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arXiv:math-ph/0005001

Classical phase space structure induced by spontaneous symmetry breaking

M. Grigorescu

Published Apr 30, 2000• 0 citations
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Abstract

The collective dynamics of a many-body system is described as a special case of low-energy quantum dynamics, occurring when the ground state breaks a continuous symmetry of the Hamiltonian. This approach is applied to the spontaneous breaking of the rotational symmetry of a nuclear Hamiltonian. It is shown that the excitation operator of the isovector low-lying angular oscillations in deformed nuclei is a linear combination between angular momentum operators, which generate static rotations, and "angle" operators, which generate the transition to a rotating frame.

DOI
10.48550/arxiv.math-ph/0005001

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The collective dynamics of a many-body system is described as a special case of low-energy quantum dynamics, occurring when the ground state breaks a continuous symmetry of the Hamiltonian. This approach is applied to the spontaneous breaking of the rotational symmetry of a nuclear Hamiltonian. It is shown that the excitation operator of the isovector low-lying angular oscillations in deformed nuclei is a linear combination between angular momentum operators, which generate static rotations, and "angle" operators, which generate the transition to a rotating frame.
A brief description of excited and ground states in two-dimensional quantum dots and rotating nuclei is presented within a mean field approach and a random-phase approximation (RPA). We discuss the procedure to restore the rotational symmetry broken at the mean field, which can be extended for other symmetry breaking cases. We propose to consider a disappearance of collective excitations in the rotating frame as a manifestation of symmetry breaking phenomena of the rotating mean field. In particular, we demonstrate that the disappearance of a collective octupole mode in the rotating frame in 162 Yb gives rise to the nonaxial octupole deformation.
As a direct generalization of our previous work, we construct a manifestly rotation invariant Hamiltonian describing a system composed of a tensor boson with spin I = 2 — such as, for example, phonon of nuclear quadrupole surface vibration — in which special cubic and quartic self-interactions are taken into account. All the ground state solutions are obtained. It is shown that these solutions can be classified into three classes: The first is the normal vacuum of boson, which is manifestly rotation symmetry. The second class of solutions can be viewed as generated from a single chosen state by rotating it into an arbitrary direction. The third one is Goldstone modes built on this chosen ground state. This chosen state — an exact ground state solution of our rotation invariant Hamiltonian — is not an eigenstate of angular momentum, but contains infinitely many different angular momentum states in it; thus, typifying “spontaneous breakdown” of “rotation symmetry”. It is further shown that this state can be naturally identified to an “intrinsic” deformed state in nuclear collective model of Bohr and Mottelson. The results are examined from the standpoint of Goldstone theorem and detailed discussions are given to explore how Goldstone theorem is modified in our finite dimensional system.
We investigate spontaneous U(1) symmetry breaking and the associated phase transitions in rotating interacting Bose gases. Using a theoretical framework that combines mean-field analysis with rotational dynamics, we analyze how rigid rotation modifies the condensate structure and critical behavior. The study identifies the emergence of Goldstone modes and clarifies their role in the low-energy excitation spectrum. The results provide insight into the interplay between symmetry, rotation, and many-body interactions, contributing to a deeper theoretical understanding of phase structures in Bose systems.
We investigate spontaneous U(1) symmetry breaking and the associated phase transitions in rotating interacting Bose gases. Using a theoretical framework that combines mean-field analysis with rotational dynamics, we analyze how rigid rotation modifies the condensate structure and critical behavior. The study identifies the emergence of Goldstone modes and clarifies their role in the low-energy excitation spectrum. The results provide insight into the interplay between symmetry, rotation, and many-body interactions, contributing to a deeper theoretical understanding of phase structures in Bose systems.We investigate spontaneous U(1) symmetry breaking and the associated phase transitions in rotating interacting Bose gases. Using a theoretical framework that combines mean-field analysis with rotational dynamics, we analyze how rigid rotation modifies the condensate structure, critical behavior, and low-energy excitation spectrum. We identify the emergence of Goldstone modes (massless rotons and massive phonons) in the symmetry-broken phase and clarify their role in mediating low-energy excitations—findings that remain robust at low momentum regardless of rotation. A key result is the angular velocity (Ω) dependence of the critical temperature (Tc) for U(1) phase transition, where Tc scales as Ω^(1/3), distinct from the Ω^(2/5) (nonrelativistic) and Ω^(1/4) (ultrarelativistic) scaling observed in noninteracting rotating Bose gases. Rotation also alters the temperature dependence of the thermodynamic potential minima, changing the characteristic factor from (1 - t) (t = T/Tc for nonrotating systems) to (1 - t³) for rotating gases. We further demonstrate that rotation preserves the second-order nature of the phase transition, while modifying the critical exponents and reducing the discontinuity in heat capacity with increasing Ω. Additionally, we define a σ meson dissociation temperature (Tdiss) characterized by mσ(Tdiss) = 2mπ(Tdiss), showing that Tdiss is always lower than Tc. Thermal mass corrections are shown to ensure the validity of Goldstone’s theorem in the rotating frame, even in the chiral limit. These results deepen our understanding of the interplay between symmetry, rotation, and many-body interactions, with implications for interpreting extreme conditions in heavy-ion collisions and compact astrophysical objects, while advancing the theoretical framework for phase structures in rotating Bose systems.
We consider a quantum many-body system on a lattice which exhibits a spontaneous symmetry breaking in its infinite volume ground states, but in which the corresponding order operator does not commute with the Hamiltonian. Typical examples are the Heisenberg antiferromagnet with a Néel order, and the Hubbard model with a (superconducting) off-diagonal long range order. In the corresponding finite system, the symmetry breaking is usually “obscured ” by “quantum fluctuation ” and one gets a symmetric ground state with a long range order. In such a situation, Horsch and von der Linden proved that the finite system has a low-lying eigenstate whose excitation energy is not more than of order N −1, where N denotes the number of the sites in the lattice. Here we study the situation where the broken symmetry is a continuous one. For a particular set of states (which are orthogonal to the ground state and with each other), we prove bounds for their energy expectation values. The bounds establish that there exist ever increasing numbers of low-lying eigenstates whose excitation energies are bounded by a constant times N −1. A crucial feature of the particular low-lying
Atomic nuclei are complex, quantum many-body systems whose structure manifests itself through intrinsic quantum states associated with different excitation modes or degrees of freedom.Collective modes (vibration and/or rotation) dominate at low energy (near the ground-state).The associated states are usually employed, within a truncated model space, as a basis in (coherent) coupled channels approaches to low-energy reaction dynamics.However, excluded non-collective states can be essential, and their effects on the open (nuclear) system dynamics are usually treated through complex potentials.Is this a complete description of open system dynamics?Does it include effects of quantum decoherence?Can decoherence be manifested in reaction observables?This paper discusses these issues and the main ideas of a coupled-channels density-matrix approach that makes it possible to quantify the role and importance of quantum decoherence in low-energy nuclear reaction dynamics.Some topical applications are highlighted.