Anomalous Correlators, Negative Frequencies, and Non-Phase-Invariant
Hamiltonians in Random Waves
A. Villois ,1,2 G. Dematteis ,2,* Y. V. Lvov ,3 M. Onorato ,2,4 and J. Shatah5
1School of Engineering, Mathematics and Physics, University of East Anglia,
Norwich Research Park, Norwich, NR4 7TJ, United Kingdom
2Dipartimento di Fisica, Universita` degli Studi di Torino–Via Pietro Giuria 1, 10125 Torino, Italy
3Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York, USA
4Istituto Nazionale di Fisica Nucleare, INFN, Sezione di Torino–Via Pietro Giuria 1, 10125 Torino, Italy
5Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012, USA
(Received 27 February 2025; accepted 2 September 2025; published 16 October 2025)
We investigate a generic non-phase-invariant Hamiltonian model that governs the dynamics of nonlinear dispersive waves. We give evidence that initial data characterized by random phases naturally evolve into phase correlations between positive and negative wave numbers, leading to the emergence of nonzero anomalous correlators and negative frequencies. Using analytical techniques, we show that anomalous correlators develop on a timescale of Oð1=ϵÞ, earlier than the kinetic timescale. Our theoretical predictions are validated through direct numerical simulations of the deterministic system.

DOI: 10.1103/3brq-j34x
Introduction—Many physical systems are characterized by the propagation of dispersive and nonlinear waves. When the number of waves is large, a deterministic description becomes impractical, necessitating a statistical approach similar to the statistical treatment of gases. In the context of interacting waves, since the late 1960s, wave turbulence (WT) theory [1,2] has been proposed as an analog to the Boltzmann theory for gases. A central role in WT theory is played by the wave kinetic equation (WKE), which describes both equilibrium thermodynamic states and out-of-equilibrium dynamics [3,4].
Despite some rigorous mathematical derivations of the WKE for specific PDEs [5–7], the applicability of WT theory remains an open issue for many partial differential equations (PDEs) describing nonlinear interacting waves. One key property that has to be proven in the derivation of the WKE is that the Fourier phases of the normal variables remain uncorrelated up to kinetic timescales—this is analogous to the problem of propagation of chaos, a concept also present in the derivation of the Boltzmann equation for hard spheres. Only for the nonlinear Schrödinger (NLS) equation in dimension d ≥ 2 has this problem been rigorously addressed [8].
When the nonlinearity is sufficiently strong, phase correlations are expected. This phenomenon is well known
*Contact author: giovanni.dematteis@unito.it
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. in integrable systems, where phase-coherent structures such as solitons or breathers can emerge and persist even from initial data characterized by random phases. For non- integrable systems, the situation is more subtle: in the limit of small nonlinearity, it is not clear if or when phase correlations will appear, and if they will survive for long times. Such correlations can modify the statistical properties of the system, potentially invalidating the assumptions behind the kinetic description, particularly when these correlations emerge from initially random phases.
A special type of phase correlation that can arise is the so-called anomalous correlator. These correlators, which first appeared in the BCS theory of superconductivity [9], have been studied in S theory [10,11] and are linked to coherent pumping in the system. More recently, anomalous correlators have been observed in nonlinear particle chains, specifically in numerical simulations of the β-FPUT system [12]. However, the origin of these correlators in nonlinear dispersive PDEs remains unclear, and a general theoretical understanding is still lacking.
In this Letter, we show that the emergence of anomalous correlators in Hamiltonian wave systems, characterized by four-wave interaction, is associated with the presence of non-phase-preserving terms; more specifically, we show that terms of the type 3 ↔ 1 are responsible for the phase correlations that ultimately lead to the formation of anomalous correlators. These interactions introduce resonances that cannot be eliminated by normal form transformations (see Appendix E).
Definition of the model—We focus our analysis of anomalous correlators on nonlinear Schrödinger (NLS) equationlike systems. The standard NLS equation describes
dispersive wave systems with four-wave interactions of the type two waves scattering into two waves, which conserves the total number of waves (or wave action). Since conserved quantities in PDEs correspond to symmetries in the system, we modify the NLS equation by introducing terms that break the phase symmetry, thereby violating wave action conservation. However, the Hamiltonian (system’s energy) remains conserved despite these modifications. The Hamiltonian for this system is given by
H ¼ Hlin þ Hnlin;
Hlin ¼	jj∂xjα=2uj2dx;
Hnlin ¼ ϵ	2ν juj4 þ μ½u3u^ þ ðu^Þ3u^ dx; ð1Þ
where u ¼ uðx; tÞ ∈ C with x ∈ ½0; L^, and α, ν, and μ are real and positive constants, with ϵ ≪ 1 serving as the parameter that scales the nonlinearity relative to dispersion. The parameter α sets the linear wave dispersion relation; see Ref. [13]. The additional terms involving μ are responsible for breaking phase invariance of the canonical complex variables [14]. The model reduces to the standard NLS equation when α ¼ 2 and μ ¼ 0. The freedom given by the parameter α is important for the goals of our research: it allows us, for example, to include the dispersion relation of surface gravity waves (α ¼ 1=2). A lot of work is being undertaken on this application [15–18], for which the preservation in time of random phases has not yet been proven.
The WT theory deals with the interaction of a large number of normal modes that, in the case of periodic boundary conditions, correspond to the Fourier modes. Using the following definition of the Fourier transform,
1 L	2π
akðtÞ ¼ 1=2 0 uðx; tÞe iLkxdx;	ð2Þ
the equations of motion associated with the Hamiltonian (1) are an infinite set of coupled ordinary differential equations (ODEs):
da1	ϵ
i ¼ωa þ

νa^2a3a4δ3142þ
2;3;4

þμ

a2a3a4δ2134

þ 3a^2 a^3 a4 δ4123

ð3Þ

dt ω1 a1 þ L where ωk ¼ jkjα, ai ¼ ak ðtÞ ∈ C with ak ≠ a^-k, δ3142……≡ δk1þk2þ^^^;k3þk4þ::. is the Kronecker delta, and the sums over the indices k2, k3 , and k4 extend from -∞ to þ∞.
For our theoretical approach, it is useful to isolate in the sums the diagonal and antidiagonal terms, i.e., those terms for which k1 ¼ k3 and k2 ¼ k4 , k1 ¼ k4 and k2 ¼ k4 (diagonal terms), and k1 ¼ -k2 and k3 ¼ -k4 (antidiagonal terms). The equations of motion now read as follows:

da1
i dt

¼ ω˜ 1 a1 þ ρ1 a^-1 þ X νa^2 a3 a4 δ12 þ
L 2;3;4
þ μ a2a3 a4δ1	þ 3a^2 a^3 a4 δ123	;

ð4Þ

where
ω˜1 ¼ ω1 þ Lϵ	f2νja2j2 þ 6μℜ½a2a-2^g; L2
ρ1 ¼ Lϵ	ð6μja2j2 þ νa2a-2Þ;	ð5Þ L2
and the sums (with a prime) are intended over all terms except for the diagonal and antidiagonal ones. Interestingly, a nonlinear correction to the frequencies beyond the standard one is arising from the non-phase-invariant contribution. In standard WT theory, the main observable is the wave action spectral density function nk ≡ nðk; tÞ, defined as nk ¼ haka^ki (in the limit of L → ∞), where the average is taken over an initial set of data with random phases and amplitudes [19], and the time evolution of nk is described by the WKE [19]. We underline that hakak^i is not the only existing second-order correlator: one can also build hak1 ak2 i. In general, this correlator may appear in nonho- mogeneous conditions. However, if one assumes statistical homogeneity, the only correlation allowed is for k1 ¼ -k2; see proof in Appendix B. This leads to the definition of the so-called anomalous correlator as mk ≡ mðk; tÞ ¼ haka-ki [11]—note that mk ∈ C.
Numerical evidence—Here, we present numerical evidence for the emergence of anomalous correlations in a non-phase-invariant system. Our direct numerical simulations integrate Eq. (4) over time, starting from the initial conditions where Fourier modes have random, independent, and identically distributed (IID) uniform phases. The initial mode amplitudes satisfy jak j2 ¼ n¯ k, with the initial smooth distribution given by n¯ k ¼ Ce-k2=ð2σ2Þ, where σ2 ¼ 40. The constant C is chosen to ensure that the field amplitude in physical space remains Oð1Þ, enforcing juðx; 0Þj2dx ¼ L. We choose the box size to be L ¼ 20π. More details on the numerical implementation can be found in Appendix A.
For the numerical results presented here, we set ϵ ¼ Lp with p ¼ -0.6 [20], along with ν ¼ 1, μ ¼ 0.1, corresponding to a weakly nonlinear regime where the ratio of linear to nonlinear contributions satisfies Hlin =Hnlin ≃ Oð102Þ. The choice of μ fulfills the stability condition μ < ν=3 (see Appendix D). Simulations were performed for various values of α, including α ¼ 1=2, all yielding qualitatively similar results. The results shown here correspond to α ¼ 2; for such value of α, resonant interactions of the type 3 ↔ 1 are present, i.e., there exist four wave numbers such that the two equations below are

satisfied simultaneously (Appendix E):
k1 ¼ k2 þ k3 þ k4; k21 ¼ k22 þ k23 þ k24: ð6Þ
This represents a key distinction between the present results and those discussed in Ref. [12], where the 3 ↔ 1 interactions are nonresonant and can be eliminated by a near-identity (normal form) transformation. Figure 1 illustrates the development of anomalous correlations from random initial data. Figure 1(a) displays the evolution of the anomalous mass, defined as k jmk j2 . The blue curves represent ensemble averages computed over 500 realizations of the initial ensemble. Initially zero, the anomalous mass rapidly grows to nonzero values. For comparison, Fig. 1(b) shows the evolution of the wave action nk, where k nk2 exhibits behavior similar to that of the anomalous mass.
Focusing on the time evolution individual modes, Fig. 1(c) shows the real part of mk for k ¼ 0, which oscillates around a nonzero value. To provide a more complete picture of the development of anomalous correlations, Fig. 1(e) shows the real part of the correlation matrix hak1 ak2 i. This matrix is obtained by averaging over different realizations and subsequently time-averaging over the simulation window. A clear signal appears along the antidiagonal, indicating the spontaneous emergence of anomalous correlations. Notably, for the phase-invariant case (μ ¼ 0), the same simulations show no such antidiagonal correlations, and the anomalous mass remains zero throughout the entire evolution.
Finally, we analyze the system in both space and time by performing a Fourier transform over the time window t ¼½0; 100^. Figure 1(f) shows the magnitude of the space-time Fourier transform, revealing not only the expected positive frequency branch associated with the linear dispersion relation, but also an additional branch corresponding to negative frequencies. The presence of negative frequencies and their connection to anomalous correlators in non-phase-invariant Hamiltonian systems will be further explored in subsequent sections.
Theoretical developments—To provide theoretical understanding of the numerical results, we start by stating the following:
Theorem—A spatially homogeneous phase-invariant Hamiltonian system with initial data distributed according to a phase-invariant measure cannot develop anomalous correlations.
We give a proof of the theorem in Appendix B. Examples of phase-invariant measures include the Gaussian or the random-phase measure (adopted in our numerical simulations). Note that this theorem is valid for any level of nonlinearity; i.e., it is not restricted to the weakly nonlinear regime.
Now, starting from Eq. (4), the goal is to derive evolution equations for the observables nkðtÞ and mkðtÞ previously defined; we will show that, if mkðt ¼ 0Þ¼0, then the anomalous correlators will arise at a timescale of order 1=ϵ (we recall that the timescale of the standard WKE, when a kinetic regime exists, is 1=ϵ2 [5,19]). A key step in the derivation is the generalization of the Wick selection rule by which the fourth-order correlators are split by including the anomalous correlators [21]:
ha^1a^2a3a4i¼n1n2ðδ31δ42 þ δ41δ32Þþm^1m3δ12δ34;
ha^1 a2a3a4i ¼ n1 m3 ðδ21 δ34 þ δ41 δ23 Þþn1 m4 δ31 δ24;

FIG. 1. (a) Time evolution of the anomalous mass Pk jmk j2 , showing its rapid growth from an initially zero state. (b) Evolution of the standard correlator k nk2, which exhibits behavior similar to the anomalous mass. (c) and (d) Evolution of the real and imaginary parts of mk for k ¼ 0, with the real part oscillating around a nonzero value and the imaginary part around zero. In these figures [from (a) to (d)], the blue lines are the result of the ensemble average of numerical simulations of Eq. (4); the orange lines are obtained from the numerical simulations of Eq. (8). The agreement is excellent, even for timescales larger than τ ¼ 1=ϵ ∼ 11. (e) Real part of the correlation matrix hak1 ak2 i, obtained by ensemble and time averaging, showing a strong signal along the antidiagonal, indicating the emergence of anomalous correlations. (f) Magnitude of the space-time Fourier transform of uðx; tÞ over the time window t ¼½0; 100^, revealing both a positive frequency branch associated with the linear dispersion relation and an additional branch corresponding to negative frequencies.

ha1a2a3a4i¼m1m3ðδ12δ34 þ δ14δ23Þþm1m4δ13δ24: ð7Þ
Assuming random phase and amplitude (RPA) initial data [19], the resulting equations at order ϵ are the following (details in Appendix C):
ddntk ¼ 2ℑ½hρkim^k^;
iddmtk ¼ 2hω˜kimk þhρkiðnk þn-kÞ;	ð8Þ
where
hω˜ i¼ω þ ϵ Xf2νn þ 6μℜ½m ^g
k k L2	2	2
hρki ¼ Lϵ	ð6μn2 þνm2Þ:	ð9Þ L2
Note that after some algebra manipulation and the assumption of isotropy in the wave action, nk ¼ n-k ,it is possible to prove the following result:
dd
n2 ¼ jm j2:	ð10Þ dt k dt k
In Fig. 1, panels (a) and (b), the orange lines represent the time evolution of Eq. (8), initialized with the same conditions as the ensemble averages obtained from Eq. (4). This shows that Eq. (8) accurately captures the evolution of nk and mk over the characteristic timescale Oð1=ϵÞ. Moreover, the similarity in behavior observed in both figures is justified by the relation given in Eq. (10). Under the RPA assumption, wehavemkðt ¼ 0Þ¼0while nkðt ¼ 0Þ ≠ 0. Consequently, if μ ¼ 0, anomalous correlators cannot develop, in agreement with the Theorem previously stated. However, at order ϵ, anomalous correlators emerge only if the system lacks phase invariance. This effect arises due to a driving term in the evolution equation for mk, specifically, the term ϵ 2ð6μn2Þðnk þ n-kÞ in Eq. (8). This term acts as a forcing mechanism on the real part of mk, solely dependent on μ and on the standard correlators nk . The presence of this driving term explains the offset in the oscillations of the anomalous correlator seen in Fig. 1(c).
Negative frequencies and anomalous correlators—The set of equations in (3) appears rather complicated; for interpretation purposes, we focus our attention on a minimal model, i.e., a simplified version of our equations that still retains sufficient information that allows us to establish a link between the negative frequencies observed numerically and the existence of anomalous correlators. We propose the following linear set of ODEs:
i ddatk ¼ ω¯kak þ ρ¯a^-k;	ð11Þ
with
ω¯ k ¼ ωk þ 2ϵνA;	ρ¯ ¼ 6ϵμA;	ð12Þ and with A ¼ ð1=LÞ l jalðt ¼ 0Þj2. In our minimal model, we have simplified the original equations in (4) and (5) by neglecting the sums in (4) and the terms in (5) proportional to aka-k . This last approximation is based on the assumption that these terms, which are associated with the anomalous correlator, are small. Moreover, we have taken A to be time-independent (fixed by the initial conditions), assuming that it evolves on a slower timescale with respect to the other terms. Although these assumptions are rather crude and not fully justified, they yield reasonable results (also in the framework of the full model), as will be shown later. The linear system above is diagonalized by the transformation from ak to zk :
ak ¼ zk - ckz^-k;	a^-k ¼ z^-k - ckzk;	ð13Þ
with
ck ¼λkþρ¯ω¯k;	λk ¼ qffiω˜ffiffiffi2kffiffiffi-ffiffiffiffiρffi¯ffiffi2ffi:	ð14Þ
We provide the analytical details of the diagonalization of Eq. (11) in Appendix D. The evolution equations for zk and z^-k are decoupled:
i ddztk ¼ λkzk;	iddz-t1 ¼ λkz^-k;	ð15Þ
with solution
zkðtÞ¼zkð0Þe-iλkt;z^-kðtÞ¼z^-kð0Þeiλkt; ð16Þ which leads to the solution for ak,
akðtÞ ¼ zkð0Þe-iλkt - ckz^-kð0Þeiλkt:	ð17Þ
The ak variables oscillate with two different frequencies, ^λk, one of which is responsible for the negative branch displayed in Fig. 1(f). We remark that if ρ˜ > ω˜ , then λk becomes imaginary, indicating exponential growth. However, the parameter values required for this to occur render the Hamiltonian unbounded from below, and therefore, unphysical (see Appendix D for further discussion).
Note also that these two branches exhibit different intensities due to the presence of two distinct coefficients in Eq. (17). Moreover, it is interesting to note that if there are no correlations at time t ¼ 0, the normal variables zk do not develop them. Conversely, if the anomalous correlator is nonzero at t ¼ 0, it will persist indefinitely.
The diagonalization provides an explanation to a simple mechanism of formation of anomalous correlation rooted in the linear dynamical coupling of the conjugate variable pair fak; a^-k g in Eq. (11). Indeed, posing that
hzkð0Þzkð0Þ^i¼nðkzÞð0Þ;	hzkð0Þz-kð0Þi ¼ 0;	ð18Þ

FIG. 2. The figure illustrates the dispersion relations obtained through spatiotemporal Fourier transform for a single realization of the dynamics shown in Fig. 1. Panel (a) shows the negative frequencies branch of dispersion relation obtained using variables ak . Panel (b) shows the absence of a negative frequency branch when variables zk are used.

FIG. 3. Panel (a) shows the evolution of the anomalous mass evaluated for the variables zk and ak , represented by the blue and red lines, respectively. The yellow line indicates the anomalous mass predicted by Eq. (8). Panels (b) and (c) display the correlation matrix evaluated at the final time for the variables

zk and ak , respectively.

using (17), we have
mkðtÞ ¼ hakðtÞa-kðtÞi ¼ -ck^nðkzÞð0Þþnð-zkÞð0Þ^; ð19Þ i.e., an anomalous correlation is created spontaneously for ak, while it will never exist for the normal variable zk , for which
mðkzÞðtÞ ¼ hzkðtÞz-kðtÞi ¼ mðkzÞð0Þ ¼ 0: ð20Þ
Further numerical validation—The analytical results presented so far are further supported by numerical simulations, as shown in Figs. 2 and 3. Figure 2(a) provides a close-up view of the negative frequency branch previously observed in Fig. 1(f). In contrast, Fig. 2(b) presents the space-time Fourier transform of the action density for the zk variables computed at each time step using Eq. (13), which represents the outcome of our minimal model. Despite the fact that such a model is not derived by a strict mathematical procedure, the numerical results show that the negative frequency branch has disappeared from the plot. To reinforce our findings, we examine the case of initial data that are uncorrelated in the zk variables. In Fig. 3(a), we compare the anomalous mass evaluated using the zk variables with the one obtained for the ak variables. As shown by the plot, the anomalous mass in the zk variables remains nearly zero throughout the time evolution. This result confirms that for uncorrelated initial data in zk, see Fig. 3(b), there is no development of anomalous correlators, as predicted by Eq. (20). On the contrary, our chosen initial data exhibit anomalous correlations in the ak fields; see Fig. 3(c). This is not only evident in Fig. 3 but is also accurately captured by Eq. (8).
Conclusions—In this Letter, we have identified the necessary criterion for the emergence of anomalous correlators in wave systems: the presence of phase-symmetrybreaking terms or the space inhomogeneity of the Hamiltonian. This criterion provides a unifying explanation for the appearance of anomalous correlators in seemingly unrelated systems, such as the FPUT model, BCS theory, and S theory, all of which exhibit explicit phase symmetry breaking in their Hamiltonians. We have provided evidence that anomalous correlators influence the standard correlator hakak^i up to a timescale of 1=ϵ. In addition to the emergence of anomalous correlators haka-ki, we have shown that non-phase-invariant systems also generate negative frequency modes. To address this, we introduced a normal variable transformation, which eliminates negative frequencies and provides a more effective framework for handling anomalous correlators. Notably, if normal variables are initially uncorrelated, this uncorrelation is preserved at times up to 1=ϵ. These findings have important implications for wave turbulence theory. Since standard derivations of the WKE rely on the assumption that random phases remain uncorrelated, the spontaneous generation of anomalous correlators presents a fundamental challenge to the validity of the WKE in systems with phase symmetry breaking. Resolving this issue is a crucial step toward a rigorous derivation of kinetic equations in such systems.
We underline that anomalous correlators are expected to appear in systems such as gravity waves, elastic plates, and capillary waves, all of which are characterized by a Hamiltonian that is not phase invariant. In all these systems, the original variables are real in physical space; however, upon introducing complex normal variables [12], our formulation can always be applied. Additionally, in optical systems like optical fibers, it is possible to engineer nonlinearities that break phase invariance. Therefore, we anticipate that this universal behavior may also be observed in such media. Looking ahead, this Letter lays the foundation for future research aimed at reconciling kinetic descriptions with the effects of phase symmetry breaking, ultimately leading to a deeper understanding of wave turbulence in non-phase-invariant systems.
Acknowledgments—M. O. was funded by Progetti di Ricerca di Interesse Nazionale (PRIN), Projects No. 2020X4T57A and No. 2022WKRYNL, project “Mathematical Methods in Non-Linear Physics” and “FieldTurb,” Istituto Nazionale di Fisica Nucleare. The Simons Foundation, United States, Award No. 652354 on Wave Turbulence is acknowledged for support.
Data availability—The data that support the findings of this article are not publicly available. The data are available from the authors upon reasonable request.
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End Matter

Appendix A: Numerical details—Numerically, we solve the following equation of motion:
∂u
i ¼ P½∂αu þ ϵðνPðjuj2ÞþμPððu^Þ2Þþ3μPðu3ÞÞu^;
∂t	x
ðA1Þ
where P is the Galerkin projector, which acts in Fourier space by filtering out all jkj modes higher than j2=3kmax j, where kmax ¼ π=Δx, with Δx the resolution in physical space. The numerical integration is performed pseudospectrally: the linear part of Eq. (A1) is solved in Fourier space, while the nonlinear part is handled using a fourth-order Runge-Kutta method. The time step is chosen to be at least 10 times smaller than the fastest linear oscillation.
Appendix B: Proof of the theorem for phase-invariant systems—Let γ ¼ fγkg, with k ∈ ð2π=LÞZ, be complexvalued IID random variables, and let fðx; γÞ¼ k γkpffin¯ffiffikffiffieikx denote a random field. Assume that the
measure of γk ðdFÞ is phase invariant,
hGðeiαzÞi ¼ hGðzÞi;	ðB1Þ where hGðzÞi ≔ R GðzÞdFðzÞ. This is the case of z ¼ z1 þ iz2 and dF ¼ ð1=pffi2ffiffiπffiffiÞeijzj2=2dz1dz2, or z ¼ eiθ and dF ¼ dθ. If uðx; t; γÞ¼ k ak ðt; γÞeikx solves a translation-invariant equation with initial value akð0;γÞ ¼ pffin¯ffiffikffiffiγk, then uðxþα;t;γÞ¼Pkakðt;γÞeikðxþαÞ ¼ k eikx ak ðt; γ Þeikα solves the equation with initial data Pk eikxpffin¯ffiffikffiffiγkeikα. Consequently, akðt;γαÞ¼eikαakðt;γÞ, with γα ¼ eikαγk. Therefore, heiðk-lÞαak ðt; γÞal^ðt; γÞi ¼ hakðt; γαÞal^ðt; γαÞi ¼ hakðt; γÞal^ðt; γÞi, where the last equivalence is true thanks to the phase invariance of the measure. Thus,
hakðt;γÞal^ðt;γÞi ¼ nkðtÞδðk - lÞ:	ðB2Þ
Similarly,
hakðt; γÞalðt; γÞi ¼ mkðtÞδðk þ lÞ:	ðB3Þ
Note that, since fγkg are IID, mkð0Þ ¼ 0, because hγkγ-ki ¼ 0. If the evolution is phase invariant, i.e.,
akðt; eiαγÞ ¼ eiαakðt; γÞ;	ðB4Þ then
e2iαmkðtÞ ¼ e2iαhakðt; γÞa-kðt; γÞi
¼ hakðt; γαÞa-kðt; γαÞi ¼ mkðtÞ:	ðB5Þ
Therefore, for fγkg IID, phase-invariant measure and phase-invariant evolution, mkðtÞ ¼ 0 for any t. Note that if fγ g are IID with phase-invariant law, but the k, evolution is not phase invariant, even if mkð0Þ ¼ 0, mkðtÞ is not necessarily zero for t>0.
Appendix C: Derivation of Eq. (8)—Here, we derive the two Eq. (8) starting from Eq. (4). Let us multiply (4) by a^1, and then subtract the complex conjugate of (4) multiplied by a1. This yields
d2	2	2	^^
i	ja1 j ¼ ω˜ 1ðja1 j - ja1 j Þþρ1 ða1 a-1 Þ - ρ a1a-1 þ o:t:

¼ 2iℑ½ρ1 ða1a-1Þ^^þo:t:

ðC1Þ

dt	1 where o.t. indicates the remainder of off-diagonal terms.
Now, let us multiply (4) by a-1, and then add (4) but replacing k1 with -k1, and multiplied by a1. We obtain
d
i dt ða1a-1Þ ¼ 2ω˜ 1 ða1a-1Þ þ ρ1 ja-1 j2 þ ρ-1ja1 j2 þ o:t:
¼ 2iℑ½ρ1ða1a-1Þ^^ þ o:t:	ðC2Þ Averaging over initial data and assuming that the system remains spatially homogeneous for later times, then we obtain Eq. (8).

du
i ¼ ωu þ ρv; dt

dv
i	¼ -ωv - ρu;

ðD1Þ

Appendix D: Diagonalization of the linear non-phaseinvariant dynamics—Rewrite the linear system (11) in simplified notation as or in matrix form as

id u ¼ ω
dt v -ρ¯

ρ	u:
-ω	v

ðD2Þ

Here, we use the following shorthand notation: u ¼ ak , v ¼ a^-k, ρ¯ ¼ 6ϵμA, ω¯ ¼ ω þ 2ϵνA. The matrix has eigenvalues ^λ, with λ ¼ ω¯ 2 - ρ¯2, with eigenvectors given, respectively, by
1	-c
e1 ¼	;e2 ¼	; with c ¼
-c	1

ρ : ðD3Þ λþω

Notice that the symmetry between e1 and e2 is due to the peculiar relationship between ρ¯ , λ, and ω¯ , by which ðλ - ω¯ Þ=ρ¯ ¼ -ρ¯ =ðλ þ ω¯ Þ¼-c. The diagonalizing change of variables is thus given by

uv

1	-c	z
-c 1	w ;

ðD4Þ

so that u ¼ z - cw and z ¼ ½1=1 - c2^ðu þ cvÞ. Finally, notice that for λ to be real when ω ¼ 0, we need the condition μ < ν=3. This is a condition for the linear stability that must be satisfied by the Hamiltonian (1). We remark that such condition is, however, less stringent than the condition for the Hamiltonian to be bounded from below, μ < ν=4. To see that, let us perform the transformation u ¼ reiθ. In this representation, the quartic part of the Hamiltonian density reduces to
νν
juj4 þ μðu3 u^ þ u^3uÞ ¼ r4	þ 2μ cos 2θ :
22
Minimizing with respect to the phase θ [i.e., when cosð2θÞ ¼ -1] yields
ν
- 2jμj:
2
Therefore, for the Hamiltonian to be bounded from below, μ must satisfy the condition
ν
μ<4:
From the linear stability analysis, however, we obtained a weaker condition, namely μ < ν=3. Thus, in the regime μ > ν=3, the Hamiltonian becomes unbounded from below, and the model is unphysical.
Appendix E: 3 ↔ 1 resonances—The non-phaseinvariant terms in (1) contain nontrivial resonances. Take three integers l, m, and n, and let k1 ¼ lðn2 þnmþm2Þ,
k2 ¼ lmðm þ nÞ, k3 ¼ lnðm þ nÞ. Note that k2 þ k3 ¼ lðn þ mÞ2. Now, using the wave number resonance condition, the fourth wave number reads k4 ¼ k1 - k2 - k3 . For the frequency resonance condition to be fulfilled, it must be k24 ¼ k21 - k22 - k23 . This is true if ðk1 - k2 - k3Þ2 - k21 þ k22 þ k23 ¼ 0, which is readily verified by direct substitution in terms of l; m; n.