Partial Dynamical Symmetries
This overview focuses on the notion of partial dynamical symmetry (PDS), for which a prescribed symmetry is obeyed by a subset of solvable eigenstates, but is not shared by the Hamiltonian. General algorithms are presented to identify interactions, of a given order, with such intermediate-symmetry structure. Explicit bosonic and fermionic Hamiltonians with PDS are constructed in the framework of models based on spectrum generating algebras. PDSs of various types are shown to be relevant to nuclear spectroscopy, quantum phase transitions and systems with mixed chaotic and regular dynamics.
Dynamical symmetry, partial symmetry, algebraic models,
quantum phase transitions, regularity and chaos,
pairing and seniority.
PACS numbers: 21.60.Fw, 03.65.Fd, 21.10.Re, 21.60.Cs, 05.45.-a
- 1 Introduction
- 2 PDS type I
- 3 PDS type II
- 4 PDS type III
- 5 Partial Solvability
- 6 PDS and Quantum Phase Transitions
- 7 PDS and Mixed Regular and Chaotic Dynamics
- 8 PDS and Higher-Order Terms
- 9 PDS and Coupled Systems
- 10 PDS in Fermion Systems
- 11 Concluding Remarks
Symmetries play an important role in dynamical systems. Constants of motion associated with a symmetry govern the integrability of a given classical system. At the quantum level, symmetries provide quantum numbers for the classification of states, determine spectral degeneracies and selection rules, and facilitate the calculation of matrix elements. An exact symmetry occurs when the Hamiltonian of the system commutes with all the generators () of the symmetry-group , . In this case, all states have good symmetry and are labeled by the irreducible representations (irreps) of . The Hamiltonian admits a block structure so that inequivalent irreps do not mix and all eigenstates in the same irrep are degenerate. In a dynamical symmetry the Hamiltonian commutes with the Casimir operator of , , the block structure of is retained, the states preserve the good symmetry but, in general, are no longer degenerate. When the symmetry is completely broken then , and none of the states have good symmetry. In-between these limiting cases there may exist intermediate symmetry structures, called partial (dynamical) symmetries, for which the symmetry is neither exact nor completely broken. This novel concept of symmetry and its implications for dynamical systems, in particular nuclei, are the focus of the present review.
Models based on spectrum generating algebras form a convenient framework to examine underlying symmetries in many-body systems and have been used extensively in diverse areas of physics . Notable examples in nuclear physics are Wigner’s spin-isospin SU(4) supermultiplets , SU(2) single- pairing , Elliott’s SU(3) model , symplectic model , pseudo SU(3) model , Ginocchio’s monopole and quadrupole pairing models , interacting boson models (IBM) for even-even nuclei  and boson-fermion models (IBFM) for odd-mass nuclei . Similar algebraic techniques have proven to be useful in the structure of molecules [10, 11] and of hadrons . In such models the Hamiltonian is expanded in elements of a Lie algebra, (), called the spectrum generating algebra. A dynamical symmetry occurs if the Hamiltonian can be written in terms of the Casimir operators of a chain of nested algebras, . The following properties are then observed. (i) All states are solvable and analytic expressions are available for energies and other observables. (ii) All states are classified by quantum numbers, , which are the labels of the irreps of the algebras in the chain. (iii) The structure of wave functions is completely dictated by symmetry and is independent of the Hamiltonian’s parameters.
A dynamical symmetry provides clarifying insights into complex dynamics and its merits are self-evident. However, in most applications to realistic systems, the predictions of an exact dynamical symmetry are rarely fulfilled and one is compelled to break it. The breaking of the symmetry is required for a number of reasons. First, one often finds that the assumed symmetry is not obeyed uniformly, i.e., is fulfilled by only some of the states but not by others. Certain degeneracies implied by the assumed symmetry are not always realized, (e.g., axially deformed nuclei rarely fulfill the IBM SU(3) requirement of degenerate and bands ). Secondly, forcing the Hamiltonian to be invariant under a symmetry group may impose constraints which are too severe and incompatible with well-known features of nuclear dynamics (e.g., the models of  require degenerate single-nucleon energies). Thirdly, in describing transitional nuclei in-between two different structural phases, e.g., spherical and deformed, the Hamiltonian by necessity mixes terms with different symmetry character. In the models mentioned above, the required symmetry breaking is achieved by including in the Hamiltonian terms associated with (two or more) different sub-algebra chains of the parent spectrum generating algebra. In general, under such circumstances, solvability is lost, there are no remaining non-trivial conserved quantum numbers and all eigenstates are expected to be mixed. A partial dynamical symmetry (PDS) corresponds to a particular symmetry breaking for which some (but not all) of the virtues of a dynamical symmetry are retained. The essential idea is to relax the stringent conditions of complete solvability so that the properties (i)–(iii) are only partially satisfied. It is then possible to identify the following types of partial dynamical symmetries
PDS type I: part of the states have all the dynamical symmetry
PDS type II: all the states have part of the dynamical symmetry
PDS type III: part of the states have part of the dynamical symmetry.
In PDS of type I, only part of the eigenspectrum is analytically solvable and retains all the dynamical symmetry (DS) quantum numbers. In PDS of type II, the entire eigenspectrum retains some of the DS quantum numbers. PDS of type III has a hybrid character, in the sense that some (solvable) eigenstates keep some of the quantum numbers.
The notion of partial dynamical symmetry generalizes the concepts of exact and dynamical symmetries. In making the transition from an exact to a dynamical symmetry, states which are degenerate in the former scheme are split but not mixed in the latter, and the block structure of the Hamiltonian is retained. Proceeding further to partial symmetry, some blocks or selected states in a block remain pure, while other states mix and lose the symmetry character. A partial dynamical symmetry lifts the remaining degeneracies, but preserves the symmetry-purity of the selected states. The hierarchy of broken symmetries is depicted in Fig. 1.
The existence of Hamiltonians with partial symmetry or partial dynamical symmetry is by no means obvious. An Hamiltonian with the above property is not invariant under the group nor does it commute with the Casimir invariants of , so that various irreps are in general mixed in its eigenstates. However, it posses a subset of solvable states, denoted by in Fig. 1, which respect the symmetry. The commutator or vanishes only when it acts on these ‘special’ states with good -symmetry.
In this review, we survey the various types of partial dynamical symmetries (PDS) and discuss algorithms for their realization in bosonic and fermionic systems. Hamiltonians with PDS are explicitly constructed, including higher-order terms. We present empirical examples of the PDS notion and demonstrate its relevance to nuclear spectroscopy, to quantum phase transitions and to mixed systems with coexisting regularity and chaos.
1.1 The interacting boson model
In order to illustrate the various notions of symmetries and consider their implications, it is beneficial to have a framework that has a rich algebraic structure and allows tractable yet detailed calculations of observables. Such a framework is provided by the interacting boson model (IBM) [8, 14, 15, 16], widely used in the description of low-lying quadrupole collective states in nuclei. The degrees of freedom of the model are one monopole boson () and five quadrupole bosons (). The bilinear combinations span a U(6) algebra. These generators can be transcribed in spherical tensor form as
where , and standard notation of angular momentum coupling is used. U(6) serves as the spectrum generating algebra and the invariant (symmetry) algebra is O(3), with generators . The IBM Hamiltonian is expanded in terms of the operators (1) and consists of Hermitian, rotational-scalar interactions which conserve the total number of - and - bosons, . Microscopic interpretation of the model suggests that for a given even-even nucleus the total number of bosons, , is fixed and is taken as the sum of valence neutron and proton particle and hole pairs counted from the nearest closed shell .
The three dynamical symmetries of the IBM are
The associated analytic solutions resemble known limits of the geometric model of nuclei , as indicated above. Each chain provides a complete basis, classified by the irreps of the corresponding algebras, which can be used for a numerical diagonalization of the Hamiltonian in the general case. In the Appendix we collect the relevant information concerning the generators and Casimir operators of the algebras in Eq. (2). Electromagnetic moments and rates can be calculated in the IBM with transition operators of appropriate rank. For example, the most general one-body E2 operator reads
A geometric visualization of the model is obtained by an energy surface
Here are quadrupole shape parameters whose values, , at the global minimum of define the equilibrium shape for a given Hamiltonian. The shape can be spherical or deformed with (prolate), (oblate), -independent, or triaxial . The latter shape requires terms of order higher than two-body in the boson Hamiltonian [21, 22]. The equilibrium deformations associated with the Casimir operators of the leading subalgebras in the dynamical symmetry chains (2) are, for U(5), for SU(3) and for O(6).
2 PDS type I
PDS of type I corresponds to a situation for which the defining properties of a dynamical symmetry (DS), namely, solvability, good quantum numbers, and symmetry-dictated structure are fulfilled exactly, but by only a subset of states. An algorithm for constructing Hamiltonians with this property has been developed in  and further elaborated in . The analysis starts from the chain of nested algebras
where, below each algebra, its associated labels of irreps are given. Eq. (6) implies that is the dynamical (spectrum generating) algebra of the system such that operators of all physical observables can be written in terms of its generators [11, 13]; a single irrep of contains all states of relevance in the problem. In contrast, is the symmetry algebra and a single of its irreps contains states that are degenerate in energy. A frequently encountered example is , the algebra of rotations in 3 dimensions, with its associated quantum number of total angular momentum . Other examples of conserved quantum numbers can be the total spin in atoms or total isospin in atomic nuclei.
The classification (6) is generally valid and does not require conservation of particle number. Although the extension from DS to PDS can be formulated under such general conditions, let us for simplicity assume in the following that particle number is conserved. All states, and hence the representation , can then be assigned a definite particle number . For identical particles the representation of the dynamical algebra is either symmetric (bosons) or antisymmetric (fermions) and will be denoted, in both cases, as . For particles that are non-identical under a given dynamical algebra , a larger algebra can be chosen such that they become identical under this larger algebra (generalized Pauli principle). The occurrence of a DS of the type (6) signifies that the Hamiltonian is written in terms of the Casimir operators of the algebras in the chain
and its eigenstates can be labeled as ; additional labels (indicated by ) are suppressed in the following. Likewise, operators can be classified according to their tensor character under (6) as .
Of specific interest in the construction of a PDS associated with the reduction (6), are the -particle annihilation operators which satisfy the property
for all possible values of contained in a given irrep of . Any -body, number-conserving normal-ordered interaction written in terms of these annihilation operators and their Hermitian conjugates (which transform as the corresponding conjugate irreps)
has a partial G-symmetry. This comes about since for arbitrary coefficients, , is not a G-scalar, hence most of its eigenstates will be a mixture of irreps of G, yet relation (8) ensures that a subset of its eigenstates , are solvable and have good quantum numbers under the chain (6). An Hamiltonian with partial dynamical symmetry is obtained by adding to the dynamical symmetry Hamiltonian, (7), still preserving the solvability of states with
If the operators span the entire irrep of G, then the annihilation condition (8) is satisfied for all -states in , if none of the irreps contained in the irrep belongs to the Kronecker product . So the problem of finding interactions that preserve solvability for part of the states (6) is reduced to carrying out a Kronecker product. In this case, although the generators of do not commute with , their commutator does vanish when it acts on the solvable states (8)
Eq. (11b) follows from and the fact that involves a linear combination of -tensor operators which satisfy Eq. (8). The arguments for choosing the special irrep in Eq. (8), which contains the solvable states, are based on physical grounds. A frequently encountered choice is the irrep which contains the ground state of the system.
The above algorithm for constructing Hamiltonians with PDS of type I is applicable to any semisimple group. It can also address more general scenarios, in which relation (8) holds only for some states in the irrep and/or some components of the tensor . In this case, the Kronecker product rule does not apply, yet the PDS Hamiltonian is still of the form as in Eqs. (9)-(10), but now the solvable states span only part of the corresponding -irrep. This is not the case in the quasi-exactly solvable Hamiltonians, introduced in , where the solvable states form complete representations. The coexistence of solvable and unsolvable states, together with the availability of an algorithm, distinguish the notion of PDS from the notion of accidental degeneracy , where all levels are arranged in degenerate multiplets.
An Hamiltonian with PDS of type I does not have good symmetry but some of its eigenstates do. The symmetry of the latter states does not follow from invariance properties of the Hamiltonian. This situation is opposite to that encountered in a spontaneous symmetry breaking, where the Hamiltonian respects the symmetry but its ground state breaks it. The notion of PDS differs also from the notion of quasi-dynamical symmetry . The latter corresponds to a situation in which selected states in a system continue to exhibit characteristic properties (e.g., energy and B(E2) ratios) of a dynamical symmetry, in the face of strong symmetry-breaking interactions. Such an “apparent” persistence of symmetry is due to the coherent nature of the mixing in the wave functions of these states. In contrast, in a PDS of type I, although the symmetry is broken (even strongly) in most states, the subset of solvable states preserve it exactly. In this sense, the symmetry is partial but exact!.
In what follows we present concrete constructions of Hamiltonians with PDS associated with the three DS chains (2) of the IBM. The partial symmetries in question involve continuous Lie groups. PDS can, however, be associated also with discrete groups which are relevant, e.g., to molecular physics. An example of a partial symmetry which involves point groups was presented in , employing a model of coupled anharmonic oscillators to describe the molecule . The partial symmetry of the Hamiltonian allowed a derivation of analytic expressions for the energies and eigenstates of a set of unique levels. Furthermore, the numerical calculations required to obtain the energies of the remaining (non-unique) levels were greatly simplified since the Hamiltonian could be diagonalized in a much smaller space.
2.1 U(5) PDS (type I)
The U(5) DS chain of the IBM and related quantum numbers are given by 
where the generators of the above groups are defined in Table 16 of the Appendix. For a given U(6) irrep , the allowed U(5) and O(5) irreps are and or , respectively. The values of contained in the O(5) irrep , are obtained by partitioning , with integers, and . The multiplicity label in the reduction, counts the maximum number of -boson triplets coupled to . The eigenstates are obtained with a Hamiltonian with U(5) DS which, for one- and two-body interactions, can be transcribed in the form
Here and are the linear and quadratic Casimir operators of U(5), respectively, and denotes the quadratic Casimir operator of , as defined in the Appendix. The Casimir operators of U(6) are omitted from Eq. (13) since they are functions of the total boson number operator, , which is a constant for all -boson states. The spectrum of is completely solvable with eigenenergies
The U(5)-DS spectrum of Eq. (14) resembles that of an anharmonic spherical vibrator, describing quadrupole excitations of a spherical shape. The splitting of states in a given U(5) multiplet, , is governed by the O(5) and O(3) terms in (13). The lowest U(5) multiplets involve states with quantum numbers , , and , .
The construction of Hamiltonians with U(5)-PDS is based on identification of -boson operators which annihilate all states in a given U(5) irrep . A physically relevant choice is the irrep which consists of the ground state, with , built of -bosons
Considering U(5) tensors, , composed of bosons of which are -bosons then, clearly, the Hermitian conjugate of such operators with will annihilate the state of Eq. (15). Explicit expressions for -boson U(5) tensors, with are shown in Table 1. From them one can construct the following one- and two-body Hamiltonian with U(5)-PDS
where means Hermitian conjugate. By construction,
Using Eq. (182), we can rewrite in the form
where is the U(5) dynamical symmetry Hamiltonian, Eq. (13), and is given by
The last property follows from the U(5) selection rules of , , and the fact that the irreps and do not contain an state. Altogether, (18) is not diagonal in the U(5) chain, but retains the following solvable U(5) basis states with known eigenvalues
As will be discussed in Section 6, this class of Hamiltonians with U(5)-PDS of type I is relevant to first-order quantum shape-phase transitions in nuclei.
A second class of Hamiltonians with U(5)-PDS can be obtained by considering the interaction
This interaction breaks the U(5) DS, however, it still has selected U(5) basis states as zero-energy eigenstates
where takes the values compatible with the reduction. These properties follow from the fact that annihilates states with () and annihilates states with . Adding the interaction to the U(5) dynamical symmetry Hamiltonian (13), we obtain the following Hamiltonian with U(5)-PDS
is not diagonal in the U(5) chain, but retains the following solvable U(5) basis states with known eigenvalues
The Hamiltonian (24) with U(5)-PDS of type I, contains terms from both the U(5) and O(6) chains (2), hence preserves the common segment of subalgebras, . As such, it exhibits also an O(5)-PDS of type II, to be discussed in Section 3. As will be shown in Section 6, this class of Hamiltonians is relevant to second-order quantum shape-phase transitions in nuclei.
2.2 SU(3) PDS (type I)
The SU(3) DS chain of the IBM and related quantum numbers are given by 
where the generators of the above groups are defined in Table 16 of the Appendix. For a given U(6) irrep , the allowed SU(3) irreps are with non-negative integers, such that, . The multiplicity label is needed for complete classification and corresponds geometrically to the projection of the angular momentum on the symmetry axis. The values of contained in the above SU(3) irreps are , where ; with the exception of for which . The states form the (non-orthogonal) Elliott basis  and the Vergados basis  is obtained from it by a standard orthogonalization procedure. The two bases coincide in the large-N limit and both are eigenstates of a Hamiltonian with SU(3) DS. The latter, for one- and two-body interactions, can be transcribed in the form
where is the quadratic Casimir operator of , as defined in the Appendix. The spectrum of is completely solvable with eigenenergies
where and . The spectrum resembles that of an axially-deformed rotovibrator and the corresponding eigenstates are arranged in SU(3) multiplets. In a given SU(3) irrep , each -value is associated with a rotational band and states with the same L, in different -bands, are degenerate. The lowest SU(3) irrep is , which describes the ground band of a prolate deformed nucleus. The first excited SU(3) irrep contains both the and bands. States in these bands with the same angular momentum are degenerate. This - degeneracy is a characteristic feature of the SU(3) limit of the IBM which, however, is not commonly observed . In most deformed nuclei the band lies above the band. In the IBM framework, with at most two-body interactions, one is therefore compelled to break SU(3) in order to conform with the experimental data. To do so, the usual approach has been to include in the Hamiltonian terms from other chains so as to lift the undesired - degeneracy. Such an approach was taken by Warner Casten and Davidson (WCD) in , where an O(6) term was added to the SU(3) Hamiltonian. However, in this procedure, the SU(3) symmetry is completely broken, all eigenstates are mixed and no analytic solutions are retained. Similar statements apply to the description in the consistent Q formalism (CQF) , where the Hamiltonian involves a non-SU(3) quadrupole operator. In contrast, partial SU(3) symmetry, to be discussed below, corresponds to breaking SU(3), but in a very particular way so that part of the states (but not all) will still be solvable with good symmetry. As such, the virtues of a dynamical symmetry (e.g., solvability) are fulfilled but by only a subset of states.
The construction of Hamiltonians with SU(3)-PDS is based on identification of -boson operators which annihilate all states in a given SU(3) irrep , chosen here to be the ground band irrep . For that purpose, we consider the following boson-pair operators with angular momentum
As seen from Table 2, these operators are proportional to two-boson SU(3) tensors, , with and
The corresponding Hermitian conjugate boson-pair annihilation operators, and , transform as under SU(3), and satisfy
Equivalently, these operators satisfy
The state is a condensate of bosons and is obtained by substituting the SU(3) equilibrium deformations in the coherent state of Eq. (5), . It is the lowest-weight state in the SU(3) irrep and serves as an intrinsic state for the SU(3) ground band . The rotational members of the band , Eq. (31), are obtained by angular momentum projection from . The relations in Eqs. (31)-(32) follow from the fact that the action of the operators leads to a state with bosons in the U(6) irrep , which does not contain the SU(3) irreps obtained from the product .
where . For , the Hamiltonian is an SU(3) scalar, related to the quadratic Casimir operator of SU(3)
For , the Hamiltonian transforms as a SU(3) tensor component. For arbitrary coefficients, is not an SU(3) scalar, nevertheless, the relations in Eqs. (31)-(32) ensure that it has a solvable zero-energy band with good SU(3) quantum numbers . When the coefficients are positive, becomes positive definite by construction, and the solvable states form its SU(3) ground band.
These relations imply that the sequence of states
are eigenstates of with eigenvalues . A comparison with Eq. (28) shows that these energies are the SU(3) eigenvalues of , Eq. (35), and identify the states to be in the SU(3) irreps . They can be further shown to be the lowest-weight states in these irreps. Furthermore, satisfies
The states (37) are deformed and serve as intrinsic states representing bands with angular momentum projection () along the symmetry axis . In particular, as noted earlier, represents the ground band () and is the -band (). The rotational members of these bands, , Eq. (39), can be obtained by O(3) projection from the corresponding intrinsic states . Relations (38) and (39) are equivalent, since the angular momentum projection operator, , is composed of O(3) generators, hence commutes with . The intrinsic states break the O(3) symmetry, but since the Hamiltonian in Eq. (34) is O(3) invariant, the projected states with good are also eigenstates of with energy and with good SU(3) symmetry. For the ground band the projected states span the entire SU(3) irrep . For excited bands , the projected states span only part of the corresponding SU(3) irreps. There are other states originally in these irreps (as well as in other irreps) which do not preserve the SU(3) symmetry and therefore get mixed. In particular, the ground , and bands retain their SU(3) character and respectively, but the first excited band is mixed. This situation corresponds precisely to that of partial SU(3) symmetry. An Hamiltonian which is not an SU(3) scalar has a subset of solvable eigenstates which continue to have good SU(3) symmetry.
All of the above discussion is applicable also to the case when we add to the Hamiltonian of Eq. (34) the Casimir operator of O(3), and by doing so, convert the partial SU(3) symmetry into partial dynamical SU(3) symmetry. The additional rotational term contributes just an splitting but does not affect the wave functions. The most general one- and two-body Hamiltonian with SU(3)-PDS can thus be written in the form
Here is the SU(3) dynamical symmetry Hamiltonian, Eq. (27), with parameters and . The term in Eq. (40) is not diagonal in the SU(3) chain (26), but Eqs. (31) and (39) ensure that it annihilates a subset of states with good SU(3) quantum numbers. Consequently, retains selected solvable bands with good SU(3) symmetry. Specifically, the solvable states are members of the ground band
The solvable states (41)-(42) are those projected from the intrinsic states of Eq. (37), and are simply selected members of the Elliott basis . In particular, the states belonging to the ground and bands are the Elliott states and respectively. Their wave functions can be expressed in terms of the orthonormal Vergados basis, . For the ground band and for members of the band with odd, the Vergados and Elliott bases are identical. The Elliott states in the band with even are mixtures of Vergados states in the SU(3) irrep
where are known coefficients which appear in the transformation between the two bases .
Since the wave functions of the solvable states are known, it is possible to obtain analytic expressions for matrix elements of observables between them. For calculating E2 rates, it is convenient to rewrite the relevant E2 operator, Eq. (3), in the form
where is the quadrupole SU(3) generator  and is a tensor under SU(3). The B(E2) values for and transitions
The Hamiltonian of Eq. (40), with SU(3)-PDS, was used in  to describe spectroscopic data of Er. The experimental spectra  of the ground , and bands in this nucleus is shown in Fig. 2, and compared with an exact DS, PDS and broken SU(3) calculations.
According to the previous discussion, the SU(3)-PDS spectrum of the solvable ground and bands is given by