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Order, Chaos and Quasi Symmetries in a First-Order Quantum Phase Transition

We study the competing order and chaos in a first-order quantum phase transition with a high barrier. The boson model Hamiltonian employed, interpolates between its U(5) (spherical) and SU(3) (deformed) limits. A classical analysis reveals regular (chaotic) dynamics at low (higher) energy in the spherical region, coexisting with a robustly regular dynamics in the deformed region. A quantum analysis discloses, amidst a complicated environment, persisting regular multiplets of states associated with partial U(5) and quasi SU(3) dynamical symmetries.


Introduction
Quantum phase transitions (QPTs) are qualitative changes in the ground state properties of a physical system induced by a variation of parameters λ in the quantum HamiltonianĤ(λ) [1,2]. Such ground-state transformations have received considerable attention in recent years and have found a variety of applications in many areas of physics and chemistry [3]. The particular type of QPT is reflected in the topology of the underlying mean-field (Landau) potential V (λ). Most studies have focused on second-order (continuous) QPTs [4], where V (λ) has a single minimum which evolves continuously into another minimum. The situation is more complex for discontinuous (first-order) QPTs, where V (λ) develops multiple minima that coexist in a range of λ values and cross at the critical point, λ = λ c . The interest in such QPTs stems from their key role in phase-coexistence phenomena at zero temperature. Examples are offered by the metal-insulator Mott transition [5], heavy-fermion superconductors [6], quantum Hall bilayers [7] and shape-coexistence in mesoscopic systems, such as atomic nuclei [8].
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. This effect has been observed in quantum optics models of N two-level atoms interacting with a single-mode radiation field [9], where the onset of chaos is triggered by continuous QPTs. In the present contribution, we address the mixed regular and chaotic dynamics associated with a first order QPT [10][11][12]. For that purpose, we employ an interacting boson model which describes such QPTs between spherical and axially-deformed nuclei. The model is amenable to both classical and quantum treatments, has a rich group structure and inherent geometry, which makes it an ideal framework for studying the intricate interplay of order and chaos and the role of symmetries in such shape-phase transitions.

Symmetries, geometry and quantum phase transitions in the IBM
The interacting boson model (IBM) [13] describes low-lying quadrupole collective states in nuclei in terms of N interacting monopole (s) and quadrupole (d) bosons representing valence nucleon pairs. The bilinear combinations G ij ≡ b † i b j = {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,Ĥ = ij ij G ij + ijk u ijk G ij G k , and consists of Hermitian |N, σ, τ, n ∆ , L γ−unstable deformed rotor (1c) The associated analytic solutions resemble known limits of the geometric model of nuclei [14], as indicated above. The basis members are classified by the irreducible representations (irreps) of the corresponding algebras. Specifically, the quantum numbers N, n d , (λ, µ), σ, τ and L label the relevant irreps of U(6), U(5), SU (3) Here (β, γ) are quadrupole shape parameters analogous to the variables of the collective model of nuclei. Their values (β eq , γ eq ) at the global minimum of V (β, γ) define the equilibrium shape for a given Hamiltonian. For one-and two-body interactions, the shape can be spherical (β eq = 0) or deformed (β eq > 0) with γ eq = 0 (prolate), γ eq = π/3 (oblate), or γ-independent. The dynamical symmetries of Eq. (1), correspond to phases of the system, and provide analytic benchmarks for the dynamics of stable nuclear shapes. Quantum phase transitions (QPTs) between such stable shapes have been studied extensively in the IBM framework [16,17] and are manifested empirically in nuclei [8]. The Hamiltonians employed mix interaction terms from different DS chains, e.g.,Ĥ(λ) = λĤ a + (1 − λ)Ĥ b . The coupling coefficient (λ) responsible for the mixing, serves as the control parameter and the surface, V (λ) ≡ V (λ; β, γ), serves as the Landau potential. 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. However, for particular symmetry breaking, some intermediate symmetry structure can survive. The latter include partial dynamical symmetry (PDS) [18] and quasi-dynamical symmetry (QDS) [19]. In a PDS, the conditions of an exact dynamical symmetry (solvability of the complete spectrum and existence of exact quantum numbers for all eigenstates) are relaxed and apply to only part of the eigenstates. In a QDS, particular states continue to exhibit selected characteristic properties (e.g., energy and B(E2) ratios) of the closest dynamical symmetry, in the face of strong-symmetry breaking interactions. This "apparent" symmetry is due to the coherent nature of the mixing. As discussed below, both PDS and QDS are relevant to quantum phase transitions [19,20].
In view of the central role of the Landau potential, V (β, γ), for QPTs, it is convenient to resolve the Hamiltonian into two parts,Ĥ =Ĥ int +Ĥ col [21]. The intrinsic part (Ĥ int ) determines the potential surface while the collective part (Ĥ col ) is composed of kinetic terms which do not affect the shape of V (β, γ). For first-order QPTs, the resolution allows the construction of an intrinsic Hamiltonian with a high-barrier [22], and by treating it separately, one avoids the complication of rotation-vibration couplings that can obscure the simple pattern of mixed dynamics, reported below. Henceforth, we confine the discussion to the dynamics of the intrinsic Hamiltonian.

Classical limit and topology of the Landau potential
The classical limit of the IBM is obtained through the use of Glauber coherent states. This amounts to replacing (s † , d † µ ) by six c-numbers (α * s , α * µ ) rescaled by √ N and taking N → ∞, with 1/N playing the role of [23]. Number conservation ensures that phase space is 10dimensional and can be phrased in terms of two shape (deformation) variables, three orientation (Euler) angles and their conjugate momenta. The shape variables can be identified with the β, γ variables introduced through Eq. (2). Setting all momenta to zero, yields the classical potential which is identical to V (β, γ) mentioned above. In the classical analysis presented below we consider, for simplicity, the dynamics of L = 0 vibrations, for which only two degrees of freedom are active. The rotational dynamics with L > 0 is examined in the subsequent quantum analysis.
For the intrinsic Hamiltonian of Eq. (3), constrained to L = 0, the above procedure yields the following classical Hamiltonian Here the coordinates β ∈ [0, √ 2], γ ∈ [0, 2π) and their canonically conjugate momenta p β ∈ [0, √ 2] and p γ ∈ [0, 1] span a compact classical phase space. The term, H d,0 = (T + β 2 )/2 with T = p 2 β + p 2 γ /β 2 , denotes the classical limit ofn d (restricted to L = 0) and forms an isotropic harmonic oscillator Hamiltonian in the β and γ variables. Notice that the classical Hamiltonian of Eq. (8) contains complicated momentum-dependent terms originating from the two-body interactions in the Hamiltonian (3), not just the usual quadratic kinetic energy T . Setting p β = p γ = 0 in Eq. (8) leads to the following classical potential The variables β and γ can be interpreted as polar coordinates in an abstract plane parametrized by Cartesian coordinates x = β cos γ and y = β sin γ. Using these relations together with p x = p β cos γ −(p γ /β) sin γ and p y = (p γ /β) cos γ +p β sin γ, one can cast the classical Hamiltonian, Eq. (8) and potential, Eq. (9), in Cartesian form. Thus, The control parameters ρ and ξ determine the landscape and extremal points of the potentials V 1 (ρ; β, γ) and V 2 (ξ; β, γ), Eq. (9). Important values of these parameters at which a pronounced change in structure is observed, are the spinodal point (ρ * ) where a second (deformed) minimum occurs, an anti-spinodal point (ξ * * ) where the first (spherical) minimum disappears and a critical point (ρ c , ξ c ) in-between, where the two minima are degenerate. For the potentials under discussion, the values of the control parameters at these points are The critical point separates the spherical and deformed phases. The spinodal and anti-spinodal points embrace it and mark the boundary of the phase coexistence region.
In general, the only γ-dependence in the potentials (9) is due to the 2 − β 2 β 3 cos 3γ term. This induces a three-fold symmetry about the origin β = 0. As a consequence, the deformed extremal points are obtained for γ = 0, 2π 3 , 4π 3 (prolate shapes), or γ = π 3 , π, 5π 3 (oblate shapes). It is therefore possible to restrict the analysis to γ = 0 and allow for both positive and negative values of β, corresponding to prolate and oblate deformations, respectively. The potentials V (β, γ = 0) = V (x, y = 0) for several values of ξ, ρ, are shown at the bottom rows of Figs. 2-4. The relevant potential in the spherical phase is V 1 (ρ; β, γ), Eq. (9a), with 0 ≤ ρ ≤ ρ c . In this case, β eq = 0 is a global minimum of the potential at an energy V sph = 0, representing the spherical equilibrium shape. The limiting value at the domain boundary is For ρ = 0, (the U(5) limit), the potential is independent of γ and has β eq = 0 as a single minimum. For ρ > 0, the potential depends on γ, and β = 0 remains a single minimum for 0 ≤ ρ < ρ * , At the spinodal point ρ * , V 1 (ρ) acquires an additional deformed local minimum at an energy V def > 0, and a barrier develops between the two minima. The spherical and deformed minima cross and become degenerate at the critical point (ρ c , ξ c ).
The relevant potential in the deformed phase is V 2 (ξ; β, γ), Eq. (9b), with ξ ≥ ξ c . In this case, [β eq = 2 √ 3 , γ eq = 0] is a global minimum of the potential at an energy V def = 0, representing the deformed equilibrium shape. The limiting value of the domain boundary is The spherical minimum turns local in V 2 (ξ) for ξ > ξ c with energy V sph = 4h 2 ξ > 0, and disappears at the anti-spinodal point ξ * * . For ξ > ξ * * , β = 0 turns into a maximum and the potential remains with a single deformed minimum, reaching the SU(3) limit at ξ = 1.
The indicated changes in the topology of the potential surfaces upon variation of the control parameters (ρ, ξ), identify three regions with distinct structure. Here ρ * , (ρ c , ξ c ), ξ * * , are the spinodal, critical and antispinodal points, respectively, with values given in Eq. (10). The deformation at the global (local) minimum of the Landau potential (9) is marked by solid (dashed) lines. β eq = 0 (β eq = 2 √ 3 ) on the spherical (deformed) side of the QPT. Region I (III) involves a single spherical (deformed) shape, while region II involves shape-coexistence.
I. The region of a stable spherical phase, ρ ∈ [0, ρ * ], where the potential has a single spherical minimum. II. The region of phase coexistence, ρ ∈ (ρ * , ρ c ] and ξ ∈ [ξ c , ξ * * ), where the potential has both spherical and deformed minima which cross and become degenerate at the critical point. III. The region of a stable deformed phase, ξ > ξ * * , where the potential has a single deformed minimum.
The potential surface in each region serves as the Landau potential of the QPT, with the equilibrium deformations as order parameters. The latter evolve as a function of the control parameters (ρ, ξ) and exhibit a discontinuity typical of a first order transition. As depicted in Fig 1, the order parameter β eq is a double-valued function in the coexistence region (in-between ρ * and ξ * * ) and a step-function outside it. In what follows, we examine the nature of the classical and quantum dynamics in each region.

Classical analysis
Chaotic properties of the IBM have been studied extensively [24], albeit, with a simplified Hamiltonian, giving rise to an extremely low barrier. A new element in the present study is the presence of a high barrier at the critical-point, V bar /h 2 = 0.268, compared to V bar /h 2 = 0.0018 in previous works. This allows the uncovering of a rich pattern of regularity and chaos across a generic first-order QPT in a wide coexistence region.
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 x and the momentum p x each time a trajectory intersects the plane [25]. 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.   Fig. 2, but for the classical Hamiltonians H 1 (ρ) (8a) with ρ * * < ρ ≤ ρ c , and H 2 (ξ) (8b) with ξ c ≤ ξ < ξ * * . The classical potentials are V 1 (ρ; x, y = 0) (9a) and V 2 (ξ; x, y = 0) (9b). The Peres lattices involve the quantum HamiltoniansĤ 1 (ρ) (3a) andĤ 2 (ξ) (3b).  H d,0 ), involves the 2D harmonic oscillator Hamiltonian and V 1 (ρ = 0) ∝ 2β 2 − β 4 /2. The system is in the U(5) DS limit and is completely integrable. The orbits are periodic and, as shown in Fig. 2, appear in the surface of section as a finite collection of points. The sections for ρ = 0.03 in Fig. 2, show the phase space portrait typical of an anharmonic (quartic) oscillator with two major regular islands, weakly perturbed by the small ρ cos 3γ term. The orbits are quasi-periodic and appear as smooth one-dimensional invariant curves. For small β, V 1 (ρ) ≈ β 2 −ρ √ 2β 3 cos 3γ and resembles the well-known Hénon-Heiles potential (HH) [26]. Correspondingly, as shown for ρ = 0.2 in Fig. 2, at low energy, the dynamics remains regular and two additional islands show up. The four major islands surround stable fixed points and unstable (hyperbolic) fixed points occur in-between. At higher energy, one observes a marked onset of chaos and an ergodic domain. The chaotic component of the dynamics increases with ρ and maximizes at the spinodal point ρ * = 0.5. The chaotic orbits densely fill two-dimensional regions of the surface of section.
The dynamics changes profoundly in region II of phase coexistence (ρ * < ρ ≤ ρ c and ξ c ≤ ξ < ξ * * ). The Poincaré sections before at and after the critical point, (ρ = 0.6, ξ c = 0, ξ = 0.1) are shown in Fig. 3. In general, the motion is predominantly regular at low energies and gradually turning chaotic as the energy increases. However, the classical dynamics evolves differently in the vicinity of the two wells. As the local deformed minimum develops, robustly regular dynamics attached to it appears. The trajectories form a single island and remain regular even at energies far exceeding the barrier height V bar . This behavior is in marked contrast to the HH-type of dynamics in the vicinity of the spherical minimum, where a change with energy from regularity to chaos is observed, 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 , the chaotic trajectories from the spherical region can penetrate into the deformed region and a layer of chaos develops, and gradually dominates the surviving regular island for E V bar . As ξ increases, the spherical minimum becomes shallower, and the HH-like dynamics diminishes. Fig. 4 displays the classical dynamics in region III (ξ * * ≤ ξ ≤ 1). The local spherical minimum and the associated HH-like dynamics disappear at the anti-spinodal point ξ * * = 1/3. The regular motion, associated with the single deformed minimum, prevails for ξ ≥ ξ * * . Here a single stable fixed point, surrounded by a family of elliptic orbits, continues to dominate the Poincaré section. In certain regions of the control parameter ξ and energy, the section landscape changes from a single to several regular islands, reflecting the sensitivity of the dynamics to local degeneracies of normal modes. A notable exception to such variation is the SU(3) DS limit (ξ = 1), for which the system is integrable and the phase space portrait is the same for any energy.

Quantum analysis
Quantum manifestations of classical chaos are often detected by statistical analyses of energy spectra [25]. In a quantum system with mixed regular and irregular states, the statistical properties of the spectrum are usually intermediate between the Poisson and the Gaussian orthogonal ensemble (GOE) statistics. Such global measures of quantum chaos are, however, insufficient to reflect the rich dynamics of an inhomogeneous phase space structure encountered in Figs. 2-4, with mixed but well-separated regular and chaotic domains. To do so, one needs to distinguish between regular and irregular subsets of eigenstates in the same energy intervals. For that purpose, we employ the spectral lattice method of Peres [27], which provides additional properties of individual energy eigenstates. The Peres lattices 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 can be shown to display an ordered pattern, while chaotic dynamics leads to disordered meshes of points [27,28].
In the present analysis we choose the Peres operator to ben d . The lattices correspond to the set of points {x i , E i }, with x i ≡ 2 i|n d |i /N and |i being the eigenstates of the quantum Hamiltonian (3). The expectation value ofn d in the condensate of Eq. (2) is related to the deformation β and the coordinate x in the classical potential (9). Accordingly, the particular choice of lattices {x i , E i } can distinguish regular from irregular states and associate them with a given region in phase space, through the classical-quantum correspondence β = x ↔ x i . The Peres lattices for L = 0 eigenstates of the intrinsic Hamiltonian (3) with N = 80, are shown on the bottom rows of Figs. 2-4, overlayed on the classical potentials V (x, y = 0) of Eq (9). For ρ = 0, the Hamiltonian (4) has U(5) dynamical symmetry with a solvable spectrum, a function of n d . For large N and replacing x i by β, the Peres lattice coincides coincides with V 1 (ρ = 0), a trend seen exactly for ρ = 0 and approximately at ρ = 0.03 in Fig. 2. For ρ = 0.2, at low energy a few lattice points still follow the potential curve V 1 (ρ), but at higher energies one observes sizeable deviations and disordered meshes of lattice points, in accord with the onset of chaos in the classical Hénon-Heiles system. The disorder in the Peres lattice enhances at the spinodal point ρ * = 0.5, where the chaotic component of the classical dynamics maximizes. As seen in Figs. 3-4, whenever a deformed minimum occurs in the potential, the Peres lattices exhibit regular sequences of states localized within and above the deformed well. They form several chains of lattice points close in energy, with the lowest chain originating at the deformed ground state. A close inspection reveals that the x i -values of these regular states, lie in the intervals of x-values occupied by the regular tori in the Poincaré sections. Similarly to the classical tori, these regular sequences persist to energies well above the barrier V bar . The lowest sequence consists of L = 0 bandhead states of the ground g(K = 0) and β n (K = 0) bands. Regular sequences at higher energy correspond to β n γ 2 (K = 0), β n γ 4 (K = 0) bands, etc. In contrast, the remaining states, including those residing in the spherical minimum, do not show any obvious patterns and lead to disordered (chaotic) meshes of points at high energy. For ξ > ξ * * , a larger number and longer sequences of regular K = 0 bandhead states are observed in the vicinity of the single deformed minimum (x ≈ 1), as its depth increases.
Peres lattices can also be used to visualize the dynamics of quantum states with nonzero angular momenta. Examples for N = 50, L = 0, 2, 3, 4, eigenstates of the critical-point Hamiltonian,Ĥ 1 (ρ c ) =Ĥ 2 (ξ c ), are shown in Fig. 5. The right column in the figure combines the separate-L lattices and overlays them on the relevant classical potential. Rotational states with L = 0, 2, 4, . . ., comprise the regular K = 0 bands mentioned above, and are accompanied by sequences L = 2, 3, 4, . . ., forming K = 2 bands. Additional K-bands (not shown in Fig. 5), corresponding to multiple β and γ vibrations about the deformed shape, can also be identified. The states in each regular band share a common intrinsic structure, as indicated by their nearly equal values of n d and a similar coherent decomposition of their wave functions in the SU(3) basis, to be discussed in Section 7. These ordered band structures show up in the vicinity of the deformed well and are not present in the disordered (chaotic) portions of the Peres lattice. Their occurrence and persistence in the spectrum throughout the coexistence region, including the critical-point, is somewhat unexpected, in view of the strong mixing and abrupt structural changes taking place.

Symmetry aspects
Away from the U(5) and SU(3) limits (ρ > 0 and ξ < 1), both dynamical symmetries are broken in the intrinsic Hamiltonian (3). The competition between terms with different symmetry character, drives the system through a first-order QPT with a characteristic pattern of mixed The purity of eigenstates with respect to a DS basis can be evaluated by means of the U(5) and SU(3) Shannon entropies defined as The states shown on the right column of Fig. 6 have a different character. They belong to the five lowest regular sequences seen in the combined Peres lattices of Fig. 5. 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) (12b) for the rotational states in each band. The ground g(K = 0) and the 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. (7) with SU(3) PDS. The non-solvable Kbands are mixed with respect to SU(3), but the mixing is similar for the different L-states in the same band. Such strong but coherent (L-independent) mixing is the hallmark of SU (3) quasi-dynamical symmetry (QDS). It results from the existence of a single intrinsic state for each such band and imprints an adiabatic motion and increased regularity [29].
The coherent decomposition characterizing SU(3) QDS, implies strong correlations between the SU(3) components of different L-states in the same band. This can be used as a criteria for the identification of rotational bands. We focus here on the L = 0, 2, 4, 6, members of K = 0 bands. Given a L = 0 + i state, among the ensemble of possible states, we associate with it those L j > 0 states which show the maximum correlation, max j {π(0 i , L j )}. Here π(0 i , L j ) is a Pearson coefficient whose values lie in the range [−1, 1]. Specifically, π(0 i , L j ) = 1, −1, 0, indicate a perfect correlation, a perfect anti-correlation, and no linear correlation, respectively, among the SU(3) components of the 0 i and L j states. To quantify the amount of coherence (hence of SU(3)-QDS) in the chosen set of states, we employ the following product of the maximum correlation coefficients [30] We consider the set of states {0 i , 2 j , 4 k , 6 } as comprising a K = 0 band with SU(3)-QDS, if C SU3 (0 i −6) ≈ 1. As expected, we find the values C SU3 (0 i −6) ≈ 1 for all the 'deformed' K-bands, shown in the right column of Fig. 6. It should be noted that the coherence property of a band of states, as measured by C SU3 (0 i −6), is independent of its purity, as measured by S SU3 (L i ). Thus, in Fig. 6, the pure g(K = 0) and γ(K = 2) bands with SU (3)   In this case, the L-states in a given K-band belong to a single SU(3) irrep, hence necessarily C SU3 (0−6) = 1. For ξ < 1, S SU3 (L = 0) > 0 acquires positive values, reflecting an SU(3) mixing. The SU(3) breaking becomes stronger at higher energies and as ξ approaches ξ c = 0 from above, resulting in higher values of S SU3 (L = 0). A notable exception to this behavior is the deformed ground state (L = 0 1 ) ofĤ 2 (ξ), which maintains S SU3 (L = 0 1 ) = 0 throughout region III (ξ * * ≤ ξ ≤ 1) and in part of region II (ξ c ≤ ξ < ξ * * ), in accord with its SU(3)-PDS property, Eq. (7a). In contrast to the lack of SU(3)-purity in all excited L = 0 states, the SU(3) correlation function maintains a value close to unity, C SU3 (0−6) ≈ 1. This indicates that the SU(3) mixing is coherent and that these L = 0 states serve as bandhead states of K = 0 bands with a pronounced SU(3) QDS. This band-structure is observed throughout region III in extended energy domains, consistent with the classical analysis, which revealed a robustly regular dynamics in this region.
In region I (0 ≤ ρ ≤ ρ * ), all states show high values of S SU3 (L = 0) ≈ 1 and C SU3 (0−6) < 1, indicating considerable SU(3) mixing and lack of SU(3) coherence. This is in line with the presence of spherical-states, at low energy, and of more complex-type of states at higher energy, and the absence of rotational bands in this region. In region II of phase coexistence (ρ * < ρ ≤ ρ c and ξ c ≤ ξ < ξ * * ), one encounters both points with C SU3 (0−6) ≈ 1, and points with C SU3 (0−6) < 1. This reflects the presence of deformed states arranged into regular bands, exemplifying SU(3) QDS, and at the same time, the presence of spherical states and other states of a different nature. These results highlight the relevance of U(5)-PDS (partial purity) and SU(3)-QDS (coherence) for clarifying the survival of regular subsets of states in the presence of more complicated type of states, a situation encountered in QPTs of non-integrable systems.