Degeneracy patterns in quantum mechanics stem from the system symmetries. In particular, the broken-symmetry phase in the well-known Lipkin-Meshkov-Glick (LMG) model is composed of doubly-degenerate states of different parity. In this work, we show that such doublets can exist even if parity is not conserved. For this purpose, our starting point is an anharmonic LMG Hamiltonian with a second-order ground-state quantum phase transition (GSQPT) and a rich spectrum, with two different excited-state quantum phase transitions. The inclusion in the Hamiltonian of a term inducing a first-order GSQPT breaks the parity symmetry but conserving the exponential degeneracy in the energy doublets. We demonstrate that this phenomenon can be traced back to the existence of a $\mathbb{Z}_2$ symmetry (reflection symmetry) in the system's classical limit phase space that leads to an anti-unitary $\mathbb{Z}_2$ symmetry in the quantum system.
Degeneracy patterns in quantum mechanics stem from the system symmetries. In particular, the broken-symmetry phase in the well-known Lipkin-Meshkov-Glick (LMG) model is composed of doubly-degenerate states of different parity. In this work, we show that such doublets can exist even if parity is not conserved. For this purpose, our starting point is an anharmonic LMG Hamiltonian with a second-order ground-state quantum phase transition (GSQPT) and a rich spectrum, with two different excited-state quantum phase transitions. The inclusion in the Hamiltonian of a term inducing a first-order GSQPT breaks the parity symmetry but conserving the exponential degeneracy in the energy doublets. We demonstrate that this phenomenon can be traced back to the existence of a Z2 symmetry (reflection symmetry) in the system's classical limit phase space that leads to an anti-unitary Z2 symmetry in the quantum system.
The Lipkin-Meshkov-Glick (LMG) model originally describes a Fermionic many-body system in nuclear physics. However, in recent years, it has been widely found in condensed matter physics, quantum information systems, and quantum optics, and it is of wider and wider interest. Previous studies on this model mainly focused on the physics under the thermal dynamical limit, such as quantum phase transitions and quantum entanglement. There are also some researches about LMG model with finite size in some special limits, but the finite-size effect on energy spectrum is not very clear yet. This is the main motivation of this work. In this paper, the exact diagonalization method and the quantum perturbation theory are used to calculate and analyze the energy-level structure of the LMG model at a finite N. To solve it, we first map this model into the angular-momentum space to obtain a reduced LMG model. By this mapping, the dimension of Hilbert space is reduced to N+1 from 2N. The exact solution of its energy levels can be obtained easily in the U(1) limit where the total spin is conserved. We find that the levels are woven into a fishing-net structure in the U(1) limit. While away from the U(1) limit, the crossings between even and odd levels will open a gap, and the system's energy levels will be grouped into pairs with an odd and an even level, forming some bound states, called doublet states, and the parity of each doublet state will oscillate as the Zeeman field increases. This work gives the values of the critical Zeeman field for the parity crossings. These critical values shift as the interacting parameters and disappear at zero in the Z2 limit. In the Z2 limit, the system energy levels form splittings near the zero Zeeman field. In this article, we analytically calculate the relationship between these energy gaps and the Zeeman field. For odd and even number N, the parity of each state has a different behavior. Specifically, the ground state and the doublet excited states of the system with odd N will suffer a parity reversion at zero Zeeman field, while the states with even N will not. By tuning the interacting parameters, we also study the crossover from the U(1) limit to the Z2 limit. The parity oscillation we find in this system is a very important physical phenomenon, which also exists in some other systems like optical cavity quantum electrodynamics and magnetic molecule system.
The standard Lipkin-Meshkov-Glick (LMG) model undergoes a second-order ground-state quantum phase transition (QPT) and an excited-state quantum phase transition (ESQPT). The inclusion of an anharmonic term in the LMG Hamiltonian gives rise to a second ESQPT that alters the static properties of the model [Gamito et al., Phys. Rev. E 106, 044125 (2022)2470-004510.1103/PhysRevE.106.044125]. In the present work, the dynamical implications associated to this new ESQPT are analyzed. For that purpose, a quantum quench protocol is defined on the system Hamiltonian that takes an initial state, usually the ground state, into a complex excited state that evolves on time. The impact of the new ESQPT on the time evolution of the survival probability and the local density of states after the quantum quench, as well as on the Loschmidt echoes and the microcanonical out-of-time-order correlator (OTOC) are discussed. The anharmonity-induced ESQPT, despite having a different physical origin, has dynamical consequences similar to those observed in the ESQPT already present in the standard LMG model.
One-dimensional quantum systems that undergo spontaneous symmetry-breaking, having a symmetric (non-degenerate) and a broken-symmetry (doubly-degenerate) phase, have been intensely studied in different branches of physics. In most cases, the spontaneously-broken symmetry is parity. However, it is possible to obtain similar phases in systems without parity symmetry, through an antiunitary symmetry that implies a two-fold symmetry either on momentum or coordinate in the system's classical limit. To illustrate this phenomenon, we use a Kerr parametric oscillator (KPO) with one- and two-photon drives that, despite the breaking of parity symmetry, may have doubly-degenerate levels. Different realizations of squeezed KPOs convey a great deal of attention, as effective Hamiltonians for driven superconducting circuits and the occurrence of degeneracy in such systems could be of practical interest in their application to obtain protected qubits in parity-breaking setups. In addition to this, the reported spectral features strongly indicate the existence of additional symmetries in the system.
The basic Lipkin-Meshkov-Glick model displays a second-order ground-state quantum phase transition and an excited-state quantum phase transition (ESQPT). The inclusion of an anharmonic term in the Hamiltonian implies a second ESQPT of a different nature. We characterize this ESQPT using the mean field limit of the model. The alternative ESQPT, associated with the changes in the boundary of the finite Hilbert space of the system, can be properly described using the order parameter of the ground-state quantum phase transition, the energy gap between adjacent states, the participation ratio, and the quantum fidelity susceptibility.
We study the dynamics of the Lipkin-Meshkov-Glick (LMG) model with a finite number of spins. In the thermodynamic limit, the ground state of the LMG model with an isotropic Hamiltonian in the broken phase breaks to a mean-field ground state with a certain direction. However, when the spin number N is finite, the exact ground state is always unique and is not given by a classical mean-field ground state. Here, we prove that when N is large but finite, through a tiny external perturbation, a localized state which is close to a mean-field ground state can be prepared, which mimics spontaneous symmetry breaking. Also, we find the localized in-plane spin polarization oscillates with two different frequencies $\ensuremath{\sim}O(1/N)$, and the lifetime of the localized state is long enough to exhibit this oscillation. We numerically test the analytical results and find that they agree very well with each other. Finally, we link the phenomena to quantum time crystals and time quasicrystals.
We introduce the notion of mixed symmetry quantum phase transition (MSQPT) as singularities in the transformation of the lowest-energy state properties of a system of identical particles inside each permutation symmetry sector μ, when some Hamiltonian control parameters λ are varied. We use a three-level Lipkin-Meshkov-Glick model, with U(3) dynamical symmetry, to exemplify our construction. After reviewing the construction of U(3) unitary irreducible representations using Young tableaux and the Gelfand basis, we first study the case of a finite number N of three-level atoms, showing that some precursors (fidelity susceptibility, level population, etc.) of MSQPTs appear in all permutation symmetry sectors. Using coherent (quasiclassical) states of U(3) as variational states, we compute the lowest-energy density for each sector μ in the thermodynamic N→∞ limit. Extending the control parameter space by μ, the phase diagram exhibits four distinct quantum phases in the λ-μ plane that coexist at a quadruple point. The ground state of the whole system belongs to the fully symmetric sector μ=1 and shows a fourfold degeneracy, due to the spontaneous breakdown of the parity symmetry of the Hamiltonian. The restoration of this discrete symmetry leads to the formation of four-component Schrödinger cat states.