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arXiv:2209.12497 · Quantum

Spontaneous symmetry emergence in a Hermitian system of coupled oscillators without symmetry

T. T. Sergeev, Evgenii S. Andrianov, A. A. Zyablovsky

Published May 15, 2025• 1 citation
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Abstract

Spontaneous symmetry breaking in systems with symmetry is a cornerstone phenomenon accompanying second-order phase transitions. Here, we predict the opposite phenomenon, namely, spontaneous symmetry emergence in a system that lacks symmetry. In the example of two coupled oscillators interacting non-symmetrically with a set of oscillators whose frequencies uniformly fill a finite frequency range, we demonstrate that the system state can acquire symmetry that is not inherent in the system Hamiltonian. The emergence of symmetry is manifested as a change in the system dynamics, which can be interpreted as a phase transition in a Hermitian system that lacks symmetry.

DOI
10.22331/q-2025-05-15-1746

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Spontaneous symmetry breaking in systems with symmetry is a cornerstone phenomenon accompanying second-order phase transitions. Here, we predict the opposite phenomenon, namely, spontaneous symmetry emergence in a system that lacks symmetry. In the example of two coupled oscillators interacting non-symmetrically with a set of oscillators whose frequencies uniformly fill a finite frequency range, we demonstrate that the system state can acquire symmetry that is not inherent in the system Hamiltonian. The emergence of symmetry is manifested as a change in the system dynamics, which can be interpreted as a phase transition in a Hermitian system that lacks symmetry.
Landau's spontaneous symmetry breaking theory is a fundamental theory that describes the collective behaviors in many-body systems. It was well known that for usual spontaneous symmetry breaking in Hermitian systems, the order-disorder phase transition with gap closing and spontaneous symmetry breaking occur at the same critical point. In this paper, we generalized the Landau's spontaneous symmetry breaking theory to the cases in non-Hermitian (NH) many-body systems with biorthogonal Z2 symmetry and tried to discover certain universal features. We were surprised to find that the effect of the NH terms splits the spontaneous biorthogonal Z2 symmetry breaking from a (biorthogonal) order-disorder phase transition with gap closing. The sudden change of similarity for two degenerate ground states indicates a new type of quantum phase transition without gap closing accompanied by spontaneous biorthogonal Z2 symmetry breaking. We will take the NH transverse Ising model as an example to investigate the anomalous spontaneous symmetry breaking. The numerical results were consistent with the theoretical predictions.
Spontaneous symmetry breaking generally circumvents one-dimensional systems with local interactions in thermal equilibrium. Here, we analyze a category of one-dimensional Hermitian models via local non-Hermitian constructions. Notably, spontaneous symmetry breaking and long-range order may emerge at finite temperatures in such systems under periodic boundary conditions, in sharp contrast to Hermitian constructions. We demonstrate clear numerical evidence, such as order parameters and specific heat, supporting phase diagrams with robust ordered phases. Non-Hermitian physics plays a vital role in prohibiting domain-wall proliferation and promoting spontaneous symmetry breaking. The fermions exhibit an exotic topological nature in their path-integral windings, which uphold nonzero integers-commonly a non-Hermitian signature-in the ordered phases, thus offering a novel and spontaneous origin for both symmetry breaking and non-Hermiticity.
By adding an imaginary interacting term proportional to ip1p2i{p}_{1}{p}_{2}ip1​p2​ to the Hamiltonian of a free anisotropic planar oscillator, we construct a model which is described by the PTPTPT-pseudo-Hermitian Hamiltonian with the permutation symmetry of two dimensions. We prove that our model is equivalent to the Pais-Uhlenbeck oscillator and thus establish a relationship between our PTPTPT-pseudo-Hermitian system and the fourth-order derivative oscillator model. We also point out the spontaneous breaking of permutation symmetry which plays a crucial role in giving a real spectrum free of interchange of positive and negative energy levels in our model. Moreover, we find that the permutation symmetry of two dimensions in our Hamiltonian corresponds to the identity (not in magnitude but in attribute) of two different frequencies in the Pais-Uhlenbeck oscillator, and reveal that the unequal-frequency condition imposed as a prerequisite upon the Pais-Uhlenbeck oscillator can reasonably be explained as the spontaneous breaking of this identity.
Abstract We study the dynamical evolution and symmetry breaking of the non-Hermitian double Jaynes–Cummings model with symmetry. The results indicate that there exist symmetry maintained and breaking phase in this model, and the non-Hermitian terms lead to different behaviors in two phases. In the symmetry phase, there exists stable Rabi oscillation caused by the interaction between photons and atoms. At the exceptional point, the system shows critical property that the probabilities in each state over time tend to be equal. The system will undergo a transition from the symmetry phase to the symmetry breaking phase, as increasing the coupling constant. In the symmetry breaking phase, complex eigenvalues appear, and the probability in each state increases monotonically in an exponential form. This study can help us to understand the effect of symmetry on the interaction between photons and atoms in non-Hermitian systems.
A cornerstone of the theory of phase transitions is the observation that many-body systems exhibiting a spontaneous symmetry breaking in the thermodynamic limit generally show extensive fluctuations of an order parameter in large but finite systems. In this work, we introduce the dynamical analog of such a theory. Specifically, we consider local dissipative dynamics preparing an equilibrium steady-state of quantum spins on a lattice exhibiting a discrete or continuous symmetry but with extensive fluctuations in a local order parameter. We show that for all such processes, there exist asymptotically stationary symmetry-breaking states, i.e., states that become stationary in the thermodynamic limit and give a finite value to the order parameter. We give results both for discrete and continuous symmetries and explicitly show how to construct the symmetry-breaking states. Our results show in a simple way that, in large systems, local dissipative dynamics satisfying detailed balance cannot uniquely and efficiently prepare states with extensive fluctuations with respect to local operators. We discuss the implications of our results for quantum simulators and dissipative state preparation.
Spontaneous symmetry breaking (SSB) is crucial to the occurrence of phase transitions. Once a phase transition occurs, a quantum system presents degenerate eigenstates that lack the symmetry of the Hamiltonian. After crossing the critical point, the system is essentially evolved to a quantum superposition of these eigenstates until decoherence sets in. Despite the fundamental importance and potential applications in quantum technologies, such quantum-mechanical SSB phenomena have not been experimentally explored in many-body systems. We here present an experimental demonstration of the SSB process in the Lipkin-Meshkov-Glick model, governed by the competition between the individual driving and intraqubit interaction. The model is realized in a circuit quantum electrodynamics system, where 6 Xmon qubits are coupled in an all-to-all manner through virtual photon exchange mediated by a resonator. The observed nonclassical correlations among these qubits in the symmetry-breaking region go beyond the conventional description of SSB, shedding new light on phase transitions for quantum many-body systems.