Fracton infrared triangle

In theories with conserved dipole moment, isolated charged particles (fractons) are immobile, but dipoles can move. We couple these dipoles to the fracton gauge theory and analyze the universal infrared structure. This uncovers an observable double kick memory effect which we relate to a novel dipole soft theorem. Together with their asymptotic symmetries this constitutes the first realization of an infrared triangle beyond Lorentz symmetry. This demonstrates the robustness of these IR structures and paves the way for their investigation in condensed matter systems and beyond.


I. INTRODUCTION
Fractons [1,2] are novel quasiparticles whose characteristic feature is their limited mobility.This restricted mobility originates from their built-in dipole symmetry which leads to conserved dipole moments d i = x i ρd 3 x.Fracton theories attract attention not only for their interesting phenomenological applications, but also for their intricate theoretical underpinnings which challenge common quantum field techniques [3][4][5].The dipole symmetry is an example of generalized symmetries, whose study has led to breakthroughs in our understanding of quantum field theories [6,7].Moreover, the underlying symmetries are closely related to Carroll symmetries [8][9][10][11], which play a fundamental role in flat-space holography [12][13][14][15].
The infrared (IR) triangle [16] is a triangular correspondence, which connects asymptotic symmetries, memory effects and soft theorems.It controls the infrared behavior of gravity and several relativistic gauge theories, and is a building block of the celestial holography program [17][18][19][20][21].In this work, we establish the infrared triangle in fracton gauge theory by demonstrating the existence of a novel and observable double kick memory effect, which we also relate to a dipole soft theorem and novel asymptotic symmetries (see Figure 1).This shows that IR triangles
One consequence of the non-lorentzian nature of the fracton gauge theory is that its degrees of freedom have two different dispersion relations and propagation speeds.As a result, the spacetime has two radiative regions (see Figure 2), in contradistinction to the single null infinity in lorentzian theories.This makes the asymptotic structure of these theories richer than that of their relativistic counterparts.These novel features also imply the need for implementing new techniques beyond the conventional ones.For example, because of the asymmetry between space and time, a radiative coordinate system of Bondi type is not directly available.
One of our main results is a novel memory effect.Memory effects refer to observables that persist in a probe system after the passage of waves.Early examples of memory effects in the context of gravity include the displacement of freely falling test masses [22][23][24][25].However, this field has witnessed a significant interest in recent years due to discovery of the relation between memory effects and fundamental properties of gravity [26][27][28], and a plethora of novel persistent wave observables have been found in gravity and gauge theory [16,[29][30][31][32][33][34][35][36][37][38][39].While fracton memory effect shares some features with gravity and some with gauge theories, it has unique properties, in particular, it leads to a double kick effect on test quasiparticles (dipoles), as is depicted in Figure 2.This double kick memory effect is a measurable infrared observable and encourages the exciting perspective to try to observe them in condensed matter systems, especially considering the ongoing experimental investigations into systems with dipole symmetry [40][41][42][43].
This work is structured as follows.In Section II we review the fracton scalar charge gauge theory.In Section III we derive the spectrum of the theory and decoupled equations obeyed by each sector.In particular, we obtain the analogue of the Liénard-Wiechert potentials, and thereby uncover the memory effects in the scattering of dipoles.We show that test dipoles experience a double kick memory effect.In Section IV we analyze asymptotic symmetries and their corresponding conservation laws, which we relate in Section V to memory effects.In Sec-arXiv:2310.16683v2[hep-th] 24 Dec 2023 tion VI we derive the dipole soft factor from the memory effect (an alternative derivation using Feynman diagrams is presented in the supplementary material A).We finish with a discussion and an outlook in Section VII.Further details will appear in our extended work [44].

II. FRACTON GAUGE THEORY
In this section, we introduce the (scalar charge) gauge theory [45,46] which describes the interactions among charged fractons.It is a higher-rank gauge theory defined by a scalar field ϕ and a symmetric tensor A ij (i, j, k are spatial indices from 1 to 3), with the Lagrangian density where The constant c has units of velocity, but is not necessarily the speed of light.The tensors E ij and F ijk are analogous to an electric and magnetic tensor, respectively.The symmetries imply F [ijk] = 0 and the useful relation and leads to the equations of motion where ρ and J ij represent the charge and current densities, respectively.Consistency of the field equations (4) leads to the continuity equation which implies that the electric charge Q and dipole mo- of a localized source are conserved.The conservation of the dipole moment implies, in particular, that isolated monopoles in this theory cannot move.

III. DECOUPLED FIELD EQUATIONS AND MEMORY EFFECT
An important consequence of the fact that the fracton gauge theory is not Lorentz invariant is that various dynamical degrees of freedom obey different dynamical equations.We decouple these equations of motion using a systematic decomposition of the gauge field A ij into representations of the rotation group.Moreover, imposing the constraint (4a), one finds where the superscripts T and TL, denoted collectively by #, refer to transverse and transverse-longitudinal projections A # ij = P # ijmn A mn with defined in terms of transverse and longitudinal projectors where ∆ −1 is the inverse of the Laplacian ∆ = ∂ i ∂ i , given by a Green function integral.Using (7) in the dynamical equation (4b) combined with Ḟijk = 2 ∂ [i E j]k , one finds the decoupled dynamical equations where is the wave operator with speed c.The unequal propagation speeds c and c = c/ √ 2 of the dynamical degrees of freedom reflect the non-lorentzian nature of this theory, whose Aristotelian symmetry structure is discussed in [8,47].Also note that gauge transformations (3) whose parameter vanishes at the boundary (i.e., when Λ = O(1/r)), are in the kernel of P T and P TL .This implies that the dynamical variables A T ij , A TL ij are gauge invariant and account for the expected 2 + 1 and 2 degrees of freedom, respectively.However, they do transform under so-called "large-gauge transformations" as will be discussed in section IV.Equations (10) are solved by (11) where □ −1 c represents the inverse of □ c using a retarded Green function integral.In deriving (11), we have used the commutativity of □ −1 c and the projectors.Asymptotic behavior.Assuming that the source is localized, the asymptotic behavior of the fields can be derived from an asymptotic expansion of (11) in the radiation zone r → ∞.One finds that and therefore where P ij = δ ij − n i n j and n i = (sin θ cos φ, sin θ sin φ, cos θ) is the radial normal vector in spherical coordinates θ A = (θ, φ).

III.1. Radiation of scattering dipoles
Since in this theory isolated monopoles cannot move, a natural setup is the scattering of moving dipoles.The current of a dipole d i moving on a path z i (t) with velocity v i = żi (t) is given by [46] We can use the current to calculate the Green integral , (15) where R ≡ x − z(t) and R, R are its norm and unit direction, respectively.This is the fracton dipole analog of the Liénard-Wiechert solution of moving point charges in electrodynamics.Inserting ( 15) into ( 11), one finds that far from the source, asymptotic fields A T ij (n) and of a dipole moving with constant velocity are given by where we use where η = 1 for outgoing and −1 for incoming dipoles.Double kick memory effect.These memory effects have observable consequences.As an example, consider a fractonic dipole situated at far distance from the source of radiation.The dipole is affected by the radiation through the generalized Lorentz force law [46] Using the asymptotic form (13) of the electric field and an analogous expression for the magnetic field, and integrating the result over time in the interval (t 0 , t f ), we find that there is a net kick effect on the dipole that is proportional to the memory effects Each of the fast and slow radiative modes cause a net kick effect on the test dipole and thus it undergoes a double kick memory effect (see Figure 2).The term proportional to δu in (19b) describes the acceleration produced by the Coulomb field of the total charge of the source, which in contrast to electromagnetism, is of the same order in the large r expansion as the wave solution.

IV. ASYMPTOTIC CONDITIONS AND BONDI ANALYSIS OF FRACTONIC WAVES
An immediate consequence of the different propagation speeds in this theory is that at very large distances, there will be a decoupling of the T and TL sectors, defining two distinct radiation zones: the fast radiation zone of T-waves, where t, r → ∞ while u := t − r/c remains finite, and the slow radiation zone of TL-waves, with ũ := t−r/c finite.Accordingly, the radiative phase space splits into two distinct phase spaces Γ T , Γ TL .The splitting is consistent, since the asymptotic observables we are interested in, such as memory effects, can be decomposed into operators with support in either of the phase spaces O = O T + O TL .A more comprehensive analysis which might be useful to study subleading radiative effects, which are not necessarily separable, would involve considering the direct product of the two phase spaces.Such analysis requires a careful analysis of the matching between the fields in the two radiative zones and goes beyond the scope of this letter.
We will therefore analyze the structure of fields and asymptotic symmetries in each of these asymptotic regions independently.To this end, we solve (10) asymptotically in the limit r → ∞.We assume that the source fields decay fast enough, so that we can implement source-free wave equations at leading orders.We use the notation (c # , u # ) to unify results that are valid in both radiative regions with their respective propagation speed and retarded time.Also, overdot refers to derivative with respect to retarded time of the region under consideration.
The transformation from Cartesian to spherical coordinates, which are more convenient for the asymptotic analysis is carried out by suitable projections with the triad (n i , re A i ) and e A i (n) = ∂n i ∂θ A .The induced metric γ AB = δ ij e A i e B j denotes the metric of the unit 2-sphere γ AB dx A dx B = dθ 2 + sin 2 θdφ 2 which is used to lower and raise A, B, . . .indices and has determinant γ.Therefore, the analysis in the preceding sections reveals the following asymptotic behavior for the electric field written as a tensor density, i.e., multiplied by r 2 √ γ The asymptotic behavior of the electric field is consistent with the following fall-off for the potentials Here, q is a constant parameter of the asymptotic solution, which after matching with the bulk solution coincides with the total charge Q.As shown in [48], this expression for the leading order of ϕ is essential for achieving both finite energy density and charges.Note that the falloff is preserved under the Aristotelian symmetries, i.e., spacetime translations and rotations.The fall-off ( 21) is preserved under gauge transformations of the form and a similar expression in the slow radiation zone by replacing (u, c) by (ũ, c) and (λ, η) by ( λ, η).The corresponding gauge transformation take the same functional form in both regions (except that (η, λ) → ( λ, η) in the TL sector) where D A is the covariant derivative with respect to γ AB , while D 2 = D A D A is the sphere Laplacian and The corresponding charges and fluxes should be worked out in each radiative region separately, as the radiative fields behave differently: Transverse sector T. The radiative field in this region is A AB (u, θ A ), while A rA and ϕ (0) are time independent functions on the sphere, we therefore find Transverse-longitudinal sector TL.The radiative field in this region is A rA (ũ, θ A ), while A AB and ϕ (0) are time independent.Accordingly, The charges corresponding to asymptotic symmetries ( 22) can be worked out using canonical or covariant phase space methods [49,50].To get finite charges, we also impose the following constraints [48], consistent with the solutions obtained in the previous section in a harmonic expansion in Y ℓ,m (θ, ϕ).Using the equation of motion together with the conditions ( 24) or ( 25), one finds that the total charge is finite and reads where the charge densities depending on the radiation zone are given by The term proportional to ϵ in (28) gives the total charge, while terms proportional to λ and η give two infinite sets of charges in each sector, where the conserved dipole moment is the ℓ = 1 in the mode expansion of P.
In the presence of radiation, charges are no longer conserved, but carried away by fractonic waves.The time evolution of the charges are specified by the following flux equations derived from the equations of motion.In the fast radiation zone (T-sector) whereas in the slow radiation zone (TL-sector) Radiation also carries away energy from the system.The field equations imply that the energy density H = which reveals the following flux-balance equations These are the fractonic analogue of Bondi's energy loss formula [51,52], showing that the energy is always decreasing in time.

V. MEMORY EFFECT AND ASYMPTOTIC SYMMETRIES
In this section, we show that fracton memory effects can be realized as a vacuum transition under fracton asymptotic symmetries.Consider a dynamical process in which the system is non-radiative before some initial time and after some final time, implying that ȦAB = 0 in the limit u → ±∞ and ȦrA = 0 in the limit ũ → ±∞.Therefore, the initial and final radiative vacua are labeled by time-independent tensors A rA (θ A ), A AB (θ A ).The memory effect discussed in section III.1 implies that the vacua are not identical before and after radiation.Rather, the transition between the vacua induces a large gauge transformation as we will see shortly.
A generic vector Y A and symmetric tensor X AB on the sphere can be decomposed as where the terms involving ϵ AB tensor have odd (magnetic) parity.However, starting from ( 17) and transforming to spherical coordinates, we find that the magnetic terms vanish and the memory terms can be expressed in terms of three scalar memory fields C S , C V , C T on the sphere The first (second) line refers to the memory accumulated during the fast (slow) radiation.

Memory as vacuum transition
In each sector, the corresponding memory leads to a change of vacuum given by a large gauge symmetry.The memory δA rA in the TL sector corresponds to a large gauge transformation given by η = −C V , λ = 0 in (23).The T sector is subtle, the memory δA AB corresponds to choosing (23).This choice also induces a change in A rA , but that is no problem since A rA is not part of the radiative phase space of the T sector. 2  These equations can be inverted to compute the memory fields using the following Green functions 1 Subscripts S, V, T refer to scalar, vector, and tensor which refer to the corresponding propagating modes. 2 Upon considering the product phase space of the slow and fast radiation modes, there could emerge a new symmetry, which is exclusively responsible for the memory in the trace mode.This situation is reminiscent of the "breathing" memory in scalartensor theories of gravity [35,53].We leave this problem to future work.
As a result, vacuum fields are given by where (17) in these equations reveals the memory fields.The change in the charges resulting from radiation flux is also exclusively determined by the memory fields.Integrating ( 30) and (31) in time, and using (36), one finds that the flux of charges through the fast radiation (T-sector) is given by while the slow radiation (TL-sector) carries Thus we have established connections between asymptotic symmetries and memory effects in the fracton infrared triangle (see Figure 1).

VI. SOFT FACTORS FROM THE MEMORY EFFECT
In this section we determine the soft factors for a scattering process of dipoles from the memory effect.
Consider a scattering process of N dipoles with momentum p α = m α v α and dipole moment d α , with the emission of one fractonic soft photon with frequency ω and polarization projectors ϵ ij # .The scattering amplitude is expected to factorize in the soft limit ω → 0 as To derive the soft factor S # ij , we will closely follow [27].We illustrate the derivation for the T sector, while the analysis for the TL sector would be similar.Starting from a Fourier mode decomposition of the gauge field A T ij , it can be shown that the radiative field A T ij = lim r→∞ (rA T ij ) can be computed using a saddle point approximation where α labels the transverse mode with polarization ϵ α ij , created by a † α .The next step is to use this result to compute the memory This equation relates the memory to the creation and annihilation of a fractonic soft (zero frequency) photon.As a result, an amplitude with an external soft photon factorizes according to (41) with soft factors Thus the soft factors S T ij , S TL ij in (41) are given up to overall factors −4πc 2  # by (A8a) and (A8b) respectively.An alternative derivation of the soft factors is detailed in the supplemental material and involves the use of Feynman diagrams within a simple effective model that describes the dynamics of dipoles coupled to the fracton gauge field.The derivation of the soft factor from the conservation of asymptotic charges will be discussed in our extended work [44].

VII. DISCUSSION AND OUTLOOK
In this work, we introduced an observable double kick memory effect (Section III) and the corresponding dipole soft theorem (Section VI), which we related to the asymptotic symmetries of fracton gauge theory.This provides the first instance of an IR triangle (Figure 1) for a theory beyond Lorentz symmetry and further evidence for the robustness of this triangular correspondence.
The tools developed and implemented in this work can be used to study radiation and IR effects beyond lorentzian symmetries, which opens the door to explore other models of relevance to condensed matter systems and beyond, e.g., [45,46,.Conversely, studying theories beyond Lorentz symmetry can improve our understanding of IR structures (similar to Nambu-Goldstone modes whose dispersion relations are richer and more intricate once Lorentz symmetry is not imposed).The double kick effect (see Figure 2) is an example of a novel memory observable which exhibits the more intricate structure that can appear in a non-lorentzian theory.
One avenue we would like to highlight involves leveraging the duality between (specific models of) fractons and vortices [82] as well as the relationship with elasticity [83,84] (see [85] for a review).These dualities open up exciting possibilities for creating experimental setups that are potentially easier to realize and which could facilitate the observation of memory effects (see also [86]).
While fractons have originated from condensed matter physics, they might play an important role in the holographic understanding of gravity in asymptotically flat spacetimes.The reason is the correspondence between the fracton algebra and the Carroll algebra which is the underlying symmetry of gravity in asymptotically flat spacetimes [9][10][11][87][88][89].Indeed, many of the discussed dualities and experiments can equally well be seen though the lens of carrollian physics, e.g., in [90] insights from fractons have been used in the context of Carroll fluids.
In this work, we have focused on the leading IR behavior.However, it might be interesting to explore subleading effects and consider how celestial holography [17][18][19][20][21] could be extended to this setup.After all Lorentz symmetry is absent, but we still recover an analog IR structure.In addition, to get closer to experiments that investigate dipole symmetry [40][41][42][43] it might also be interesting to introduce and study boundaries at finite distances.
Motivated by the historically prolific interdisciplinary dialogue between high-energy and condensed matter physics, we are also excited by the prospect that the inaugural experimental validation of memory effects may manifest within the domain of condensed matter systems.We hope that this work will serve as an initial stepping stone for this promising endeavor.

FIG. 2 .
FIG. 2. (a): Since the waves propagate with speeds c and c the theory has two different radiation zones (two "null infinities").(b): It follows that a dipole in the far region will receive two kicks, but the orientation ⃗ d of the dipole stays inert.