A model field theory with (ψlnψ ) 2 potential: Kinks with super-exponential profiles
AA model field theory with ( ψ ln ψ ) potential: Kinks withsuper-exponential profiles Pradeep Kumar (1) , Avinash Khare (2) and Avadh Saxena (3)(1)
Department of Physics, P.O. Box 118408, University of Florida, Gainesville, Florida 32605, USA (2)
Physics Department, Savitribai Phule Pune University, Pune, India 411007 (3)
Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos,New Mexico 87545, USA
Abstract:
We study a (1+1)-dimensional field theory based on ( ψ ln ψ ) potential. There are threedegenerate minima at ψ = 0 and ψ = ±
1. There are novel, asymmetric kink solutions of the form ψ = ∓ exp( − exp( ± x )) connecting the minima at ψ = 0 and ψ = ∓
1. The domains with ψ = 0 repel thelinear excitations, the waves (e.g. phonons). Topology restricts the domain sequences and therefore theordering of the domain walls. Collisions between domain walls are rich for properties such as transmissionof kinks and particle conversion, etc. To our knowledge this is the first example of kinks with super-exponential profiles and super-exponential tails. Finally, we provide a comparison of these results with the φ model and its half-kink solution. 1 a r X i v : . [ n li n . PS ] A ug Introduction
Among the (topological) kink-bearing theories [1], there are the well-known polynomials, the best knownof them being the φ field theory followed by several others as well. There are also the trigonometric(sine Gordon and Double sine Gordon [2, 3, 4, 5]) and other transcendental functions. In each case, theyhave a potential with degenerate minima giving rise to generally stationary domains. The kinks are thedomain walls between these minima. These are solvable for their analytical profiles. Often the propertiesof kink-kink and kink-antikink interactions can be discerned from the properties of the potential. Thesurprises sometimes come from a numerical simulation of the collisions, such as the transparency of kinkand antikink in a sine-Gordon theory [5] and particle conversion [2, 3, 4, 5] in the double sine-Gordontheory.In the following, we present a study of a one (space plus one time) dimensional scalar field theorywith a potential term of the form ( ψ ln ψ ) . The provenance [6] here is from a theory of higher orderphase transitions in the limit of infinite order. Note that most Gaussian field theories start with a termquadratic in the field as well as in its gradient. The next terms contain the essential physical specificse.g. quartic or higher order nonlinearities. Higher order phase transitions [6, 7, 8] contain progressivelyweaker nonlinearities beyond the quadratic term. Our model potential represents the minimal nonlinearity in the form ( ψ ln ψ ) . In one dimension, it supports an analytically solvable, unexpected kink solution withmultiple flavors and unique interactions.The objective of this paper is to discuss these solutions, which are kinks in a minimal-nonlinearityfield theory described by a ( ψ ln ψ ) potential. We begin by noting (see Fig. 1) that there are threedegenerate minima in this potential, ψ = 0 and ψ = ±
1. The potential is defined for ψ < ψ = ln ψ . The functional form of the (0,1) kink is super-exponential, i.e. of the form exp( − exp( − x )),which arises in the context of extremal statistics and is known as Gumbel distribution [9]. A stabilityanalysis shows that the kinks are stable, the solid body motion takes place in a manner similar to otherkinks, at zero frequency and the corresponding wave function is the gradient of the kink profile. Thereis a continuum of propagating states that exist only in the ψ (cid:54) = 0 domain. They are repelled from the ψ = 0 domain. The kink-phonon interaction is described by a Morse-like potential that has an eigenvalue2pectrum more akin to the Bargmann potentials [10] of an entirely different context.Adjacent configurations of kinks and antikinks are subject to topological constraints thus resulting insequences that are specific. Interactions are repulsive between the kinks (and super-exponential) and areattractive between a kink-antikink pair (which are either exponential or super-exponential).Most of the known kink solutions for a variety of potentials harbor kinks with exponential tails exceptthe φ model [11], which has a kink with a power-law tail. Recently, a whole family of potentials has beendiscovered in which kinks have a power-law tail [12, 13, 14, 15]. However, the kink solutions obtained inthis paper represent the very first example of a super-exponential tail.This paper is organized as follows. Sec. 2 describes the details of the field theory based on the ( ψ ln ψ ) potential. Sec. 3 contains the kink solutions and a discussion of their stability. Sec. 4 includes thetopological considerations that determine the special configurations of infinite sequences. This section alsodiscusses the nature of various kink-kink and kink-antikink interactions. Sec. 5 contains a comparison ofthe results with that of a φ model with a half-kink solution. Finally, Sec. 6 provides a summary of theresults and speculations on possible future directions. We consider a field theory described by a free energy functional F ( ψ ( x )) whose minimum provides theEuler-Lagrange equation for the field ψ ( x ): F = f (cid:90) (cid:104) c | ∂ x ψ | + d ( ψ ln ψ ) (cid:105) . (1)Here f sets the energy scale and c and d are positive, material dependent constants. As shown in Fig. 1,the potential has three degenerate minima, at ψ = 0 , ±
1. Here we use ln ψ = ln ψ , so that negative ψ can be accommodated. The three minima are separated by two maxima at ψ = ± /e .We will focus here on a specific dynamics (with γ being an effective mobility) described by γ ∂ ψ∂t = − ∂F∂ψ . (2)The frequency dispersion of small amplitude oscillations of wave number q are described by γω = V (cid:48)(cid:48) ( ψ ) + cq . (3)3 - - ψ , ϕ - V Figure 1: Potential V ( ψ ) associated with the ( ψ ln ψ ) interaction (blue). Note that V ( ψ ) is smooth at ψ = 0 and there is no cusp. Comparison with the φ model potential V ( φ ) = φ (1 − φ ) (red). Theconspicuous difference between the two potentials near V (0) leads to super-exponential vs. exponentialkink tails.Here V (cid:48) = 2 dψ ln ψ (ln ψ + 1) is the slope and V (cid:48)(cid:48) = 2 d [(ln ψ ) + 3 ln ψ + 1] is the curvature of the potential V ( ψ ) = d ( ψ ln ψ ) . Note that V (cid:48)(cid:48) (1) = 2 d = − V (cid:48)(cid:48) (1 /e ).Despite the apparent equivalence of the three degenerate minima, there are differences. This is firstapparent here in that the potential curvature is divergent at ψ = 0. Moreover, there are other relatedfeatures that appear (and are discussed below) such as a complete blocking of linear excitations, e.g.phonons, from the region ψ = 0. We will defer that discussion until later. The Euler-Lagrange equation for a static kink solution becomes − c∂ xx ψ + dψ ln ψ (ln ψ + 1) = 0 . (4)4 - x ψ , ϕ Figure 2: Asymmetric kink profile for ψ ( x ), Eq. (6), with super-exponential ( x <
0) and exponential( x >
0) asymptotes (blue). Comparison with the φ asymmetric half-kink, Eq. (11), which has anexponential tail on either side (red).The usual practice is to transfer the length scale so that the independent variable is y = x (cid:112) d/c . Theoutside energy scale factor then becomes f √ cd . After one integration of the equation of motion, we get ψ y = ± ψ ln ψ . A. The kink solution connecting ψ = − y = −∞ to ψ = 0 at y = ∞ with the center at y = 0 and ψ (0) = − /e is ψ A ( y ) = − exp( − exp y ) . (5)Its energy is E A = f o √ cd (1 / √ ψ = 0 at y = −∞ to ψ = − y = ∞ is ψ A ( y ) (cid:48) = − exp( − exp( − y )).5 . The kink solution connecting ψ = 0 at y = −∞ to ψ = 1 at y = ∞ is of the same energy but theprofile is given by ψ B ( y ) = exp( − exp( − y )) . (6)The corresponding antikink solution (with the same energy) is given by ψ B ( y ) (cid:48) = exp( − exp( y )). Notethat ψ B ( y ) (cid:48) = − ψ A ( y ) and ψ A ( y ) (cid:48) = − ψ B ( y ).These solutions break the parity symmetry. This is related to the asymptotic forms of the solution:For y → −∞ , the kink profile for ψ A is − y ), while for y → ∞ , it remains a super-exponential.Between ( −
1, 0), the center at y = 0 has the value ψ = − /e . Similarly, for y → ∞ , the kink profile for ψ B is 1 − exp( − y ), while for y → −∞ , it remains a super-exponential. Between (0, 1), the center at y = 0has the value ψ = 1 /e and the kink is asymmetric (Fig. 2).The fluctuations around a kink solution also have novel properties. These are the linear waves (e.g.phonons) interacting with the kinks and are described by a linearized version of Eq. (2).Assuming δψ ( y, t ) = ψ ( y, t ) − ψ A = δψ ( y ) exp( iωt ), we have, γω δψ = − δψ yy + 2( e y − e y + 1) δψ. (7)This is an eigenvalue problem for a variant of the Morse potential. It contains a repulsive and an attractiveterm. The former is square of the latter and the attractive part has a coefficient of 3 (instead of 2 in aregular Morse potential). This potential has only one bound state at ω = 0, the translational Goldstonemode for the ψ A ( y ) kink, which is given by, Ψ ( y ) = e y e − e y . (8)The wave function has no nodes and therefore this must be the lowest eigenvalue. Since there are nonegative eigenvalues here these kinks are therefore stable to emission of linear waves. There is a continuumof propagating states ( γω = 1 + k ), starting with a frequency gap at ω = 1. These wave functions areperfectly reflected, there is no transmitted component on the right .Similarly, assuming δψ ( y, t ) = ψ ( y, t ) − ψ B = δψ ( y ) exp( iωt ), we have, γω δψ = − δψ yy + 2( e − y − e − y + 1) δψ. (9)6his is also an eigenvalue problem for a variant of the Morse potential (shown in Fig. 3). This potentialalso has only one bound state at ω = 0, the translational Goldstone mode for the ψ B ( y ) kink is given by,Ψ ( y ) = e − y e − e − y . (10)These wave functions are also perfectly reflected, there is no transmitted component on the left . The wavefunction for the bottom of the continuum spectrum has a node at y = ln 2.The super-exponential function, ψ B ( y ) = exp( − exp( − y )), is known in the general area of extremevalue statistics and distributions as Gumbel distribution [9]. Presumably, this form arises here due to thefield accessing the ψ = 0 domain with significantly suppressed probability (since V (cid:48)(cid:48) (0) diverges). - x - V k ψ , V k ϕ Figure 3: Potential associated with phonons (see Eq. (9)) interacting with a type B kink for the logarithmicpotential (blue). For x → −∞ it diverges whereas for x → ∞ it asymptotes to 2. Comparison with thepotential corresponding to phonons (Eq. (12)) of the half-kink of the φ model (red). In this case bothasymptotes are finite, i.e. 1 and 4, respectively. 7 K-K’ interactions and collisions
As we have seen above, this field theory has three degenerate minima (Fig. 1) and therefore that manydomains and two types of kinks ( A and B ) subject to topological constraints. Let us consider the interactionbetween them. For example, the system may evolve in the state ψ = 1 in one part and ψ = − ψ = 0. A pathdrawn from the center of the domain ψ = 1 and ending at the center of domain ψ = − B (cid:48) and A (cid:48) .Figure 4 shows that there are six possible configurations, ( A, A (cid:48) ) is a bubble of ψ = 0 amid ψ = − B , B (cid:48) ) has a bubble of ψ = 1 with ψ = 0 on the outside. ( A , B ) connects ψ = − ψ = 1 on the right with ψ = 0 bubble in the middle. Similarly, ( B (cid:48) , A (cid:48) ) connects ψ = 1 on the rightto ψ = − ψ = 0 bubble in the middle. Note that a pair ( A (cid:48) , A ) describes a bubble of ψ = − ψ = 0 and ( B (cid:48) , B ) has a bubble of ψ = 0 in a sea of ψ = 1. 𝐴′ 𝐴 𝐵′ 𝐵 𝐵′ 𝐴′𝐴 𝐴′ 𝐵 𝐵′ 𝐴 𝐵
Figure 4: Six distinct configurations of various kinks ( A , A (cid:48) , B , B (cid:48) ). Topologically, the φ model kinkconfigurations are the same. AB kink-kink interaction is repulsive and super-exponential. On the other hand, AA (cid:48) kink-antikinkas well as B (cid:48) B (antikink-kink) interactions are attractive and super-exponential. However, A (cid:48) A and BB (cid:48) interactions are attractive and exponential. The asymptotic interaction between any of these pairs isinferred from their asymptotic profiles. Note that the asymptotes are exponential corrections when thekink is approaching ψ = 1. When ψ = 0, the approach is super-exponential. When the overlapping part8f the profile is ψ = 1, the interaction can be easily calculated and is given by E KK (cid:48) = − − d ),where d is the kink-antikink separation. 𝑛 = 1𝑛 = 0𝑛 = −1 𝐴 𝐵 𝐵′ 𝐵 𝐵′ 𝐴′ 𝐵′ 𝐵 𝐵′ 𝐴′ 𝐴 𝐵𝑛 = 1𝑛 = 0𝑛 = −1 𝐵 𝐵′ 𝐴′ 𝐴 𝐴′ 𝐴 𝐴′ 𝐴 𝐵 𝐵′ Figure 5: Selected representative examples of general kink configurations. Note that there are many morepossibilities than represented here.There are a limited number of kink-antikink sequences that are possible in an infinite chain (Fig. 5).These are topological restrictions. Consider a kink A , it must be followed either by an antikink A (cid:48) or bya kink B . The kink B must be followed by the antikink B (cid:48) . This is easier seen in a three-rung ladder of ψ = − ψ = 0 and ψ = 1. When the state is ψ = −
1, it can go to ψ = 0 (in kink A ) and back (in antikink A (cid:48) ) an infinite number of times. That would be an infinite sequence of ( A , A (cid:48) ). Likewise from the state ψ = 1, one can insert an infinite sequence of ( B (cid:48) , B ). From state ψ = 0, one has the choice of antikink A (cid:48) or kink B .The kinematics of collisions between allowed pairs of kinks or antikinks has some surprising features(Fig. 6). Intuitively, it follows that a collision between the pair ( B (cid:48) , B ) refers to a bubble shrinking underthe mutual attractive interaction. The pair may have been launched with some initial kinetic energy. Thefinal state would be the pair reversing their original direction of motion.Along similar lines, collisions between ( B , B (cid:48) ) or ( A (cid:48) , A ) have conversion properties. Here the finalstate can be either the original pair reversing their direction of motion or more likely emerging as an ( A (cid:48) , A ) pair. 9 𝐴 𝐴′ -1 0 -1
𝐴 𝐴′
𝐵 𝐵′ -1 0 -1
𝐴 𝐴′ -1-1 𝑛 < −1
𝐵′ 𝐴′𝐵𝐴 -1 -100 10 0
𝐴′ 𝐴 -1 Figure 6: Three different collision scenarios depending on the kinetic energy of the colliding kinks. φ model A well-known model that has three degenerate minima is the φ model [16, 17] with the potential V ( φ ) =(1 / φ (1 − φ ) (see Fig. 1). The minima are at φ = 0, ± ± / √ φ ( x ) = ∓ / (cid:113) [1 + exp( ± x )] (11)with φ (+ ∞ ) = 0, φ ( −∞ ) = − φ (0) = − / √
2. For the other kink (preceded by + sign) φ ( −∞ ) = 0, φ (+ ∞ ) = 1 (see Fig. 2). As can be easily checked, all the tails have an exponential fall off. Topologically,the different types of kinks, antikinks and other configurations are similar in the two cases (see Figs. 4, 5,6), i.e. φ versus logarithmic.The analog of Eq. (9) for the stability of the half-kink solution of the φ model in Eq. (11), assuming10 φ ( x, t ) = φ ( x, t ) − φ K = δφ ( x ) exp( iωt ), is γω δφ = − δφ xx + 4 − e − x + e − x (1 + e − x ) δφ , (12)which is quite different in that in both limits x → ±∞ the kink potential attains a finite value (see Fig.3). Nevertheless, just like the logarithmic case (see Eqs. 8, 10), this potential also has only one boundstate at ω = 0, the translational Goldstone mode for the half-kink, given byΦ ( x ) = e − x (1 + e − x ) / . (13)However, there are some important differences: (i) the asymptotes are different, in the present log-arithmic case, one tail is exponential while the other one (near ψ = 0) is super-exponential, and moreimportantly (ii) V (cid:48)(cid:48) ( ψ = 0) is divergent whereas V (cid:48)(cid:48) ( φ = 0) = 1, which is finite. This completely changeshow the ψ = 0 domain behaves in contrast to the φ = 0 domain. (iii) The latter has a well-definedphonon dispersion whereas the former expels phonons (near ψ = 0). The exponential tails interact in theusual way, i.e. exponentially ´a l`a Manton [18]. However, the super-exponential tail leads to a weaker,super-exponential interaction, which is novel.There have been significant studies of collision among φ kinks and antikinks [19]. Specifically, as mightbe expected, results for B - B (cid:48) [i.e. collisions (0,1)+(1,0)] are very different from those of B (cid:48) - B [i.e. (1,0)+(0,1)] collisions. In particular, for B - B (cid:48) collisions, till the velocity v < v p about 0.289, the kink pairalways remained trapped. For v > v p , the collisions yielded a reflected pair of kinks, i.e. B + B (cid:48) → A (cid:48) + A ,i.e. (0,1)+(10) → (0, − − B (cid:48) - B collisions revealed a very different picture (as explainedin detail in Fig. 1 of ref. [19]). For v = v cr < . v > v cr the kinks have enough energy to always separate. According to ref. [19] this is similar towhat happens for collisions in the φ case [20].Ref. [19] attributes this difference between B (cid:48) - B and B - B (cid:48) collisions to the fact that the stabilitypotential is not symmetrical w.r.t reflections: x → − x . So the potential faced by B - B (cid:48) is different thanthat faced by B (cid:48) - B . For small velocities, the adiabatic approximation is valid and using it one can estimatethe spectrum of small fluctuations about the potentials experienced by B - B (cid:48) and B (cid:48) - B configurations. Fromthe plots of the potentials [19] one sees that the two potentials are different. Further, the small fluctuation11nalysis shows that while in the B - B (cid:48) collisions one has only two zero modes, in the B (cid:48) - B case, apart fromthe zero mode, there are meson states as in the φ case and hence similarities with φ collision case [20].The full gamut of possibilities in logarithmic kink collisions are likely similar but will need to be studiednumerically beyond what Figs. 4 - 6 suggest. Because V (cid:48)(cid:48) ( ψ = 0) is divergent in the present case, thecollision dynamics is expected to be quite different than in the φ case. We have here a field theory based on a potential ( ψ ln ψ ) . The nonlinear Euler-Lagrange equation canbe solved analytically for the kink profile as well as the linear fluctuations eigenvalues and wave functions.The kinks are novel, super-exponential in analytical form and are described by a Gumbel distribution ( ψ B kink in particular) from extreme value statistics [9]. The fluctuations are described by the solutions of aMorse-like potential with asymmetric transmission (fluctuations can only survive in the ψ = 1 domain).A comparison with the φ model and its attendant kinks as well as phonons provides useful insights.The possible kink-antikink pair sequences have topological constraints. These sequences can alterna-tively be imagined as bubbles in an onion dome structure. A domain with ψ = 0 may be surrounded by ψ = 1 (or ψ = − ψ = 0 again. Inside an ψ = − ψ = 1)domain there can be bubbles of ψ = 0 only. The inside bubbles may shrink corresponding to a collision.Only a ψ = 0 domain may contain two possible domains ψ = 1 or ψ = −
1. A dynamic state where a ψ = 0domain surrounds a kernel that breathes between ψ = +1 and ψ = − Acknowledgment
We acknowledge valuable discussions with YuXuan Wang, Shizeng Lin and Ayhan Duzgun. A.K. is gratefulto Indian National Science Academy (INSA) for the award of INSA Scientist position at Savitribai PhulePune University. This work was supported in part by the U.S. Department of Energy.
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