Simultaneous symmetry breaking in spontaneous Floquet states: temporal Floquet-Nambu-Goldstone modes, Floquet thermodynamics, and the time operator

Juan Ramón Muñoz de Nova; Fernando Sols

doi:10.22331/q-2025-09-05-1850

Simultaneous symmetry breaking in spontaneous Floquet states: temporal Floquet-Nambu-Goldstone modes, Floquet thermodynamics, and the time operator

Juan Ramón Muñoz de Nova and Fernando Sols

Departamento de Física de Materiales, Universidad Complutense de Madrid, E-28040 Madrid, Spain

Published:	2025-09-05, volume 9, page 1850
Editor:	Ivan Contreras
Eprint:	arXiv:2402.10784v4
Doi:	https://doi.org/10.22331/q-2025-09-05-1850
Citation:	Quantum 9, 1850 (2025).

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Abstract

We study simultaneous symmetry breaking in spontaneous Floquet states, focusing on the specific case of an atomic condensate. We first describe the quantization of the Nambu-Goldstone (NG) modes for a stationary state simultaneously breaking several symmetries of the Hamiltonian by invoking the generalized Gibbs ensemble, which enables a thermodynamical description of the problem. The quantization procedure involves a Berry-Gibbs connection, which depends on the macroscopic conserved charges associated to each broken symmetry and whose curvature is not invariant under generalized gauge transformations. We extend the formalism to Floquet states, where Goldstone theorem translates into the emergence of Floquet-Nambu-Goldstone (FNG) modes with zero quasi-energy. In the case of a spontaneous Floquet state, there is a genuine temporal FNG mode arising from the continuous time-translation symmetry breaking, whose quantum amplitude provides a rare realization of a time operator in Quantum Mechanics. Furthermore, since they conserve energy, spontaneous Floquet states can be shown to possess a conserved Floquet charge. Conventional Floquet systems also admit a thermodynamic description in terms of the Floquet enthalpy, the Legendre transform of the energy with respect to the Floquet charge, as these operate at fixed frequency. We apply our formalism to a particular realization of spontaneous Floquet state, the CES state, which breaks $U(1)$ and time-translation symmetries, representing a time supersolid. We numerically compute its density-density correlations, predicted to be dominated by the temporal FNG mode at long times, observing a remarkable agreement between simulation and theory. Based on these results, we propose a feasible experimental scheme to observe the temporal FNG mode of the CES state.

Featured image: Upper: An attractive delta potential $V(x)=-Z\delta(x)$ is suddenly introduced at $t=0$ within a homogeneous condensate flowing with initial velocity $v$. Experimentally, this can be mimicked by sweeping an attractive localized potential with velocity $v$ against the homogeneous bulk of a condensate confined by a boxlike potential. Lower: The dynamical phase diagram for the final state of the system as a function of $(Z,v)$ only displays two states: the true stationary ground state (GS) or the CES state. In the latter case, the density $|\Psi(x,t)|^2$ becomes asymptotically periodic, thus self-consistently giving rise to a spontaneous Floquet state. The corresponding density-density correlation function $G(x,x',t)$ is dominated at late times by the ballistic growth of the quantum amplitude of the temporal FNG mode, displaying a quasi-periodic time dependence. A remarkable agreement is found between theory and simulation.

Popular summary

Spontaneous Floquet states are states of a time-independent Hamiltonian that oscillate like a Floquet state due to interactions. Since they spontaneously break time-translation symmetry, it is of central interest to study spontaneous symmetry breaking and the associated spectrum of Nambu-Goldstone (NG) modes in such states, especially when other continuous symmetries are also spontaneously broken. In this work, we study simultaneous symmetry breaking in spontaneous Floquet states, focusing on atomic condensates for illustrative purposes.

We first address the general case of a stationary state spontaneously breaking several symmetries of the underlying Hamiltonian. We find that the generalized Gibbs ensemble provides a most elegant solution to the problem due to its equitable treatment of each broken symmetry, enabling also a thermodynamic description. The quantization of the resulting NG modes involves a Berry connection whose variables are the macroscopic conserved charges associated to each broken symmetry, thus denoted as the Berry-Gibbs connection. Remarkably, the corresponding Berry-Gibbs curvature is not invariant under generalized gauge transformations, which extend the usual phase transformations to more general symmetries, such as spatial or time translations.

We extend the developed formalism to Floquet states, where Goldstone theorem translates into the emergence of Floquet-Nambu-Goldstone (FNG) modes with zero quasi-energy. We show that Floquet states can be characterized through a thermodynamic description completely analogous to that of stationary states. In the case of spontaneous Floquet states, this description is possible because they conserve energy as the underlying Hamiltonian is time-independent, giving rise to a novel conserved charge, denoted as the Floquet charge. In turn, the thermodynamic conjugate variable of the Floquet charge is the oscillation frequency. As a result, conventional Floquet systems also admit a thermodynamic description in terms of the Floquet enthalpy, the Legendre transform of the energy with respect to the Floquet charge, as these operate at a frequency fixed by the environment, in analogy to isothermal or isobaric systems.

In the case of a spontaneous Floquet state, there is a genuine temporal FNG mode arising from the continuous time-translation symmetry breaking, whose quantum amplitude behaves as an effective time operator, accounting for the quantum fluctuations of the time origin of the spontaneous Floquet state, which is not fixed here due to the absence of external driving.

We apply our formalism to a particular realization of spontaneous Floquet state in an atomic condensate, the so-called CES state, which breaks $U(1)$ and time-translation symmetries, representing a time supersolid. We propose a feasible experimental scheme to observe its temporal FNG mode through the measurement of density-density correlations, numerically validated by our simulations. As a result, the CES state provides a unique and tangible condensed-matter setup where to study a realization of time operator.

Due to its fundamental character and broad interdisciplinary scope, containing results of interest for vastly different fields ranging from quantum solitons or supersolids to Floquet systems or time crystals, the article is written within a pedagogical approach, and the developed formalism is explained in great detail.

► BibTeX data

@article{MunozdeNova2025simultaneous,
  doi = {10.22331/q-2025-09-05-1850},
  url = {https://doi.org/10.22331/q-2025-09-05-1850},
  title = {Simultaneous symmetry breaking in spontaneous {F}loquet states: temporal {F}loquet-{N}ambu-{G}oldstone modes, {F}loquet thermodynamics, and the time operator},
  author = {Mu{\~{n}}oz de Nova, Juan Ram{\'{o}}n and Sols, Fernando},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1850},
  month = sep,
  year = {2025}
}

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