A Class of Simple Hamiltonians with Degenerate Ground State. II A Model of Nuclear Rotation: Spontaneous Breakdown of Rotation Symmetry and Goldstone Theorem for Finite Dimensional System
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.
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 discuss the rotational excitations of highly quantum many-body systems (for example, polyatomic molecules or microdroplets of helium). For a general system, the states Fί, where and ί is a rotationally invariant ground or vibrational state, are shown to be eigenfunctions of L 2 and Lz , with eigenvalues L(L+1)ħ 2 and Lħ (for even L). These wavefunctions preserve the translational invariance and the permutation and inversion symmetries of the N-particle state ί. For harmonic pair interactions, the f = 1 wavefunctions are shown to be exact eigenstates of the N-body hamiltonian. For large N, the states Fί(f=1) represent surface oscillations of the type first proposed by Bohr. An inequality for the rotational excitation energy is obtained variationally; it depends on two, three, and four-particle correlations. Other translationally invariant angular momentum eigenfunctions are also discussed.
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.
Abstract It is generally accepted that phonons in a superfluid Bose gas are Goldstone bosons. This is justified by spontaneous symmetry breaking (SSB), which is usually defined as follows: the Hamiltonian of the system is invariant under the U ( 1 ) transformation Ψ ^ ( r , t ) → e i α Ψ ^ ( r , t ) , whereas the order parameter Ψ ( r , t ) is not. However, the strict definition of SSB is different: the Hamiltonian and the boundary conditions are invariant under a symmetry transformation, while the ground state is not. Based on the latter criterion, we study a finite system of spinless, weakly interacting bosons using three approaches: the standard Bogoliubov method, the particle-number-conserving Bogoliubov method, and the approach based on the exact ground-state wave function. Our results show that the answer to the question in the title is ‘no’. Thus, phonons in a real-world (finite) superfluid Bose gas are similar to sound in a classical gas: they are not Goldstone bosons, but quantized collective vibrational modes arising from the interaction between atoms. In the case of an infinite Bose gas, however, the picture becomes paradoxical: the ground state can be regarded as either infinitely degenerate or non-degenerate, making the phonon both similar to a Goldstone boson and different from it.
In accord with Nambu-Goldstone theorem spontaneous breaking of a global continuous symmetry of the system by a mean-field parameter causes appearance of massless Goldstone bosons, called also gapless Goldstone modes. It is demonstrated below that the theorem can be violated, i.e. the Goldstone modes possess finite gap, in the case of a novel (quantum ordered) state of interacting fermions, that was proposed recently {mukhin 2011}. This result may lead to the important consequences when considered in combination with another peculiar properties of the quantum parameter (QOP) that where recently found: QOP's Green's function is finite and periodic along the Matsubara's axis, but Wick-rotated to the axis of real frequencies, possesses infinite periodic chain of \it{second poles} along this axis {mukhin 2011}. This property of QOP leads to a remarkable physical consequence: \it{QOP does not scatter anything in Minkowski world} ! Hence, the present theory, we believe, may naturally indicate that two effects: hidden order revealed in ARPES spectra of fermi-excitations and neutron resonance feature at finite frequency in the magnetic (bosonic) excitations spectrum in lightly hole-doped copper oxides, could be emerging fingerprints of the Euclidian crystal state. A peculiar manifestation of the QOP self-consistent solution in the form of non-classical phonon states is described for an Euclidian predicted recently {mukhin,fistul 2013} in the array of Josephson junctions coupled via electromagnetic field in the superconducting resonator. It is demonstrated that it may constitute a stable state of the system, that, for a particular quantum disordering model, exists inside the stability domain of the classical photonic condensate delimited by the critical value of the quantum disordering strength.
The subject of condensed matter physics is very rich — there are an infinite number of parameters producing a diversity of exciting phenomena. As a theorist, my goal is to distill general principles out of this complexity — to construct theories that can coherently explain many known examples altogether.This thesis is composed of several attempts to develop such theories in topics related to spontaneously symmetry breaking. A remarkable feature of many-body interacting systems is that although they are described by equations respecting various symmetries, they may spontaneously organize into a state that explicitly breaks symmetries. Examples are numerous: various types of crystalline and magnetic orders, Bose-Einstein condensates of cold atoms, superfluids of liquid helium, chiral symmetry in QCD, neutron stars, and cosmic inflation.These systems with spontaneously broken continuous symmetries have gapless excitations, so called Nambu-Goldstone bosons (NGBs). Although the properties of NGBs are well understood in Lorentz-invariant systems, surprisingly, some basic properties of NGBs such as their number and dispersion in nonrelativistic systems have not been discussed from a general perspective. In the first part of this thesis, we solve this issue by developing and analyzing an effective Lagrangian that coherently captures the low-energy, long-distance physics of many different symmetry-breaking states all at once.Next, we examine whether these NGBs originating from spontaneous symmetry breaking remain to be well-defined excitations inside a metal, where low-energy electrons near Fermi surface can collide with them. Our result is a one equation criterion that specifies whether the interactions between electrons and NGBs can be ignored, or whether it completely changes their character. In the latter case, unusual phases of matter such as non-Fermi liquids may arise; in that case, NGBs are overdamped and cannot form particle-like excitations in spite of the assumed symmetry breaking.In the last part of this thesis, we investigate the possibility of spontaneously breaking of Hamiltonian itself. The homogeneity of time is one of the most fundamental symmetries of nature, underlying the conservation of the energy. The question is whether this symmetry can be spontaneously broken, as suggested recently by Wilczek, in analogy with ordinary crystals. Contrary to his proposal that attracted a significant attention and stimulated many further studies, we prove a no-go theorem that rules out spontaneously breaking of time translation, in the ground state or in the canonical ensemble of a general Hamiltonian.
The subject of condensed matter physics is very rich — there are an infinite number of parameters producing a diversity of exciting phenomena. As a theorist, my goal is to distill general principles out of this complexity — to construct theories that can coherently explain many known examples altogether.This thesis is composed of several attempts to develop such theories in topics related to spontaneously symmetry breaking. A remarkable feature of many-body interacting systems is that although they are described by equations respecting various symmetries, they may spontaneously organize into a state that explicitly breaks symmetries. Examples are numerous: various types of crystalline and magnetic orders, Bose-Einstein condensates of cold atoms, superfluids of liquid helium, chiral symmetry in QCD, neutron stars, and cosmic inflation.These systems with spontaneously broken continuous symmetries have gapless excitations, so called Nambu-Goldstone bosons (NGBs). Although the properties of NGBs are well understood in Lorentz-invariant systems, surprisingly, some basic properties of NGBs such as their number and dispersion in nonrelativistic systems have not been discussed from a general perspective. In the first part of this thesis, we solve this issue by developing and analyzing an effective Lagrangian that coherently captures the low-energy, long-distance physics of many different symmetry-breaking states all at once.Next, we examine whether these NGBs originating from spontaneous symmetry breaking remain to be well-defined excitations inside a metal, where low-energy electrons near Fermi surface can collide with them. Our result is a one equation criterion that specifies whether the interactions between electrons and NGBs can be ignored, or whether it completely changes their character. In the latter case, unusual phases of matter such as non-Fermi liquids may arise; in that case, NGBs are overdamped and cannot form particle-like excitations in spite of the assumed symmetry breaking.In the last part of this thesis, we investigate the possibility of spontaneously breaking of Hamiltonian itself. The homogeneity of time is one of the most fundamental symmetries of nature, underlying the conservation of the energy. The question is whether this symmetry can be spontaneously broken, as suggested recently by Wilczek, in analogy with ordinary crystals. Contrary to his proposal that attracted a significant attention and stimulated many further studies, we prove a no-go theorem that rules out spontaneously breaking of time translation, in the ground state or in the canonical ensemble of a general Hamiltonian.