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Gravitational wave analogues in spin nematics and cold atoms

Leilee Chojnacki leilee.chojnacki@oist.jp Theory of Quantum Matter Unit, Okinawa Institute of Science and Technology Graduate University, Onna-son, Okinawa 904-0412, Japan    Rico Pohle Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan Graduate School of Science and Technology, Keio University, Yokohama 223-8522, Japan    Han Yan Department of Physics & Astronomy, Rice University, Houston, TX 77005, USA Smalley-Curl Institute, Rice University, Houston, TX 77005, USA Institute for Solid State Physics, University of Tokyo, Kashiwa, 277-8581 Chiba, Japan    Yutaka Akagi Department of Physics, Graduate School of Science, The University of Tokyo, Hongo, Tokyo 113-0033, Japan    Nic Shannon Theory of Quantum Matter Unit, Okinawa Institute of Science and Technology Graduate University, Onna-son, Okinawa 904-0412, Japan
(July 2, 2025)
Abstract

Many large-scale phenomena in our Universe, such as gravitational waves, are challenging to reproduce in laboratory settings. However, parallels with condensed matter systems can provide alternative routes for experimental accessibility. Here we show how spin nematic phases provide a low-energy avenue for accessing the physics of linearized gravity, and in particular that their Goldstone modes are relativistically-dispersing massless spin-2 excitations, analogous to gravitational waves. We show at the level of the action that the low-energy effective field theory describing a spin nematic is in correspondence with that of linearized gravity. We then explicitly identify a microscopic model of a spin-1 magnet whose excitations in the low energy limit are relativistically dispersing, massless spin-2 Bosons which are in one-to-one correspondence with gravitational waves and, supported by simulation, outline a procedure for directly observing these analogue waves in a cold gas of 23Na atoms.

pacs:
74.20.Mn, 75.10.Jm

Introduction.– Light has been a natural companion of humanity since our earliest days, shaping civilization as we know it. However, our attention to astrophysical gravitational waves is, by comparison, still in its infancy. The experimental detection of gravitational waves by the LIGO collaboration [1] marked the beginning of a new age of observational astronomy. That said, production of measurable gravitational radiation is far from being feasible due to the energy scales involved, unlike its photonic counterpart. Alternatives that provide laboratory access to such massless spin-2 waves would therefore provide many new opportunities.

Thus far, several condensed matter systems have been suggested to mimic features of gravity, with much focus on reproducing the effects of curved spacetimes. Acoustic analogues of gravitational phenomena were first suggested by Unruh [2] and later measured [3], with many further promising experimental candidates in superfluids [4, 5], in semimetals [6, 7], in quantum Hall systems [8], in optics [9, 10] and in cold atoms [11, 12, 13, 14]. In the theory domain, connections between elasticity and emergent gravitational phenomena have been studied by Kleinert et al. [15, 16, 17, 18, 19, 20] and independently in the context of fracton models [21, 22, 23, 24, 25, 26], and related aspects of geometry also arise in magnetic models [27] and graphene [28]. However, no experimentally viable platform has yet been realized that provides direct access to massless spin-2 Bosons akin to gravitational waves, even in the flat spacetime analogue.

In this Letter we identify a parallel between gravitational waves and quadrupolar waves in quantum spin nematics, and suggest two routes for their experimental realization. We first review the description of gravitational waves within linearized gravity. We then show that an identical set of equations arises in the low-energy continuum field theory describing spin nematics. Through numerical simulation, we explore the real-time dynamics of a microscopic model with spin nematic order, showing how quadrupolar waves—equivalent to gravitational waves—are generated through the annihilation of topological defects. We conclude by suggesting an experimental protocol for the creation and observation of analogue gravitational waves in spin nematic phases, realized in either magnetic insulators or cold atoms.

Linearized gravity and gravitational waves.– We now briefly summarize the key features of linearized gravity, leading up to gravitational waves. This treatment follows the conventions of standard textbooks, e.g., Refs [29, 30, 31]. General relativity (GR) is a geometrical theory, describing the curvature of a 4-dimensional spacetime. Fundamental to this is the metric tensor, gμ​νg_{\mu\nu}, a symmetric rank-2 tensor, which allows the definition of distance. Here the Greek indices μ\mu, ν\nu run over all four spacetime dimensions. In linearized gravity, spacetime is assumed flat up to small fluctuations, hμ​νh_{\mu\nu}, such that

gμ​ν=ημ​ν+hμ​ν,\displaystyle g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}, (1)

where ημ​ν=diag​(−1,1,1,1)\eta_{\mu\nu}=\text{diag}(-1,1,1,1) is the Minkowski metric for a flat spacetime. The linearized theory is invariant under transformations

x′⁣μ\displaystyle x^{\prime\mu} =\displaystyle= xμ+ξμ​(x),\displaystyle x^{\mu}+\xi^{\mu}(x)\;, (2a)
hμ​ν′\displaystyle h_{\mu\nu}^{\prime} =\displaystyle= hμ​ν−∂νξμ−∂μξν,\displaystyle h_{\mu\nu}-\partial_{\nu}\xi_{\mu}-\partial_{\mu}\xi_{\nu}\;, (2b)

where xμx^{\mu} denotes spacetime coordinates, and ξμ\xi_{\mu} corresponds to an infinitesimal coordinate transformation. The existence of these transformations implies that not all degrees of freedom are independent, and in deriving a theory for gravitational waves, it is conventional to make the choice

hμμ​(xσ)=0​[traceless],\displaystyle h^{\mu}_{\ \mu}(x^{\sigma})=0\;[\textrm{traceless}]\;, (3a)
h0​μ​(xσ)=0​[no scalar or vector components],\displaystyle h_{0\mu}(x^{\sigma})=0\;[\textrm{no scalar or vector components}]\;, (3b)
∂nhn​m​(xσ)=0,\displaystyle\partial^{n}h_{nm}(x^{\sigma})=0\;\;,
⟹\displaystyle\implies kn​hn​m​(kσ)=0​[no longitudinal dynamics].\displaystyle k^{n}h_{nm}(k^{\sigma})=0\;[\textrm{no longitudinal dynamics}]\;. (3c)

Here, Roman indices nn, mm denote the spatial components, and the Einstein summation convention for repeated indices is assumed. Implementing these constraints, we arrive at a theory expressed in terms of a symmetric, traceless, rank-2 tensor [31], with dynamics governed by the action

𝒮𝖫𝖦=−c316​π​G​∫d4​x​[∂αhμ​ν​∂αhμ​ν],\displaystyle\mathcal{S}_{\mathsf{LG}}=-\frac{c^{3}}{16\pi G}\int d^{4}x\;\bigg{[}\partial^{\alpha}h^{\mu\nu}\partial_{\alpha}h_{\mu\nu}\bigg{]}\;, (4)

where cc is the speed of light, and GG the gravitational constant. This leads to the equation of motion for massless waves

1c2​∂t∂thμ​ν−∂n∂nhμ​ν=0,\displaystyle\frac{1}{c^{2}}\partial_{t}\partial^{t}h_{\mu\nu}-\partial_{n}\partial^{n}h_{\mu\nu}=0\;, (5)

where implicitly, only two of the 16 components of hμ​νh_{\mu\nu} have non-trivial independent dynamics. Once quantized [32], the solutions of this wave equation are spin-2 Bosons (gravitons), with dispersion

ω​(𝒌)=c​|𝒌|,\displaystyle\omega({\bf\it k})=c|{\bf\it k}|\;, (6)

and two independent polarizations, σ=+,×\sigma=+,\times, such that

hμ​ν​(t,𝒙)\displaystyle h_{\mu\nu}(t,{\bf\it x}) =\displaystyle= ∑σ=+,×∫d3k1ω​(𝒌)[ϵμ​νσaσ†(𝒌)ei​kρ​xρ\displaystyle\sum_{\sigma=+,\times}\int d^{3}k\frac{1}{\sqrt{\omega({\bf\it k})}}\big{[}\epsilon^{\sigma}_{\mu\nu}a^{\dagger}_{\sigma}({\bf\it k})e^{ik_{\rho}x^{\rho}} (7)
+(ϵμ​νσ)∗aσ(𝒌)e−i​kρ​xρ],\displaystyle+\quad\left(\epsilon^{\sigma}_{\mu\nu}\right)^{*}a^{\phantom{\dagger}}_{\sigma}({\bf\it k})e^{-ik_{\rho}x^{\rho}}\big{]}\;,

where ϵμ​νσ\epsilon^{\sigma}_{\mu\nu} is a tensor encoding information about polarization, and aσ​(𝒌)a^{\phantom{\dagger}}_{\sigma}({\bf\it k}) satisfies

[aσ​(𝒌),aσ′†​(𝒌′)]=δσ​σ′​δ​(𝒌−𝒌′).\displaystyle[a^{\phantom{\dagger}}_{\sigma}({\bf\it k}),a^{\dagger}_{\sigma^{\prime}}({\bf\it k^{\prime}})]=\delta_{\sigma\sigma^{\prime}}\delta({\bf\it k}-{\bf\it k^{\prime}})\;. (8)

For a wave with linear polarization, propagating along the z−z-direction, ϵμ​νσ\epsilon^{\sigma}_{\mu\nu} takes the specific form

ϵ+=12​(0000010000−100000),ϵ×=12​(0000001001000000).\displaystyle{\bf\it\epsilon}^{+}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&0&0\\ 0&1&0&0\\ 0&0&-1&0\\ 0&0&0&0\end{pmatrix},\ {\bf\it\epsilon}^{\times}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&0&0\\ 0&0&1&0\\ 0&1&0&0\\ 0&0&0&0\end{pmatrix}\;. (9)

Physically, this corresponds to a quadrupolar distortion of space, in which compression and dilation alternate.

We define the strain (squared) to be

V​(t,𝒙)=hm​n​(t,𝒙)​xm​xn|𝒙|2.\displaystyle V(t,{\bf\it x})=\frac{h_{mn}(t,{\bf\it x})x^{m}x^{n}}{|{\bf\it x}|^{2}}. (10)

Here again, m,n=1,2,3m,n=1,2,3 run over the spatial components. In Fig. 1 we visualize the equal strain surface in the x1−x2x^{1}-x^{2} plane (notation replaced by f1,f2f_{1},\ f_{2}), for a gravitational wave traveling in the z−z-direction, defined by

V​(t,f1,f2,z)=±const.V(t,f_{1},f_{2},z)=\pm\text{const.} (11)
Refer to caption
Figure 1: Quadrupolar nature of gravitational waves, and Goldstone modes of spin-nematic order, visualized through the associated distortions of spacetime, or the spin-nematic ground state. Results are shown for a wave of wavelength λ\lambda and period τ\tau, with polarization ϵ+{\bf\it\epsilon}^{+} [Eq. (9)], propagating along the z–axis. In the case of gravitational waves [Eq. (7)], f1,f2f_{1},f_{2}, represent xx and yy axes of spacetime, and the quantity plotted is a surface of constant strain [Eq. (10,11)]. In the case of spin–nematic order [Eq. (19)], f1,f2f_{1},f_{2}, represent spin components SxS^{x} and SyS^{y}, and the quantity plotted is the change in the spin–nematic order parameter [Eq. (20,21)]. Blue surfaces denote positive strain/deformation, while orange surfaces denote negative strain/deformation. An animated version of this figure is available in the Supplemental Materials [33].

Linearized gravity analogue in spin nematics.– In the discussion above, we have seen how small fluctuations of the metric gμ​νg_{\mu\nu}, [Eq. (1)], give rise to gravitational waves, which are linearly-dispersing massless spin-2 Bosons, described by the action 𝒮𝖫𝖦\mathcal{S}_{\mathsf{LG}} [Eq. (4)]. In this sense, the search for analogues of gravitational waves can be cast as the search for a physical system which can be described in terms of a symmetric, traceless rank-2 tensor, with linearly-dispersing excitations governed by an action of the form 𝒮𝖫𝖦\mathcal{S}_{\mathsf{LG}}.

Our strategy here is to map the gravitational perturbations on a flat background spacetime onto the Hilbert space of a quantum system whose ground state meets this description. Analogues of gravitational waves can then be found in the Goldstone modes of this symmetry-broken state.

Classically, the order parameter for a nematic liquid crystal is a symmetric, traceless rank-2 tensor [34]. We here consider a form of quantum liquid crystal known as a “quantum spin nematic”, originally introduced as a magnetic state [35, 36, 37] which preserves time-reversal symmetry, but breaks spin-rotation symmetry through the quadrupole operators

𝒬m​n=12​(Sm​Sn+Sn​Sm)−13​δm​n​Sn​Sn.\displaystyle{\mathcal{Q}}^{mn}=\frac{1}{2}\left(S^{m}S^{n}+S^{n}S^{m}\right)-\frac{1}{3}\delta_{mn}S^{n}S^{n}\;. (12)

Here SmS^{m} is a spin operator with components m=x,y,zm=x,y,z, satisfying the usual SU(2) commutation relations.

The simplest form of a quantum spin nematic is the “ferroquadrupolar” (FQ) state, a uniaxial nematic liquid crystal in which all quadrupole moments are aligned [Fig. 2a]. As in conventional liquid crystals [38], such a state can be characterized by a director 𝒅{{\bf\it d}}, and its symmetry dictates that it supports two, degenerate Goldstone modes [39], which have the character of massless, spin-2 Bosons [40, 41, 42]. We will now show how these correspond to the massless spin–2 Bosons found in linearised gravity.

We start by promoting 𝒬m​n{\mathcal{Q}}^{mn} to a tensor field Qμ​νQ^{\mu\nu} providing a low energy effective description, identifying Qm​n=𝒬m​nQ^{mn}={\mathcal{Q}}^{mn}, where Qm​n=Qm​nQ^{mn}=Q_{mn}, and by setting components Q0​μ=Qμ​0=0Q_{0\mu}=Q_{\mu 0}=0. In analogy with Eq. (1), we consider fluctuations Qμ​ν𝖤Q^{\sf E}_{\mu\nu} about a state with uniform spin nematic order Qμ​ν𝖦𝖲{Q}^{\sf GS}_{\mu\nu}, viz

Qμ​ν=Qμ​ν𝖦𝖲+Qμ​ν𝖤,\displaystyle Q_{\mu\nu}=Q^{\sf GS}_{\mu\nu}+Q^{\sf E}_{\mu\nu}\;, (13)

requiring that these fluctuations occur in the transverse channel, i.e., that the change affects the direction but not the magnitude of the quadrupolar order. This assumption is appropriate for the low-energy physics of spin nematics [40, 43, 44].

What remains is to match the fluctuations of quadrupolar order, which occur in spin–space, to the changes in spacetime coordinates appropriate for a gravitational wave. This can be accomplished by a unitary transformation

Q~μ​ν(𝒌)=Cμ​ν(𝒌,𝒅)ρ​σQρ​σ𝖤(𝒅),\displaystyle\tilde{Q}_{\mu\nu}({\bf\it k})=C_{\mu\nu}{}^{\rho\sigma}({\bf\it k},{\bf\it d})Q^{\sf E}_{\rho\sigma}({\bf\it d})\;, (14)

where Cμ​ν(𝒌,𝒅)ρ​σC_{\mu\nu}{}^{\rho\sigma}({\bf\it k},{\bf\it d}) acts on quadrupole excitations with wave vector 𝒌{\bf\it k} about a FQ state characterized by director 𝒅{\bf\it d}. Further details of this transformation are given in the Supplemental Material [33].

For an appropriate choice of Cμ​ν(𝒌,𝒅)ρ​σC_{\mu\nu}{}^{\rho\sigma}({\bf\it k},{\bf\it d}), Q~μ​ν\tilde{Q}_{\mu\nu} satisfies the conditions

Q~μμ=Qμμ=0​[traceless],\displaystyle\tilde{Q}^{\mu}_{\ \mu}=Q^{\mu}_{\ \mu}=0\;\textrm{[traceless]}, (15a)
Q~0​μ=Q0​μ=0​[no scalar or vector components],\displaystyle\tilde{Q}_{0\mu}=Q_{0\mu}=0\;\textrm{[no scalar or vector components]}, (15b)
km​Q~m​n​(kσ)=0​[no longitudinal dynamics].\displaystyle k_{m}\tilde{Q}_{mn}(k^{\sigma})=0\;\textrm{[no longitudinal dynamics]}. (15c)

The low-energy fluctuations of the spin nematic can be described in terms of a quantum non-linear sigma model [40, 43, 44]. Given these physical constraints, and the decomposition described by Eq. (13), we arrive at an action which exactly parallels linearized gravity [Eq. (4)], via,

𝒮𝖥𝖰=−12∫dtddx[χ⟂​(∂tQ~μ​ν​∂tQ~μ​ν)−ρs(∂nQ~μ​ν∂nQ~μ​ν)],\displaystyle\begin{split}\mathcal{S}_{\mathsf{FQ}}=-\frac{1}{2}\int dtd^{d}x\bigg{[}&\chi_{\perp}(\partial^{t}\tilde{Q}^{\mu\nu}\partial_{t}\tilde{Q}_{\mu\nu})\\ &-\rho_{s}(\partial^{n}\tilde{Q}^{\mu\nu}\partial_{n}\tilde{Q}_{\mu\nu})\bigg{]},\end{split} (16)

where χ⟂\chi_{\perp} is the transverse susceptibility, and ρs\rho_{s} the stiffness, associated with spin-nematic order [43, 44].

The low-lying excitations of this theory are massless spin-2 Bosons, satisfying the wave equation [cf. Eq. (5)]

1v2​∂t∂tQ~μ​ν\displaystyle\frac{1}{v^{2}}\partial_{t}\partial^{t}\tilde{Q}_{\mu\nu} −\displaystyle- ∂n∂nQ~μ​ν=0,\displaystyle\partial_{n}\partial^{n}\tilde{Q}_{\mu\nu}=0\;, (17)

with v=ρs/χ⟂v=\sqrt{{\rho_{s}}/{\chi_{\perp}}}, and with dispersion

ω​(𝒌)=v​|𝒌|.\displaystyle\omega({\bf\it k})=v|{\bf\it k}|\;. (18)

The solutions to Eq. (17) have exactly the same structure as those for gravitons [cf. Eq. (7)]

Q~μ​ν(t,𝒙)=∑σ=+,×∫d3k1ω​(𝒌)[ϵμ​νσ​bσ†​(𝒌)​ei​kρ​xρ+(ϵμ​νσ)∗bσ(𝒌)e−i​kρ​xρ],\begin{split}\tilde{Q}_{\mu\nu}(t,{\bf\it x})=\sum_{\sigma=+,\times}\int d^{3}k\frac{1}{\sqrt{\omega({\bf\it k})}}\big{[}&\epsilon^{\sigma}_{\mu\nu}b^{\dagger}_{\sigma}({\bf\it k})e^{ik_{\rho}x^{\rho}}\\ &+\left(\epsilon^{\sigma}_{\mu\nu}\right)^{*}b^{\phantom{\dagger}}_{\sigma}({\bf\it k})e^{-ik_{\rho}x^{\rho}}\big{]}\;,\end{split} (19)

where bσ​(𝒌)b^{\phantom{\dagger}}_{\sigma}({\bf\it k}) satisfies Bosonic commutation relations [Eq. (8)] and, the tensors ϵμ​νσ{\bf\it\epsilon}^{\sigma}_{\mu\nu} are given by Eq. (9).

The quadrupolar excitations can be visualized in terms of surfaces proportional to the wavefunction amplitudes of the propagating mode, in analogy with Eq. (10)

V​(𝑺,(t,𝒙))=Sm​Q~m​n​(t,𝒙)​Sn|𝑺|2.\displaystyle V({\bf\it S},(t,{\bf\it x}))=\frac{S^{m}\tilde{Q}_{mn}(t,{\bf\it x})S^{n}}{|{\bf\it S}|^{2}}. (20)

In Fig. 1 we plot the equal amplitude surface in the Sx−SyS^{x}-S^{y} plane (notation changed to f1,f2f_{1},\ f_{2}), for a quadrupole wave propagating in the z−z-direction, defined as

V​(𝑺=(f1,f2,0),(t,0,0,z))=±const.V({\bf\it S}=(f_{1},f_{2},0),(t,0,0,z))=\pm\text{const.} (21)

as shown in Fig. 1.

It follows that, from a mathematical point of view, the quadrupolar waves in a quantum spin nematic are in one-to-one correspondence with quantized gravitational waves (gravitons) in a flat, 4-dimensional spacetime. However, there is a critical distinction regarding the spaces these waves arise in, which has important implications for realizing them in experiment. Gravitational waves involve quadrupolar distortions of space, transverse to the direction of propagation. This implies that a minimum of three spatial dimensions is required to support a gravitational wave. In contrast, the quadrupolar waves found in spin nematics arise in a spin–space which is automatically three–dimensional, regardless of the number of spatial dimensions. For this reason, it is possible to explore analogues of gravitational waves in 2-dimensional spin systems. It is this subject which we turn to next.

Refer to caption
(a) FQ ground state
Refer to caption
(b) S𝖰​(𝒌,ω)S_{\sf Q}({\bf\it k},\omega)
Figure 2: Spin–nematic state on a triangular lattice, and its spin-2 excitations. (a) Ferroquadrupolar (FQ) ground state, in which quadrupole moments of spin align with a common axis. (b) Dispersion of excitations about the FQ state, showing linear character ω=v​|𝒌|\omega=v|{\bf\it k}| at long wavelength (black dashed line). This linear dispersion is consistent with the predictions of the field theory [Eq. (16)]. The spin–2 nature of the long-wavelength excitations can be inferred from the quadrupolar structure factor SQ​(𝒌,ω)S_{\rm Q}({\bf\it k},\omega) [Eq. (23)], overlaid on the plot. Results are shown for a spin-1 bilinear biquadratic (BBQ) model [Eq. (22)], with parameters J1=0,J2=−1J_{1}=0,J_{2}=-1, as described in [42].
Refer to caption
Figure 3: Numerical simulation of vortices within a spin-nematic state, showing how quadrupole waves, analogous to gravitational waves, are created when a pair of vortices in-spiral and annihilate. Individual frames are taken from dynamical simulation of a ferroquadrupolar state (FQ) in the spin-1 bilinear biquadratic (BBQ) model on a triangular lattice [Eq. (22)], with further detail given in the Supplemental Material [33]. An animated version of this result is also available [33].

Simulation using cold atoms.– The idea of using cold atoms to simulate a quantum spin nematic has a long history [45, 46, 47, 48, 49]. The majority of proposals build on “spinor condensates” of atoms, such as 23Na, 39K or 87Rb, whose internal hyperfine states mimic the magnetic basis of a spin-1 moment [45, 50, 51]. The interactions between these effective spin-1 moments depend on the details of their scattering and, where attractive, can lead to spin-nematic order [45]. Condensates described by the order parameter 𝒬α​β{\mathcal{Q}}^{\alpha\beta} [Eq. (12)] have already been observed in experiment [52]. On symmetry grounds, the Goldstone modes of these systems must be described by 𝒮𝖥𝖰\mathcal{S}_{\mathsf{FQ}} [Eq. (16)], making them analogues of linearized gravity.

Optical lattices can be arranged in a wide array of geometries, including triangular lattices [53], and cold atom experiments with 23Na atoms are carried out in many laboratories, e.g., [54, 49, 55]. Realizing an analogue of gravitational waves on a lattice therefore also seems a realistic possibility.

Realization of gravitational waves in a microscopic lattice model.– In addition to realization of analogue gravitational waves using quantum fluids as suggested above, spin-nematic phases can also be found in solid state magnetic systems. The simplest microscopic model supporting a quantum spin-nematic state is the spin-1 bilinear biquadratic (BBQ) model

ℋ𝖡𝖡𝖰=J1​∑⟨i​j⟩𝑺i⋅𝑺j+J2​∑⟨i​j⟩(𝑺i⋅𝑺j)2,\displaystyle{\mathcal{H}}_{\mathsf{BBQ}}=J_{1}\sum_{\langle ij\rangle}{\bf\it S}_{i}\cdot{\bf\it S}_{j}+J_{2}\sum_{\langle ij\rangle}\big{(}{\bf\it S}_{i}\cdot{\bf\it S}_{j}\big{)}^{2}\;, (22)

known to support FQ order for a wide range of J2<0J_{2}<0, irrespective of lattice geometry [35, 56, 57]. Particular attention has been paid to the BBQ model on a triangular lattice [41, 58, 59, 60, 43, 61, 42], where studies have been motivated by e.g. NiGa2S4 [62, 63], and FeI2 [64]. It has also been argued that 23Na atoms in an optical lattice could be used to realize the BBQ model [Eq. (22)], with parameters falling into the range relevant to FQ order [47, 65].

Explicit calculations of FQ dynamics within the BBQ model reveal two, degenerate bands of excitations, with linear dispersion at long wavelength [41, 61, 42]. The quadrupolar (spin-2) nature of these excitations at low energy is manifest in the dynamical structure factor for quadrupole moments

S𝒬​(𝒌,ω)=∑α,β∫d​t2​π​ei​ω​t​⟨𝒬α​β​(𝒌,t)​𝒬α​β​(−𝒌,0)⟩,S_{\mathcal{Q}}({\bf\it k},\omega)=\sum_{\alpha,\beta}\int\frac{dt}{2\pi}e^{i\omega t}\langle{\mathcal{Q}}^{\alpha\beta}({\bf\it k},t){\mathcal{Q}}^{\alpha\beta}({\bf\it-k},0)\rangle\;, (23)

shown in Fig. 2b, for calculations carried out at a semiclassical level [42]. Starting from Eq. (22), it is also possible to parameterize the continuum field theory Eq. (16), obtaining results in quantitative agreement with the microscopic model, as shown in Fig. 2b.

Quench dynamics, simulation and measurement.– We now turn to the question of how gravitational-wave analogues could be created and observed in experiment. For concreteness, we consider a FQ state in an explicitly 2-dimensional system, which we model as set of spin-1 moments on a lattice [cf. Eq. (22)]. Consistent with the Mermin-Wagner theorem, for low-dimensional systems to exhibit anything besides exponentially-decaying correlations at low temperature, they must undergo topological phase transitions e.g. of the BKT type [66, 67]. The FQ state in a 2D magnet is known to be connected to the high temperature magnetic phase via a vortex-induced topological phase transition [68].

The excitations which mediate this phase transition are no longer the integer vortices of the Berezinskii-Kosterlitz-Thouless (BKT) transition [69], but rather 𝒵2\mathcal{Z}_{2} vortices of homotopy group π1\pi_{1}, specific to the nematic order parameter [38]. Cooling rapidly through the transition (quenching) leads to a state rich in pairs of 𝒵2\mathcal{Z}_{2} vortices, which are subject to attractive interactions, and spiral towards one another in much the same way as gravitating masses. In the process, vortices radiate energy in the form of quadrupolar waves, Eq. (19), and eventually annihilate. This process is clearly visible in simulations of the BBQ model [Eq. (22)], as illustrated in Fig. 3 [33], and the accompanying animation [33].

As can be seen from these simulations, the dynamics of vortices is very slow compared to that of quadrupolar waves, and the timescale associated with the annihilation of 𝒵2\mathcal{Z}_{2} vortices is of order 102​J2−110^{2}J_{2}^{-1}. Observing vortices in experiment will therefore typically demand long–lived condensates. None the less, successful imaging of conventional magnetic vortices within a spinor–condensate of 23Na ions has already been realized, over timescales of ∼1​s\sim 1\text{s} [70]. An experimental protocol for observing excitations in the quadrupolar channel of a spinor condensate has also already been implemented [71], and could be used to make real–space images of nematic correlations, through their imprint on the electric polarisability of the atoms [72]. Proposals also exist for imaging the quadrupolar correlations of spin-1 moments through, e.g., Raman scattering [73].

Taken together, this all jointly suggests that it is a realistic possibility to realize and observe such spin nematic gravitational waves analogues.

Conclusions. – Linearized gravity is of fundamental interest, but hard to study in experiment, because the energy scales of excitations are so large, and their amplitudes so small. In this Letter we have shown how a theory in direct correspondence to linearized gravity arises in systems with spin-nematic order. The Goldstone modes of this spin-nematic state are massless spin-2 Bosons, which behave as exact analogues of quantized gravitational waves (gravitons). These results imply that it is possible to simulate various aspects of linearized gravity, including gravitons and topological excitations, in magnets or assemblies of cold atoms which realize a spin-nematic state [74, 75, 76, 77, 78].

Acknowledgements.
Acknowledgments – The authors are pleased to acknowledge helpful discussions with Yuki Amari, Andrew Smerald and Hiroaki T. Ueda. This work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grants No. JP19H05822, JP19H05825, JP20K14411, JP20H05154 and JP22H04469, MEXT "Program for Promoting Researches on the Supercomputer Fugaku", Grant No. JPMXP1020230411, JST PRESTO Grant No. JPMJPR2251, and by the Theory of Quantum Matter Unit, OIST. Numerical calculations were carried out using HPC facilities provided by OIST and the Supercomputer Center of the Institute of Solid State Physics, the University of Tokyo.

References