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arXiv:1808.07636

Diffusive Nambu-Goldstone modes in quantum time-crystals

Tomoya Hayata, Yoshimasa Hidaka

Published Aug 23, 2018• 9 citations
arXivHTMLPDF

Abstract

We study the Nambu-Goldstone (NG) modes associated with spontaneous breaking of the continuous time-translation symmetry. To discuss a quantum time-crystal with the spontaneously-broken continuous time-translation symmetry, we introduce the van der Pol type nonlinear-friction to open quantum systems. By considering small fluctuations around a time-periodic mean-field solution, we show that a gapless collective mode necessarily appears; this is nothing but the NG mode associated with a time crystal. We show that its dispersion relation becomes $ω=-iC\bm p^2$. We also show that noncommutative breaking of the time-translation and U(1) symmetries results in mixing of the NG modes, and the (typically) propagating NG mode appears, whose dispersion relation becomes $ω=(\pm C_1-iC_2)\bm p^2$.

DOI
10.48550/arxiv.1808.07636

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We study the Nambu-Goldstone (NG) modes associated with spontaneous breaking of the continuous time-translation symmetry. To discuss a quantum time-crystal with the spontaneously-broken continuous time-translation symmetry, we introduce the van der Pol type nonlinear-friction to open quantum systems. By considering small fluctuations around a time-periodic mean-field solution, we show that a gapless collective mode necessarily appears; this is nothing but the NG mode associated with a time crystal. We show that its dispersion relation becomes ω=−iCp2ω=-iC\bm p^2ω=−iCp2. We also show that noncommutative breaking of the time-translation and U(1) symmetries results in mixing of the NG modes, and the (typically) propagating NG mode appears, whose dispersion relation becomes ω=(±C1−iC2)p2ω=(\pm C_1-iC_2)\bm p^2ω=(±C1​−iC2​)p2.
Abstract We discuss spontaneous symmetry breaking of open classical and quantum systems. When a continuous symmetry is spontaneously broken in an open system, a gapless excitation mode appears corresponding to the Nambu–Goldstone mode. Unlike isolated systems, the gapless mode is not always a propagation mode, but it is a diffusion one. Using the Ward–Takahashi identity and the effective action formalism, we establish the Nambu–Goldstone theorem in open systems, and derive the low-energy coefficients that determine the dispersion relation of Nambu–Goldstone modes. Using these coefficients, we classify the Nambu–Goldstone modes into four types: type-A propagation, type-A diffusion, type-B propagation, and type-B diffusion modes.
When global continuous symmetries are spontaneously broken, there appear gapless collective excitations called Nambu-Goldstone modes (NGMs) that govern the low-energy property of the system. The application of this famous theorem ranges from high-energy, particle physics to condensed matter and atomic physics. When a symmetry breaking occurs in systems that lack the Lorentz invariance to start with, as is usually the case in condensed matter systems, the number of resulting NGMs can be fewer than that of broken symmetry generators, and the dispersion of NGMs is not necessarily linear. In this article, we review recently established formulas for NGMs associated with broken internal symmetries that work equally for relativistic and nonrelativistic systems. We also discuss complexities of NGMs originating from space-time symmetry breaking. In the process we cover many illuminating examples from various context. We also present a complementary point of view from the Lieb-Schultz-Mattis theorem.
We identify a geometric pairing mechanism in systems with spontaneously broken spacetime symmetries, whereby pairs of Goldstone fields become canonically conjugate and hybridize into a single type-B Nambu--Goldstone mode (NGM). The mechanism originates from deformations of the local spacetime volume element induced by Goldstone fluctuations, which endow the Goldstone manifold with a Berry curvature. Integrating this pairing with existing reductions, we propose a general counting formula for the number of gapless NGMs in many-body systems. We demonstrate the mechanism in a microscopic model of a two-component Bose--Einstein condensate, where the dilaton and the U(1)U(1)U(1) Goldstone field combine into a single NGM with quadratic dispersion.
We study an O(N)\mathrm{O}(N)O(N) scalar model under shear flow and its Nambu-Goldstone modes associated with spontaneous symmetry breaking $\mathrm{O}(N)\ensuremath{\rightarrow}\mathrm{O}(N\ensuremath{-}1)$. We find that the Nambu-Goldstone mode splits into an infinite number of gapless modes, which we call the rainbow Nambu-Goldstone modes. They have different group velocities and the fractional dispersion relation $\ensuremath{\omega}\ensuremath{\sim}{k}_{1}^{2/3}$, where k1{k}_{1}k1​ is the wave number along the flow. Such behaviors do not have counterparts in an equilibrium state.
The Nambu-Goldstone (NG) modes in a nonrelativistic system can be classified into two types from their characteristic features: being of either an odd (type I) or an even (type II) power energy-momentum dispersion. Conventionally, the type-II NG modes may universally arise from spontaneous breaking of noncommutative symmetry pairs. Here, we predict a novel type of quadratically dispersed NG modes that emerges in mixed s and p band Bose superfluids in an optical lattice and, unlike the conventional type-II NG modes, cannot be solely interpreted with the celebrated symmetry-based argument. Instead, we show that the existence of such modes has a profound connection to the topological transition on projective complex order-parameter space. The detection scheme is also proposed. Our Letter reveals a new universal mechanism for emergence of type-II NG modes, which bridges intrinsically the Landau symmetry-breaking and topological theories.
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.