Partial and quasi dynamical symmetries in quantum many-body systems

We introduce the notions of partial dynamical symmetry (PDS) and quasi dynamical symmetry (QDS) and demonstrate their relevance to nuclear spectroscopy, to quantum phase transitions and to mixed systems with regularity and chaos. The analysis serves to highlight the potential role of PDS and QDS towards understanding the emergent"simplicity out of complexity"exhibited by complex many-body systems.


Introduction
The concept of dynamical symmetry (DS) is now widely recognized to be of central importance in our understanding of complex many-body systems. It had major impact on developments in diverse areas of physics, including, hadrons, nuclei and molecules [1][2][3]. Its basic paradigm is to write the Hamiltonian of the system under consideration in terms of Casimir operators of a chain of nested algebras, G 0 ⊃ G 1 ⊃ . . . ⊃ G n . 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, |α 0 , α 1 , . . . , α n , which are the labels of the irreducible representations (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.
The merits of a DS are self-evident. However, in most applications to realistic systems, the predictions of an exact DS are rarely fulfilled and one is compelled to break it. More often one finds that, in a given system, the assumed symmetry is not obeyed uniformly, i.e., is fulfilled by only some states but not by others. In describing a transition between different structural phases, the relevant Hamiltonian, in general, involves competing interactions with incompatible symmetries. The need to address such situations has led to the introduction of partial dynamical symmetry (PDS) [4][5][6] and quasi dynamical symmetry (QDS) [7][8][9][10]. These intermediate-symmetry notions and their implications for dynamical systems, are the subject matter of the present contribution.

The interacting boson model
In order to illustrate the various notions of symmetries and demonstrate their relevance, we employ the interacting boson model (IBM) [2], widely used in the description of low-lying quadrupole collective states in nuclei in terms of N monopole (s) and quadrupole (d) bosons representing valence nucleon pairs. The bilinear combinations {s † s, s † d m , d † m s, d † m d m } span a U(6) algebra, which serves as the spectrum generating algebra. The IBM Hamiltonian is expanded in terms of these generators and consists of Hermitian, rotational-scalar interactions which conserve the total number of s-and d-bosons,N =n s +n d = s † s + m d † m d m . The three dynamical symmetries of the IBM are where, below each algebra, its associated labels of irreps are given.

Partial dynamical symmetry (PDS)
In algebraic models, such as the IBM, 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) [4][5][6] 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) of a DS, mentioned above, are only partially satisfied. It is then possible to identify the following types of partial dynamical symmetries [6] • PDS type I: some of the states have all the dynamical symmetry • PDS type II: all the states have part of the dynamical symmetry • PDS type III: some 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. In what follows we discuss algorithms for constructing Hamiltonians with partial dynamical symmetries of various types and demonstrate their relevance to quantum many-body systems.

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 PDS has been developed in [4] and further elaborated in [5]. The analysis starts from the chain of nested algebras Eq. (5) implies that G dyn is the dynamical (spectrum generating) algebra of the system such that operators of all physical observables can be written in terms of its generators; a single irrep of G dyn contains all states of relevance in the problem. In contrast, G sym is the symmetry algebra and a single of its irreps contains states that are degenerate in energy. Assuming, for simplicity, that particle number is conserved, then all states, and hence the representation Of specific interest in the construction of a PDS associated with the reduction (5), are the n-particle annihilation operatorsT which satisfy the propertŷ for all possible values of Λ contained in a given irrep Σ 0 of G. Equivalently, this condition can be phrased in terms of the action on a lowest weight (LW) state of the G-irrep Σ 0 , T [hn] σ λ |LW ; [h N ] Σ 0 = 0, from which states of good Λ can be obtained by projection. Any n-body, number-conserving normal-ordered interaction,Ĥ = α,β A αβT † αTβ , written in terms of these annihilation operators and their Hermitian conjugates (which transform as the corresponding conjugate irreps), can be added to the Hamiltonian with a DS (5), while still preserving the solvability of states with Σ = Σ 0 . If the operatorsT [hn] σ λ span the entire irrep σ of G, then the annihilation condition (6) is satisfied for all Λ-states in Σ 0 , if none of the G irreps Σ contained in the G dyn irrep [h N −n ] belongs to the G Kronecker product σ × Σ 0 . So the problem of finding interactions that preserve solvability for part of the states (5) is reduced to carrying out a Kronecker product. The arguments for choosing the special irrep Σ = Σ 0 in Eq. (6), 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.

SU(3) PDS (type I) in nuclei
The SU(3) DS chain of the IBM and related quantum numbers are given in Eq. (1b). The DS Hamiltonian involves the Casimir operators of SU(3) and O(3), with eigenvalues λ 2 +µ 2 +λµ+3(λ + µ) and L(L + 1), respectively. The spectrum resembles that of an axiallydeformed rotovibrator and the corresponding eigenstates are arranged in SU(3) multiplets. The label K corresponds geometrically to the projection of the angular momentum on the symmetry axis. In a given SU(3) irrep (λ, µ), each K-value is associated with a rotational band and states with the same angular momentum L, in different K-bands, are degenerate. The lowest SU(3) irrep is (2N, 0), which describes the ground band g(K = 0) of a prolate deformed nucleus. The first excited SU(3) irrep (2N − 4, 2) contains degenerate β(K = 0) and γ(K = 2) bands. This β-γ degeneracy is a characteristic feature of the SU(3) limit which, however, is not commonly observed. In most deformed nuclei the β band lies above the γ 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.

PDS (type II and type III)
PDS of type II corresponds to a situation for which all the states of the system preserve part of the dynamical symmetry, In this case, there are no analytic solutions, yet selected quantum numbers (of the conserved symmetries) are retained. This occurs, for example, when the Hamiltonian contains interaction terms from two different chains with a common symmetry subalgebra, e.g., If G 1 and G 1 are incompatible, i.e., do not commute, then their irreps are mixed in the eigenstates of the Hamiltonian. On the other hand, since G 2 and its subalgebras are common to both chains, then the labels of their irreps remain as good quantum numbers.
An alternative situation where PDS of type II occurs is when the Hamiltonian preserves only some of the symmetries G i in the DS chain and only their irreps are unmixed. Let G 1 ⊃ G 2 ⊃ G 3 be a set of nested algebras which may occur anywhere in the chain, in-between the spectrum generating algebra G 0 and the invariant symmetry algebra G n . A systematic procedure [22] for identifying interactions with PDS of type II, is based on writing the Hamiltonian in terms of generators, g i , of G 1 , which do not belong to its subalgebra G 2 . By construction, such Hamiltonian preserves the G 1 symmetry but, in general, not the G 2 symmetry, and hence will have the G 1 labels as good quantum numbers but will mix different irreps of G 2 . The Hamiltonians can still conserve the G 3 labels e.g., by choosing it to be a scalar of G 3 . The procedure involves the identification of the tensor character under G 2 and G 3 of the operators g i and their products, g i g j . . . g k . The Hamiltonians obtained in this manner belong to the integrity basis of G 3 -scalar operators in the enveloping algebra of G 1 and, hence, their existence is correlated with their order.
In the IBM, such a scenario can be realized by considering an interaction term of the form PDS of type III has a hybrid character, for which some of the states of the system under study preserve part of the dynamical symmetry [23]. In relation to the dynamical symmetry chain of Eq. (5), with associated basis, |[h N ] Σ Λ , this can be accomplished by relaxing the condition of Eq. (6), so that it holds only for selected states Λ contained in a given irrep Σ 0 of G and/or selected (combinations of) components λ of the tensorT [hn] σ λ . Under such circumstances, let G = G sym be a subalgebra of G in the aforementioned chain, G ⊃ G . In general, the Hamiltonians, constructed from these tensors, in the manner described in Section 3, are not invariant under G nor G . Nevertheless, they do possess the subset of solvable states, |[h N ] Σ 0 Λ , with good G-symmetry Σ 0 (which now span only part of the corresponding Girrep), while other states are mixed. At the same time, the symmetry associated with the subalgebra G , is broken in all states (including the solvable ones). Thus, part of the eigenstates preserve part of the symmetry. These are precisely the requirements of PDS of type III.
In the IBM, such a generalized partial symmetry associated with the O(6) chain of Eq. (1c), can be realized by an Hamiltonian constructed of boson-pair operators which are not invariant under O(6) nor O(5), but annihilate the coherent state, |β = 1, γ = 0; N , of Eq. (3), which has σ = N [23]. Such an Hamiltonian has a solvable ground band with good O(6) symmetry, which is not preserved by other states. All eigenstates, including the solvable ones, break the O (5) symmetry. An empirical manifestation of such type of O(6)-PDS is presented in Section 5.1.

Measures of PDS
The PDS notion reflects the purity of selected eigenstates with respect to a DS basis. The above algorithms provide a procedure for an explicit construction of Hamiltonians with such property. More general (and realistic) Hamiltonians often exhibit features of a PDS to a certain approximation. In such cases, one needs to assess the quality and applicability of the PDS notion. In what follows, we discuss two quantitative measures of PDS, based on wave-function entropy and quantum number fluctuations.
Consider an eigenfunction of the IBM Hamiltonian, |L , with angular momentum L.
The The degree of a symmetry of a state |L can also be inferred from the fluctuations of the corresponding quantum number. As an example, for the O(6) symmetry of the IBM, the fluctuations in σ can be calculated as [25] where the sum is over all basis states in the chain, Eq. (1c). If |L carries an exact O(6) quantum number, σ fluctuations are zero, ∆σ L = 0. If |L contains basis states with different O(6) quantum numbers, then ∆σ L > 0, indicating that the O(6) symmetry is broken. Note that ∆σ L also vanishes for a state with a mixture of components with the same σ but different O(5) quantum numbers τ , corresponding to a |L with good O(6) but mixed O(5) character. ∆σ L has the same physical content as S O6 (L) (13c) and both can be used as measures of O(6)-PDS.

O(6)-PDS (type III) in nuclei
A recent study [25] has examined the fluctuations ∆σ L , Eq. (14), for the entire parameter space of the ECQF Hamiltonian (4). Results of this calculation for the ground state, |L = 0 + 1 , with N = 14, are shown in Fig. 4. At the O(6) DS limit (ξ = 1, χ = 0), ∆σ 0 ≡ ∆σ L=0 1 vanishes per construction whereas it is greater than zero for all other parameter pairs. Towards the U(    (14), for rare earth nuclei in the vicinity of the identified region of approximate ground-state-O(6) symmetry [25]. Also shown are the fraction f (L) σ=N of O(6) basis states with σ = N contained in the L = 0, 2, 4 states, members of the ground band. The structure parameters ξ and χ are taken from [26].  [26], allow one to relate the structure of collective nuclei to the parameter space of the ECQF Hamiltonian (4). From the extracted (ξ, χ) parameters one can calculate the fluctuations ∆σ L and the fractions f σ=N of squared σ = N amplitude. Nuclei with ∆σ 0 < 0.5 and f σ=N > 95% in the ground-state (L = 0 + 1 ) are listed in Table 1. These quantities are also calculated for yrast states with L > 0 and exhibit similar values in each nucleus. It is evident that a large set of rotational rare earth nuclei, such as 160 Gd, are located in the valley of small σ fluctuations. They can be identified as candidate nuclei with an approximate O(6)-PDS of type III not only for the ground state, but also for the members of the band built on top of it.

Quasi dynamical symmetry (QDS)
A second kind of intermediate-symmetry occurring in algebraic modeling of dynamical systems, is that of quasi dynamical symmetry (QDS) [7][8][9][10]. While QDS can be defined mathematically in terms of embedded representations [27,28], its physical meaning is that several observables associated with particular eigenstates, may be consistent with a certain symmetry which in fact is broken in the Hamiltonian. This typically occurs for a Hamiltonian transitional between two DS limitsĤ H(α) involves competing incompatible (non-commuting) symmetries. For α = 0 or α = 1, one recovers the limiting symmetries. For 0 < α < 1, both symmetries are broken and mixing occurs. A detailed study [8][9][10] of such Hamiltonians has found that for most values of α, selected states continue to exhibit characteristic properties (e.g., energy and B(E2) ratios) of the closest DS limit. Such an "apparent" persistence of symmetry in the face of strong symmetry-breaking interactions, defines a QDS. The indicated persistence is clearly evident in the spectrum shown in Fig. 6, for an IBM Hamiltonian,Ĥ(α), interpolating between the G 1 = U (5) and G 2 = SU (3) DS limits, relevant to shape-phase transitions between spherical and axially-deformed nuclei [21]. The "apparent" symmetry is due to the coherent nature of the mixing. As seen in Fig. 7, the mixing of SU(3) irreps is large, but is approximately independent of the angular momentum of the yrast states, i.e., the SU(3) expansion coefficients C  [31] and employ the following product of the maximum correlation coefficients: The set of states {0 i , 2 j , 4 k , 6 } is considered as comprising a K = 0 band with SU(3)-QDS, if C SU3 (0 i −6) ≈ 1.

PDS and QDS in the symmetry triangle
Recently, a comprehensive analysis of the PDS and QDS properties ofĤ ECQF , Eq. (4), was carried out [30], employing the symmetry measures discussed above. Representative results are shown in Fig. 8

Linking PDS and QDS
The concept of PDS reflects the purity of selected states, hence is different from the concept of QDS which reflects a coherent mixing. Nevertheless, a link between these two hitherto unrelated symmetry concepts can be established and shown to be empirically manifested in rotational nuclei [25].
The experimental spectrum of 160 Gd, along with its ECQF description (4), is shown in the left panel of Fig. 9. The middle and right panels show the decomposition into O(6) and SU(3) basis states, respectively, for yrast states with L = 0, 2, 4. It is evident that the SU(3) symmetry is broken, as significant contributions of basis states with different SU(3) quantum numbers (λ, µ) occur. It is also clear from Fig. 9c that this mixing occurs in a coherent manner with similar patterns for the different members of the ground-state band. As explained in Section 6,

Impact of PDS and QDS on mixed regular and chaotic dynamics
Hamiltonians with a dynamical symmetry are always completely integrable. The Casimir invariants of the algebras in the chain provide a set of constants of the motion in involution. The classical motion is purely regular. A symmetry-breaking is connected to non-integrability and may give rise to chaotic motion. Hamiltonians with PDS and QDS are not completely integrable, hence can exhibit stochastic behavior, nor are they completely chaotic, since some eigenstates preserve the symmetry exactly in a PDS or mix in a coherent fashion in a QDS. Consequently, Hamiltonians with such intermediate symmetries are optimally suitable to the study of mixed systems with coexisting regularity and chaos.
The dynamics of a generic classical Hamiltonian system is mixed; KAM islands of regular motion and chaotic regions coexist in phase space. In the associated quantum system, the statistical properties of the spectrum are usually intermediate between the Poisson and the Gaussian orthogonal ensemble (GOE) statistics. In a PDS, the symmetry of the subset of solvable states is exact, yet does not arise from invariance properties of the Hamiltonian. Several works have shown that a PDS is strongly correlated with suppression of chaos [33,34]. This enhancement of regularity was seen in both the classical measures of chaos, e.g., the fraction of chaotic volume and the average largest Lyapunov exponent, and in quantum measures of chaos, e.g., the nearest neighbors level spacing distribution, whose parameter interpolates between the Poisson and GOE statistics. The reduction in chaos occurs even when the fraction of solvable states approaches zero in the classical limit, suggesting that the existence of a PDS increases the purity of other neighbouring states in the system.
The coherent mixing common to a set of states, characterizing a QDS, results from the existence of a single intrinsic state for each such band and imprints an adiabatic motion and increased regularity [35]. This was verified for low- [10] and high-lying [31] rotational bands using the ECQF Hamiltonian, Eq. (4). SU(3) QDS has been proposed [36] to underly the "arc of regularity" [37], a narrow zone of enhanced regularity in the parameter-space ofĤ ECQF . The arc, shown by a blue dotted line in Fig. 1, resides in the interior of the symmetry triangle and connects the U(5) and SU(3) vertices.
9. PDS and QDS in a first-order quantum phase transition Quantum phase transitions (QPTs) are qualitative changes in the properties of a physical system induced by a variation of parameters in the quantum Hamiltonian. Such structural changes are currently of great interest in different branches of physics [38]. The competing interactions in the Hamiltonian that drive these ground-state phase transitions can affect dramatically the nature of the dynamics and, in some cases, lead to the emergence of quantum chaos [39][40][41][42]. Here we show that PDS and QDS can characterize the remaining regularity in a system undergoing a QPT, amidst a complicated environment of other states [42].
Focusing on the dynamics at the critical-point of a first-order QPT between spherical and deformed shapes, the relevant IBM Hamiltonian [43], upto a scale, can be taken to be the second term of Eq. (10),Ĥ cri = P † 2 ·P 2 . The latter has the SU(3) basis states of Eq. (11) and the following U(5) basis states as solvable eigenstates, while all other states are mixed with respect to both U(5) and SU (3). As such,Ĥ cri exhibits a coexistence of SU(3)-PDS and U(5)-PDS [44]. The classical limit of the Hamiltonian is obtained through the use of Glauber coherent states. This amounts to replacing (s † , d † µ ) by c-numbers (α * s , α * µ ) rescaled by √ N and taking N → ∞, with 1/N playing the role of [45]. Setting all momenta to zero, yields the classical potential V (x, y), which coincides with the surface of Eq. (2), with (β, γ) as polar coordinates. The classical dynamics constraint to L = 0, can be depicted conveniently via Poincaré surfaces of sections in the plane y = 0, plotting the values of (p x , x) each time a trajectory intersects the plane. Regular trajectories are bound to toroidal manifolds within the phase space and their intersections with the plane of section lie on 1D curves (ovals). In contrast, chaotic trajectories randomly cover kinematically accessible areas of the section.
The Poincaré sections associated with the classical critical-point Hamiltonian are shown in Fig. 10 for representative energies. The bottom panel displays the classical potential which has two degenerate spherical and deformed minima. The dynamics in the region of the deformed minimum is robustly regular. The trajectories form a single island and remain regular even at energies far exceeding the barrier height V bar . In contrast, the dynamics in the region of the spherical minimum shows a change with energy from regularity to chaos, until complete chaoticity is reached near the barrier top. The clear separation between regular and chaotic dynamics, associated with the two minima, persists all the way to the barrier energy, E = V bar , where the two regions just touch. At E > V bar , a layer of chaos develops in the deformed region and gradually dominates the surviving regular island for E V b . The quantum manifestations of such an inhomogeneous phase space structure, can be studied by the method of Peres lattices [46]. The latter are constructed by plotting the expectation values O i = i|Ô|i of an arbitrary operator, [Ô,Ĥ] = 0, versus the energy E i = i|Ĥ|i of the Hamiltonian eigenstates |i . The lattices {O i , E i } corresponding to regular dynamics display an ordered pattern, while chaotic dynamics leads to disordered meshes of points [46,47]. In the present analysis, we choose the Peres operator to ben d , whose expectation value is related to the coordinate x in the classical potential. The lattices {x i , E i }, with x i ≡ 2 i|n d |i /N , can then distinguish regular from irregular states and associate them with a given region in phase space.
The Peres lattices corresponding to (N = 50, L = 0, 2, 3, 4) eigenstates ofĤ cri are shown in Fig. 11, overlayed on the classical potential. They disclose regular sequences of states localized within and above the deformed well. They are comprised of rotational states with L = 0, 2, 4, . . . forming regular K = 0 bands and sequences L = 2, 3, 4, . . . forming K = 2 bands. Additional Kbands (not shown in Fig. 11), corresponding to multiple β and γ vibrations about the deformed shape, can also be identified. The states in each regular band share a common structure, to be discussed below. Such ordered band-structures persist to energies above the barrier and are not present in the disordered (chaotic) portions of the Peres lattice. At low-energy, in the vicinity of the spherical well, one can also detect multiplets of states with L = 0, L = 2 and L = 0, 2, 4, typical of quadrupole excitations of a spherical shape.
An important clue on the nature of the surviving regular sequences of selected states, in the presence of more complicated type of eigenstates, comes from a symmetry analysis of their wave functions. The left column of Fig. 12 shows the U(5) n d -probabilities, P (L i ) n d (13a), for eigenstates ofĤ cri , selected on the basis of having the largest components with n d = 0, 1, 2, 3, 4, within the given L spectra. The states are arranged into panels labeled by 'n d ' to conform with the structure of the n d -multiplets of the U(5) DS limit. The normalized U(5) Shannon entropy S U5 (L i ), Eq. (13a), is indicated for representative eigenstates. In particular, the zero-energy L = 0 + 2 state is seen to be a pure n d = 0 state, with S U5 = 0, which is the solvable U(5)-PDS eigenstate of Eq. (16a). The state 2 + 2 has a pronounced n d = 1 component (96%) and the states (L = 0 + 4 , 2 + 5 , 4 + 3 ) in the third panel, have a pronounced n d = 2 component and a low value of S U5 < 0.15. All the above states with 'n d ≤ 2 have a dominant single n d component, and hence qualify as 'spherical' type of states. These are the lowest left-most states in the Peres lattices of Fig. 11, mentioned above. In contrast, the states in the panels 'n d = 3' and 'n d = 4' of Fig. 12, are significantly fragmented. A notable exception is the L = 3 + 2 state, which is the solvable U(5)-PDS state of Eq. (16b) with n d = 3. The existence in the spectrum of specific spherical-type of states with either P The states shown on the right column of Fig. 12 have a different character. They belong to the five lowest regular sequences seen in the Peres lattices of Fig. 11, in the region x ≥ 1. They have a broad n d -distribution, hence are qualified as 'deformed'-type of states, forming rotational bands: g(K = 0), β(K = 0), β 2 (K = 0), β 3 (K = 0) and γ(K = 2). Each panel depicts the SU(3) (λ, µ)-distribution, P (L i ) (λ,µ) for the band members, the normalized SU(3) Shannon entropy S SU3 (L) (13b) for the bandhead state, and the Pearson correlator C SU3 (0 i −6) defined in Section 6. The ground g(K = 0) and γ(K = 2) bands are pure [S SU3 = 0] with (λ, µ) = (2N, 0) and (2N − 4, 2) SU3) character, respectively. These are the solvable bands of Eq. (11) with SU(3) PDS. The nonsolvable K-bands are mixed with respect to SU(3) in a coherent, L-independent, manner, hence exemplify SU(3)-QDS. As expected, we find C SU3 (0 i −6) ≈ 1 for these K-bands. The persisting regular U(5)-like [SU(3)-like] multiplets reflect the geometry of the classical Landau potential, as they are associated with the different spherical (deformed) minimum. One can use the corresponding measures of PDS and QDS as fingerprints of the QPT, not only at the critical point, but also throughout the coexistence region, where the two minima interchange [42].

Concluding remarks
The many examples of PDS and QDS, discussed in the present contribution, demonstrate that these intermediate-symmetries are more abundant than previously recognized. Contrary to naive expectations, the symmetry triangle appears to encompass important elements of symmetry and "not all is lost" inside it. Although, the examples considered were presented in the framework of a bosonic model, it is important to emphasize that these symmetry concepts are applicable to any many-body problem (bosons and fermions) endowed with an algebraic structure. Examples of many-body Hamiltonians with fermionic PDS and QDS are known [48][49][50]. The PDS algorithms discussed, for constructing Hamiltonians with PDS, are applicable to any semi-simple algebra and can be extended to coupled algebraic structure, G 1 × G 2 [51,52]. Attempts are under way to extend the PDS notion to Bose-Fermi symmetries and supersymmetries [53].
An important virtue of the PDS algorithms is their ability to incorporate and provide a selection criterion for higher-order terms [54]. Such terms are needed for an accurate description of the data and for extensions of ab-initio and beyond-mean-field methods to larger systems, which necessitate a strategy to deal with A-body effective interactions and proliferation of parameters. On one hand, the PDS approach allows more flexibility by relaxing the constrains of an exact dynamical symmetry (DS). On the other hand, the PDS approach picks particular symmetry-breaking terms which do not destroy results previously obtained with a DS for a segment the spectrum. The PDS construction is implemented order by order, yet the scheme is non-perturbative in the sense that the non-solvable states can experience strong symmetrybreaking. These virtues generate an efficient tool which can greatly enhance the scope of algebraic modeling of dynamical systems.
Correlated quantum many-body systems often display an astonishing regular excitation patterns which raises a fundamental question, namely, how simplicity emerges out of complexity in such circumstances. The simple patterns show up amidst a complicated environment of other states. It is natural to associate the "simple" states with a symmetry that protects their purity and special character. This symmetry, however, is shared by only a subset of states, and is broken in the remaining eigenstates of the same Hamiltonian. It thus appears that realistic quantum many-body Hamiltonians can accommodate simultaneously eigenstates with different symmetry character. The symmetry in question cannot be exact but only partial or "apparent". These are precisely the defining ingredients of PDS and QDS. These novel concepts of symmetries can thus offer a possible clue in addressing the "simplicity out of complexity" challenge.