Semiclassical catastrophe theory of simple bifurcations

Semiclassical periodic-orbit theory (POT) is a powerful tool for the study of shell structures in the single-particle level density of finite fermionic systems GUTZpr ; GUTZbook90 ; BBann72 ; sclbook . This theory relates the oscillating level density and shell-correction energy to the sum over contributions of classical periodic orbits. It thus gives the correspondence between the fluctuation properties of the quantum dynamics and the characteristics of the periodic motion embedded in the classical dynamics.

Gutzwiller GUTZbook90 suggested the semiclassical evaluation of the Green’s function in the Feynman path integral representation to derive the POT trace formula for the level density if the single-particle Hamiltonian has no continuous symmetries other than the time-translational invariance. In this case, the energy $E$ is the single integral of motion for a particle dynamics in the mean-field potential. For a given $E$, all the generic periodic orbits (POs) are isolated, i.e., any variation of the initial condition perpendicular to the PO will violate its periodicity. The original version of the POT was extended for Hamiltonians with continuous symmetries (extended Gutzwiller approach), in particular for the rotational [$\mbox{O}(n)$] and oscillator-type [$\mbox{SU}(n)$] symmetries SMepp76 ; SMODzp77 ; MAGosc78 ; CREAGHpra97 . Berry and Tabor BT76 derived the POT for integrable systems by applying the Poisson-summation method through the semiclassical torus-quantization condition. It is also helpful in the case of a high classical degeneracy of POs. Here the classical degeneracy is defined by the number of independent parameters $\mathcal{K}$ for a continuous family of classical periodic orbits at a given energy of the particle.

Some applications of the POT to the deformations of nuclei and metallic clusters by using phase-space variables were presented in Refs. BT76 ; SMODzp77 ; CREAGHpra97 ; sclbook ; migdalrev ; MKApan16 . The pronounced shell effects caused by the deformations have been discussed. Within the improved stationary-phase method migdalrev ; MFAMBpre01 ; MAFMptp02 ; MAFptp06 ; MFAMMSBptp99 ; KKMABps15 ; MKApan16 (ISPM), the divergences and discontinuities of the standard stationary-phase method (SSPM) BBann72 ; SMepp76 ; BT76 ; CREAGHpra97 ; sclbook near the symmetry-breaking and bifurcation points were removed.

Bifurcations of the isolated POs in non-integrable Hamiltonian systems are classified by the normal-form theory Ozoriobook ; OH87 , based on a pioneering work of Meyer on the differentiable symplectic mappings Meyer . The change in the number of POs depends on the types of bifurcations: from zero to two in the isochronous or saddle-node bifurcations, from one to three in the period-doubling or pitchfork bifurcations, and so on. In the integrable systems, the classical phase space is entirely covered by the tori. The classical orbits on the rational tori (in which the frequencies of the independent motions are commensurable) form degenerate families of POs. In such systems, the bifurcations mostly occur at a surface of the physical phase-space volume occupied by the classical trajectories. For brevity, below, such a surface will be called the end point. Note that even if the commensurability of the frequencies is not fully satisfied at the end point, one has a PO there because the mode with incommensurable frequencies has zero amplitude. In the action-angle representation $(I_{n},\Theta_{n})$ for the $n$th mode, it corresponds to $I_{n}=0$, where the variation of $\Theta_{n}$ will not generate a family. Thus, the end-point POs have smaller degeneracies than those inside the physical region. With varying potential parameter, a new PO family appears in the transition from an unphysical to a physical region for a rational torus where the frequencies of the end-point PO become commensurable. It is considered as the bifurcation of the end-point PO generating a new orbit family. Such bifurcations take place repeatedly with varying potential parameter and a new family of orbits with a higher degeneracy and different frequency ratio is generated at each bifurcation point. Typical examples are the equatorial orbits with $\mathcal{K}=1$ in the spheroidal cavity where the generic family has $\mathcal{K}=2$, and the circular orbits with $\mathcal{K}=2$ in a spherical potential where the generic family appears with $\mathcal{K}=3$.

Our ISPM is based on the catastrophe theory by Fedoriuk and Maslov for solving problems with caustic and turning points in calculations of the integrals by using the saddle-point method FEDoryuk_pr ; MASLOV ; FEDoryuk_book1 ; MAFptp06 . In the SSPM, the catastrophe integrals are evaluated by an expansion of the action integral in the exponent up to second-order terms and amplitude up to zeroth-order terms near the stationary point, and the integration limits are extended to an infinite interval. The trace formula based on the SSPM encounters a divergence or discontinuity at the bifurcation point. Such catastrophe problems are due to the zeros (caustics) or infinities (turning points) of the second-order derivatives of the action integral in this expansion. Fedoriuk was the first to prove FEDoryuk_pr ; FEDoryuk_book1 the so-called Maslov theorem MASLOV : Each simple caustic or turning point (having a finite nonzero third-order derivative of the action integral) leads to a shift of the phase in the exponent of the catastrophe integral by $-\pi/2$ along the classical trajectory. This provides an extension of the one-dimensional WKB formula to higher dimensions. Thus, in the asymptotic region far from the caustic and turning points, one can use second-order expansions of the action integral (and zeroth-order expansions of the amplitude) in the semiclassical Green’s function taking into account the shift of the phase according to the Maslov theorem. However, a proper foundation for an extension of the Fedoriuk-Maslov catastrophe theory (FMCT) to the derivations of the trace formulas near the PO bifurcations, where one has to treat the semiclassical propagators near the catastrophe points, is still an open question.

Uniform approximations based on the normal-form theories give alternative ways of solving the bifurcation problems, see Refs.  CREAGHpra97 ; sclbook ; Ozoriobook ; Ullmo ; SIEjpa97 ; SSUNjpa97 ; SCHOjpa98 ; ABjpa02 ; KAijmpe04 ; BO05 ; ABjpa08 ; KAprc12 ; MO15 . The trace formula valid near the bifurcation points is derived by calculating the catastrophe integral in local uniform approximations. It gives an indistinct combination of contributions of bifurcating POs, and the asymptotic regions (where each of the POs has a separate contribution to the trace integral) are connected by a kind of the interpolation through the bifurcations in the global uniform approximations. The ISPM provides much simpler semiclassical formula in which one needs no artificial interpolation procedures over the bifurcations. Within the simplest ISPM, the contributions of the bifurcating orbits are given separately through the bifurcation and they are basically independent of the type of bifurcation. This allows one to give analytic expressions for the Gaussian-averaged level densities and the energy shell corrections STRUTscm ; FUHIscm1972 ; SMepp76 ; sclbook .

In the present work we apply the FMCT to solve the bifurcation problems that arise for some parameters of the mean-field potential for the particle motion in the end-point phase space. For the simplest but a rich exemplary case that nevertheless includes all of the necessary points of the general behavior of the integrable systems, we will consider the spherical radial power-law (RPL) potential $V\propto r^{\alpha}$ as a function of the radial power parameter $\alpha$, which controls the surface diffuseness of the system KAps16 ; KAijmpe04 ; KAprc12 . Some of these results are general for any integrable and nonintegrable Hamiltonian systems, and can be applied to a more realistic nuclear mean-field potential having the deformation. The spherical RPL model has already been analyzed within the ISPM in Ref. MKApan16 . Good agreement between the semiclassical and quantum shell structures was shown in the level-density and energy shell corrections for several values of the surface diffuseness parameter including its symmetry-breaking and bifurcation values. Quantum-classical correspondences in the deformed RPL models with and without spin-orbit coupling were also studied and various properties of the nuclear shape dynamics, such as the origins of exotic deformations and the prolate-oblate asymmetries, have been clarified KAprc12 ; AMprc14 ; KAps16 ; MKApan16 ; KAps17 . Therefore, a proper study of the general aspects of the bifurcation problem within the ISPM, even taking the simplest spherical RPL potential as an example for which one can achieve much progress in analytical derivations, is expected to be helpful.

This article is organized in the following way. In Sec. II we present a general semiclassical phase-space trace formula for the level density as a typical catastrophe integral. Section III shows the Fedoriuk-Maslov method for solving the simple caustic- and turning-point singularity problems. Section IV is devoted to the application of the FMCT to the local PO bifurcations for more general Hamiltonian systems with continuous symmetries. Section V presents the specific application of the ISPM trace formula to the bifurcations in the spherical RPL potential model. Contributions of the bifurcating orbits to the trace formula are discussed. In Sec. VI, we compare our semiclassical results, obtained at a bifurcation point and asymptotically far from bifurcations, with the quantum calculations. These results are summarized in Sec. VII. A more precise trace formula based on the third-order expansions of the action integral is presented in the Appendix.

The general semiclassical expression for the level density, $g(E)=\sum_{i}\delta(E-E_{i})$, is determined by the energy levels $E_{i}$ for the single-particle Hamiltonian $\hat{H}=\hat{T}+V$ of $\mathcal{D}$ degrees of freedom. The specific expression generic to integrable and nonintegrable systems can be obtained by the following trace formula in the $2\mathcal{D}$-dimensional phase space MFAMBpre01 ; MAFMptp02 ; MAFptp06 ; MFAMMSBptp99 ; MKApan16 :

Here $H(\mathbf{r},\mathbf{p})$ is the classical Hamiltonian in the phase-space variables $\mathbf{r},\mathbf{p}$ and $\Phi_{\rm CT}$ is the phase integral

(see the derivations in Ref. MKApan16 ). In Eq. (II), the sum is taken over all discrete classical trajectories (CTs) for a particle motion from the initial point $(\mathbf{r}^{\prime},{\bf p}^{\prime})$ to the final point $(\mathbf{r}^{\prime\prime},{\bf p}^{\prime\prime})$ with a given energy $E$ MAFptp06 . A CT can uniquely be specified by fixing, for instance, the final coordinate $\mathbf{r}^{\prime\prime}$ and the initial momentum ${\bf p}^{\prime}$ for a given time $t_{\rm CT}$ of the motion along a CT. Here $S_{\rm CT}(\mathbf{p}^{\prime},\mathbf{p}^{\prime\prime},t_{\rm CT})$ is the classical action in the momentum representation,

Integration by parts relates Eq. (3) to the classical action in the coordinate space,

by the Legendre transformation [Eq. (2)]. The factor $\mu_{\rm CT}$ is the number of conjugate points along a CT with respect to the initial phase-space point $(\mathbf{r}^{\prime},\mathbf{p}^{\prime})$. They are, e.g., the focal and caustic points where the main curvatures of the energy surface (second derivatives of the phase integral $\Phi_{\rm CT}$) vanish. In addition, there are the turning points where these curvatures become divergent. The number of conjugate points evaluated along a PO is called the Maslov index MASLOV . An extra phase component $\phi_{\mathcal{D}}$, which is independent of the individual CT, is determined by the dimension of the system and the classical degeneracy $\mathcal{K}$ [$\phi_{\mathcal{D}}$ is zero when all orbits are isolated ($\mathcal{K}=0$), as defined in Ref. GUTZbook90 ].

In Eq. (II) we introduced the local phase-space variables that consist of the three-dimensional ($\mathcal{D}=3$) coordinate $\mathbf{r}=\{x,y,z\}$ and momentum $\,\mathbf{p}=\{p_{x},p_{y},p_{z}\}$. It is determined locally along a reference CT so that the variables $(r_{\parallel}=x,p_{\parallel}=p_{x})$ are parallel and $(\mathbf{r}_{\perp}=\{y,z\},\mathbf{p}_{\perp}=\{p_{y},p_{z}\})$ are perpendicular to the CT GUTZbook90 ; SMODzp77 ; sclbook . Here $\mathcal{J}_{\rm CT}(\mathbf{p}^{\prime}_{\perp},\mathbf{p}^{\prime\prime}_{\perp})$ is the Jacobian for the transformation of the momentum component perpendicular to a CT from the initial value $\mathbf{p}_{\perp}^{\prime}$ to the final value $\mathbf{p}_{\perp}^{\prime\prime}$.

For calculations of the trace integral by the stationary-phase method (SPM), one may write the stationary-phase conditions for both $\mathbf{p}^{\prime}$ and $\mathbf{r}^{\prime\prime}$ variables. According to the definitions (2) and (3), the stationary-phase conditions are given by

The asterisk indicates that quantities in large parentheses are taken at the stationary point. Equations (II) express that the stationary-phase conditions are equivalent to the PO equations $(\mathbf{r}^{\prime\prime},\mathbf{p}^{\prime\prime})^{*}=(\mathbf{r}^{\prime},{\bf p}^{\prime})^{*}$. One of the SPM integrations in Eq. (II), e.g., over the parallel momentum $p_{\parallel}^{\prime}$ in the local Cartesian coordinate system introduced above, is the identity because of the energy conservation $E=H(\mathbf{r}^{\prime\prime},\mathbf{p}^{\prime\prime})=H(\mathbf{r}^{\prime},{\bf p}^{\prime})$. Therefore, it can be taken exactly. If the system has continuous symmetries, the integrations with respect to the corresponding cyclic variables can be carried out exactly. Notice that the exact integration is performed finally also along a parallel spatial coordinate (along the PO).

Applying the ISPM with the PO equations (II), accounting for the bifurcations and the breaking of symmetries, one may arrive at the trace formula in terms of the sum over POs migdalrev ; sclbook ; MKApan16 . The total ISPM trace formula is the sum over all of POs [families with the classical degeneracy ${\cal K}\geq 1$ and isolated orbits (${\cal K}=0$)],

where

The amplitude $A_{\rm PO}$ depends on the classical degeneracy $\mathcal{K}$ and the stability of the PO. In the exponent phase, $S_{\rm PO}(E)=\oint\mathbf{p}\cdot d\mathbf{r}$ is the action and $\mu_{\rm PO}$ is the Maslov index GUTZbook90 ; SMepp76 ; sclbook ; migdalrev ; MKApan16 .

In this section we present the essence of the ISPM, following basically the Fedoriuk-Maslov catastrophe theory (see Refs. FEDoryuk_pr ; MASLOV ; FEDoryuk_book1 ).

Assuming a convergence of the expansion [Eq. (10)] to the second order, as shown in the preceding section, one can use the same approach within the simplest ISPM2 of second order²²2We call the $n$th-order ISPM the ISPM$n$, in which we use the expansion of the phase integral $\Phi$ up to the $n$th-order terms and the amplitude up to the $(n-2)$th order near the stationary point. The simplest ISPM for $n=2$ is called the ISPM2. to investigate the symmetry-breaking and bifurcation problems in the POT. Therefore, one arrives at a sum over the separate contributions of different kinds of isolated and degenerated orbits, as in the derivation of the Maslov theorem but within the finite integration limits. The latter is important because the bifurcation point is located at a boundary of the classically accessible region. This section is devoted to the extension of the FMCT to the bifurcation catastrophe problems.

In the presence of continuous symmetries, the stationary points form a family of POs that cover a $(\mathcal{K}$$+$$1)$-dimensional submanifold $\mathcal{Q}_{\rm PO}$ of phase space, whereby $\mathcal{K}$ is the classical degeneracy of the PO family. The integration over $\mathcal{Q}_{\rm PO}$ must be performed exactly. In any systems with continuous symmetries, it is an advantage to transform the phase-space variables from the Cartesian to the corresponding action-angle variables (see, e.g., Ref. BT76 ; CREAGHpra97 ). Then the action $\Phi_{\rm CT}$ in Eq. (II) is independent of the angle variables conjugate to the conserving action variables and the integrations over these cyclic angle variables are exactly carried out. For the integrable case, for instance, integrating over the remaining action variables and using the standard SPM, one obtains the so-called Berry-Tabor trace formula BT76 . Under the existence of additional symmetries such as SU(3) or O(4), some of the integrations over the action variables can also be performed exactly because of a higher degeneracy. For partially integrable systems, the integrations over partial set of cyclic variables also greatly simplify the ISPM derivations of the trace formula (II) near the bifurcations.

To solve the bifurcation problems, some of the SPM integrations have to be done in a more exact way. For definiteness, we will consider first a simple bifurcation defined as a caustic point of the first order [Eqs. (10) and (11)] where the degeneracy parameter $\mathcal{K}$ is locally increased by one. In the SPM, after performing exact integrations over a submanifold $\mathcal{Q}_{\rm PO}$, one uses an expansion of the action phase $\Phi_{\rm CT}$ in phase space variables $\xi=\{\mathbf{r}^{\prime\prime},\mathbf{p}^{\prime}\}_{\perp}$, perpendicular to $\mathcal{Q}_{\rm PO}$ in the integrand of Eq. (II) over $\xi$ near the stationary point $\xi^{\ast}$,

where

$\xi^{\ast}$ is the stationary point, $\xi^{\ast}=\xi_{\rm PO}$, and $\Phi^{\ast}_{\rm CT}=\Phi_{\rm CT}\left(\xi^{\ast}\right)=\Phi_{\rm PO}$. To demonstrate the key point of our derivations of the trace formula, we focus on one of the phase-space variables in Eq. (II), denoted by $\xi$, which is associated with a catastrophe behavior. In the standard SPM, the above expansion is truncated at the second-order term and the integration over the variable $\xi$ is extended to $\pm\infty$. The integration can be performed analytically and yields a Fresnel integral (see, e.g., Refs. sclbook ; migdalrev ).

However, one encounters a singularity in the SSPM that is related to the zero or infinite value of $\Phi_{\rm PO}^{\prime\prime}(\xi^{\ast})$, while $\Phi_{\rm PO}^{\prime\prime\prime}(\xi^{\ast})$ remains finite in the simplest case under consideration. This singularity occurs when a PO (isolated or degenerated) undergoes a simple bifurcation at the stationary point $\xi^{\ast}$ under the variation of a parameter of the potential (e.g., energy, deformation, or surface diffuseness). The SSPM approximation to the Fresnel (error) functions by the Gaussian integrals breaks down because one has a divergence.

Notice that the bifurcation problem is similar to the caustic singularity considered by Fedoriuk within the catastrophe theory (see Sec. III and Refs. FEDoryuk_pr ; FEDoryuk_book1 ). The FMCT is adopted, however, for the specific position of such a singularity at the end point in the phase-space volume accessible for a classical motion (see the Introduction and also Ref. MAFptp06 ).

In systems with continuous symmetries, the orbit at the end point causes the bifurcation where it coincides with one of the rational tori that appears in the transition from the unphysical to the physical region. The contribution of this end-point orbit is derived using a local phase-space variable $\xi$ along it, independently of the torus orbits. Near the bifurcation point, the contribution of these end-point orbits is mostly included to the newborn orbit term. Therefore, we should consider a kind of separation of the phase space occupied by the newborn and end-point orbits to evaluate their contributions to the trace integral near the bifurcation point. Below we will call the latter an end-point manifold. By definition of the end-point manifold, its measure is zero at the bifurcation limit where the minimal $\xi_{-}$ and maximal $\xi_{+}$ coincide with the stationary point $\xi^{\ast}$, i.e.,

Thus, although the contributions of POs participating in the bifurcation are considered separately, the parent orbit contribution vanishes at the bifurcation point and there no risk of double counting. To describe the transition from the bifurcation point to the asymptotic region, one should properly define the end-point manifold. We need it to extract the additional contribution by the end-point orbit that is not covered by the term for the newborn orbit. The detailed treatment of such a transition is still open, but is beyond the scope of the present paper.

We are ready now to employ what we call the improved stationary-phase method MFAMMSBptp99 ; MFAMBpre01 ; MAFMptp02 ; MAFptp06 evaluating the trace integral for the semiclassical level density. Hereby, the integration over $\xi$ in Eq. (II) is restricted to the finite limits defined by the classically allowed phase-space region through the energy-conserving $\delta$ function in the integrand of Eq. (II). The phase and amplitude are expanded around the stationary point up to the second- and zeroth-order terms in $\xi-\xi^{*}$, respectively, and to higher-order terms if necessary.

In the simplest version of the ISPM (ISPM2), the expansion of the phase is truncated at second order, keeping the finite integration limits $\xi_{-}$ and $\xi_{+}$ given by the accessible region of the classical motion in Eq. (II). It will lead to a factor like³³3 For the case of several variables $\xi$ for which we find find zeros and infinities in eigenvalues of the matrix with second-order derivatives of $\Phi_{\rm PO}(\xi)$ at $\xi=\xi^{*}$, we diagonalize this matrix and reduce the Fresnel-like integrals to products of error functions similar to Eq. (IV).

where $\mathop{\rm erf}\nolimits(z_{1},z_{2})$ is the generalized error function,

with the complex arguments

Note that the expression (IV) has no divergence at the bifurcation point where $\Phi_{\rm PO}^{\prime\prime}(\xi^{\ast})=0$, since the error function (32) also goes to zero as

Thus, the factor $\sqrt{\Phi_{\rm PO}^{\prime\prime}}$ in the denominator of Eq. (IV) is canceled with the same in the numerator. In addition, also taking into account Eq. (30) and the discussion around it, one finds the zero contribution of the end-point term in the trace formula at the bifurcation point, which seems to be a consistent semiclassical picture.

This procedure is proved to be valid in the semiclassical limit $\kappa\rightarrow\infty$ by the FMCT FEDoryuk_pr ; MASLOV ; FEDoryuk_book1 . In this way, we can derive the separate PO contributions that are free of divergences, discontinuities, and double counting at any bifurcation point. The oscillating part of the level density can be approximated by the semiclassical trace formula (6). In Eq. (6), the sum runs over all periodic orbits (isolated or degenerated) in the classical system. The term $S_{\rm PO}(E)$ is the action integral along a PO. The amplitude $A_{\rm PO}(E)$ (which, in general, is complex) is of the order of the phase-space volume occupied by CTs. The factor given in Eq. (IV) depends on the degeneracies and stabilities of the POs, respectively (see Sec. III).

Notice that any additional exact integration in Eq. (II) with respect to a bifurcation (catastrophe) variable of the improved SPM can lead to an enhancement of the amplitude $A_{\rm PO}$ in the transition from the bifurcation point to the asymptotic region. This enhancement is of the order $1/\hbar^{1/2}$ as compared to the result of the standard SPM integration. In particular, for the newborn family with the extra degeneracy $\Delta\mathcal{K}$ higher than that of the parent PO, one has such enhancements of the order $1/\hbar^{\Delta\mathcal{K}/2}$ near the bifurcation.

The trace formula (6) thus relates the quantum oscillations in the level density to quantities that are purely determined by the classical system. Therefore, one can understand the shell effects in terms of classical pictures. The sum over POs in Eq. (6) is asymptotically correct to the leading order in $1/\hbar^{1/2}$ and it is hampered by convergence problems GUTZbook90 . However, one is free from those problems by taking the coarse-grained level density

where $\Gamma$ is an averaging width. The value of $\Gamma$ is sufficiently smaller than the distance between the major shells near the Fermi surface (see Refs. BBann72 ; SMepp76 ; SMODzp77 ; sclbook ; migdalrev ; MKApan16 ). Here $t_{\rm PO}=\partial S_{\rm PO}(E)/\partial E$ is the period of the particle motion along a PO taking into account its repetition number. We see that, depending on the smoothing width $\Gamma$, longer orbits are automatically suppressed in the above expressions and the PO sum converges, which it usually does not GUTZbook90 for nonintegrable systems in the limit $\Gamma\rightarrow 0$. Thus, one can highlight the major-shell structure in the level density using a smoothing width that is much larger than the mean single-particle level spacing but smaller than the main shell spacing (the distance between major shells) near the Fermi surface. Alternatively, a finer shell structure can be considered by using essentially smaller smoothing widths, which is important for studying the symmetry-breaking (bifurcation) phenomenon associated with longer POs.

It is an advantage of this approach that the major-shell effects in $\delta g(E)$ can often be explained semiclassically in terms of only a few of the shortest POs in the system. Examples will be given in Sec. V. However, if one wants to study a finer shell structure, specifically at large deformations, some longer orbits have to be included. Hereby, bifurcations of POs play a crucial role, as it will be exemplified in Sec. V.

Let us apply the general FMCT to the RPL potential as an analytically solvable example.