mocsy, I. N. Mishustin and P. J. Ellis, Phys. Rev. C , 015204 (2004).[8] L. F. Palhares and E. S. Fraga, Phys. Rev. D , 025013(2008).[9] L. F. Palhares, M.Sc. Thesis (Instituto de F´ısica, Uni-versidade Federal do Rio de Janeiro, 2008).[10] E. S. Fraga, L. F. Palhares and M. B. Pinto, Phys. Rev.D , 065026 (2009).[11] J. K. Boomsma and D. Boer, Phys. Rev. D , 034019(2009).[12] A. J. Mizher, M. N. Chernodub and E. S. Fraga, Phys.Rev. D , 105016 (2010).[13] V. Skokov, B. Friman, E. Nakano, K. Redlich and B. -J. Schaefer, Phys. Rev. D , 034029 (2010).[14] L. F. Palhares and E. S. Fraga, Phys. Rev. D , 125018(2010).[15] J. O. Andersen, R. Khan and L. T. Kyllingstad,arXiv:1102.2779 [hep-ph].[16] B. W. Mintz, R. Stiele, R. O. Ramos and J. Schaffner-Bielich, Phys. Rev. D , 036004 (2013).[17] J. Berges, N. Tetradis and C. Wetterich, Phys. Rept. , 223 (2002); J. M. Pawlowski, Annals Phys. ,2831 (2007); H. Gies, Lect. Notes Phys. , 287 (2012).[18] A. Bessa, F. T. Brandt, C. A. A. de Carvalho andE. S. Fraga, Phys. Rev. D , 065010 (2010). [19] A. Bessa, F. T. Brandt, C. A. A. de Carvalho andE. S. Fraga, Phys. Rev. D , 085024 (2011).[20] A. Bessa, C. A. A. de Carvalho, E. S. Fraga and F. Gelis,Phys. Rev. D , 125016 (2011).[21] L. Dolan and R. Jackiw, Phys. Rev. D , 3320 (1974).[22] A. Bessa, C. A. A. de Carvalho, E. S. Fraga and F. Gelis,JHEP , 007 (2007).[23] C. A. A. de Carvalho and R. M. Cavalcanti, Braz. J.Phys. , 373 (1997).[24] C. A. A. de Carvalho, R. M. Cavalcanti, E. S. Fraga andS. E. Jor´as, Phys. Rev. E , 056112 (2002).[25] R. P. Feynman and A. R. Hibbs, Quantum Mechanicsand Path Integrals (McGraw-Hill, 1965). R. P. Feynman,
Statistical Mechanics (Addison-Wesley, 1972).[26] L. S. Schulman,
Techniques and Applications of Path In-tegration (Dover Publications, Inc., 2005)[27] C. A. A. de Carvalho, R. M. Cavalcanti, E. S. Fraga andS. E. Jor´as, Annals Phys. , 146 (1999).[28] C. A. A. de Carvalho, R. M. Cavalcanti, E. S. Fraga andS. E. Jor´as, Phys. Rev. E , 6392 (2000).[29] B. J. Harrington, Phys. Rev. D , 2982 (1978).[30] L. Dolan and J. E. Kiskis, Phys. Rev. D , 505 (1979).[31] A. Cuccoli, R. Giachetti, V. Tognetti, R. Vaia, and P.Verrucchi, J. Phys.: Condensed Matter , 7891 (1995).[32] M. Bachmann, H. Kleinert and A. Pelster, Phys. Rev. A , 3429 (1999).[33] P. T. Saunders, An introduction to Catastrophe Theory (Cambridge Univ. Press, 1980).[34] P. F. Byrd and M. D. Friedman,
Handbook of EllipticIntegrals for Engineers and Physicists (Springer-Verlag,Berlin, 1954).[35] I. S. Gradhsteyn and I. M. Ryzhik,