Noncommutative Field Theory of the Tkachenko Mode: Symmetries and Decay Rate

We construct an effective field theory describing the collective Tkachenko oscillation mode of a vortex lattice in a two-dimensional rotating Bose-Einstein condensate in the long-wavelength regime. The theory has the form of a noncommutative field theory of a Nambu-Goldstone boson, which exhibits a noncommutative version of dipole symmetry. From the effective field theory, we show that, at zero temperature, the decay width $\Gamma$ of the Tkachenko mode scales with its energy $E$ as $\Gamma\sim E^3$ in the low-energy limit. We also discuss the width of the Tkachenko mode at a small temperature.

Introduction.-When a superfluid rotates, a lattice of quantized vortices forms.The oscillations of the vortex lattice, the so-called Tkachenko mode [1][2][3] (for a recent review, see Ref. [4]), has many distinctive properties.Unlike ordinary sound waves in a solid, at low momenta, the Tkachenko wave has a quadratic dispersion relation ω ∼ q 2 and only one polarization [5][6][7].The Tkachenko mode is a consequence of a rather intricate realization of spontaneous symmetry breaking: there are many symmetries broken by the superfluid vortex lattice, but only one Nambu-Goldstone boson (NGB) [8,9].The Tkachenko mode should exist in rotating superfluid 4 He, but it has been observed most conclusively in the rotating Bose-Einstein condensate of ultracold atoms [10].At a much larger length scale, the Tkachenko mode has been suggested to be the source of an oscillation mode of the Crab pulsar [11].
As the Tkachenko mode is the only low-energy degree of freedom, one expects that it can be described by an effective field theory (EFT) which involves a single field.However, up to now, a complete understanding of the structure of such a theory has yet to be achieved.At the quadratic level, the effective Lagrangian [8] coincides with that for a Lifshitz scalar [12], but the form of the interaction terms in the Lagrangian and how they are constrained by symmetries are not known.These interaction terms are needed to calculate the decay rate of the Tkachenko mode [13].
In this Letter, we show that noncommutative field theory (see, e.g., Refs.[14,15]) provides a convenient framework for constructing the effective field theory of the Tkachenko mode.That noncommutative field theory (NCFT) may be applicable to the problem is intuitively understandable-rotating a nonrelativistic system is formally equivalent to placing it in a magnetic field, and on the lowest Landau level (LLL) the guiding-center coordinates do not commute.Because of that, NCFT has often been invoked in the context of the quantum Hall effect [16][17][18][19][20][21][22].Vortex lattices can also be realized on the LLL [23][24][25].As we will see, in the case of the Tkachenko mode, NCFT provides a way to organize terms in the Lagrangian consistent with symmetries.Following the formalism, we are able to determine the general structure of the interacting Lagrangian, and from there, that the decay rate of a Tkachenko mode (at zero temperature) scales like the cube of its energy, Γ ∼ E 3 . ( This implies, in particular, that the Tkachenko mode becomes a more and more well-defined quasiparticle (i.e., Γ/E → 0) as the energy E approaches zero.We will also establish a connection between the Tkachenko mode and the "dipole" symmetry, which recently became a popular topic (see, e.g., Refs [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]).The Tkachenko mode realizes a more complex version of dipole symmetry: the magnetic translations, which form a nonabelian group.
Tkachenko mode as a noncommutative Nambu-Goldstone boson.-Onecan arrive at the theory of the Tkachenko mode from microscopic considerations, taking, for example, as the starting point the microscopic theory of bosons with short-range repulsive interactions and then eliminating all redundant degrees of freedom [8].It is instructive, however, to derive the most general form of the effective Lagrangian, relying solely on symmetries.Such an approach has the advantage of being applicable for strongly correlated rotating superfluids where microscopic calculations are not reliable, e.g., close to a quantum melting transition of the vortex crystal [43].
We first note that the lattice of vortices can be described, as a two-dimensional solid, by two (a = 1, 2) scalar fields X a (t, x i ); they present the coordinates frozen in the solid [44,45].In this description, in Cartesian coordinates the lattice displacement u a is related to X a by X a = x a − u a .The vortex current is related to X a by arXiv:2212.08671v2 [cond-mat.str-el]14 Feb 2024 where n 0 is the equilibrium vortex density.For a superfluid under rotation with angular velocity Ω, n 0 = 1 2π B, where the effective magnetic field B = 2mΩ with m being the mass of the elementary boson.
In a superfluid, the vortices carry charge with respect to the u(1) dynamical gauge field a µ dual to the superfluid Nambu-Goldstone boson [46,47].The boson particle number current is expressed as j µ = 1 2π ϵ µνλ ∂ ν a λ .The Lagrangian of the system contains terms that describe the coupling of the vortex current with the dual u(1) gauge field and the kinetic and potential terms of the latter, where e i = ∂ 0 a i − ∂ i a 0 , and b = ϵ ij ∂ i a j .In the above equation, me 2 /(4πb) represents the kinetic energy of the superfluid condensate, ϵ(b/2π) is the internal energy as a function of the density, and A µ is the gauge potential of the external effective magnetic field B. In the lowest Landau level limit m → 0, the kinetic term vanishes.
In fact, this term can be dropped if one is eventually interested in the limit ω ∼ q 2 : in this regime e 2 ≪ b 2 .
Without the e 2 term, variation with respect to a 0 give a constraint That means the map from x i to X a is area-preserving.
To linear order in the displacement u i , Eq. (4) implies ∂ i u i = 0, i.e., the displacement is divergence free: the Tkachenko mode is a transverse sound.
The quadratic theory of this transverse sound is analyzed in the Supplement Material (SM) [9].Here we would like to resolve the constraint (4) at the nonlinear level.This can be done iteratively, as worked out in the SM [9].Here we take a more elegant approach: on the LLL, one expects the spatial coordinates x and y to become noncommutative (see e.g., Refs.[16,20]), ( where ℓ = 1/ √ B is the magnetic length.The quantum version of Eq. ( 4) then can be written as We then conclude that Xa and xa are related by a unitary transformation, Xa = e i φ xa e −i φ, (7) where the operator φ is an arbitrary function of the two noncommuting coordinates x and ŷ.In noncommutative field theories [14,15], any operator corresponds to a Weyl symbol which is a function in space, and the above equation becomes Here the star product is defined as . To linear order in ϕ the displacement u a is then where {{•, •}} denotes the Moyal bracket, {{f, g}} As expected, to this order, the displacement is purely transverse.To all orders in ϕ, Eq. ( 8) can be written as where U = e iϕ , U −1 = e −iϕ and Thus, we identify the Tkachenko mode with a Nambu-Goldstone boson of a noncommutative field theory.We now show that this field is a compact scalar that shifts under the particle number U(1) symmetry.Magnetic translations as noncommutative dipole symmetry-On the LLL, translations are magnetic translations and do not commute: where Q denotes the boson particle number operator.
In our case, the Tkachenko mode is the only low-energy degree of freedom, so it should provide a nontrivial representation of magnetic translations.In the noncommutative theory, translations are realized as a special class of unitary transformations that are exponents of a linear function of coordinate.Acting on Xa , such a transformation changes the Weyl symbol of the latter as with ξ i = −θϵ ij α j .This is a spatial translation by ⃗ ξ.Viewing X a as fields in a field theory, such a translation is supposed to be generated by Xa → e Thus, we can identify the magnetic translation as [48] Pi According to Eqs. ( 8) and ( 13) and the associativity of the star product, magnetic translations by ⃗ c act on the Tkachenko field ϕ as multiplication on the left This allows us to interpret the action of magnetic translations on a Tkachenko field as a noncommutative version of a dipole symmetry.Expanding in ϕ, Eq. ( 15) reads To leading order, these are simply a dipole symmetry transformation ϕ → ϕ + α i x i with the parameter α i = θ −1 ϵ ij c j , but in addition, there are an infinite number of terms composed of derivatives acting on fields ϕ.These terms make the magnetic translations noncommuting, as in Eq. ( 12).Knowing the transformation law for ϕ under magnetic translations, we can find the transformation law for ϕ under particle number U (1) symmetry.Apply four translations on e iϕ , one after another: e −iβ Py e −iα Px e iβ Py e iα Px , from Eq. ( 15) we see that e iϕ becomes e i(ϕ+ αβ ℓ 2 ) .But we also know from Eq. ( 12) that the product of the four translations is e i αβ ℓ 2 Q.Thus, under U(1) charge, the Tkachenko field transforms exactly as the phase of the superfluid condensate: ϕ → ϕ + c.The Tkachenko field, therefore, has a dual role: it is the condensate phase, but at the same time, its gradient is the lattice displacement.Such a dual role is possible, of course, because at low energy, the condensate phase is entirely determined by the configuration of the vortex lattice.By the "condensate phase," one should have in mind the regular part of the phase where the singular contributions from the vortices have been subtracted away.
As a condensate phase, ϕ then should be a compact scalar field with periodicity 2π: ϕ ∼ ϕ + 2π.The periodicity of ϕ can also be seen from the following argument.Let us put the system on a torus of size L x × L y .Then the magnetic field breaks translation symmetry along the x direction to a discrete group of finite translations generated by x → x + 2πℓ 2 /L y (which can be seen by computing the Wilson line of the gauge field along a curve wrapping the torus along the y direction at fixed x).This discrete translation is generated by the operator e 2πiŷ/Ly under which ϕ → ϕ + 2πy/L y + • • • .This is allowed only when the identification ϕ ∼ ϕ + 2π is valid.
Ingredients for a Lagrangian.-We now write down the most general Lagrangian consistent with symmetries for the field U = e iϕ .The symmetries include global U (1) phase rotations U → e iα U , global magnetic translations U → e i⃗ α•⃗ x ⋆ U (noncommutative dipole symmetry),and global rotation U → e i 2 ωx 2 ⋆ U .The structures that are covariant (i.e., transforming like O → e i⃗ α•⃗ x ⋆ O ⋆ e −i⃗ α•⃗ x , etc.) under these transformations are where D i ϕ is defined as in Eq. (11).Note that D ab ϕ is symmetric and traceless [49].These can be expanded infinite series over ϕ.These series have the property that, at the order ϕ n with a given integer n, the leading terms (in derivatives) have 2n derivatives if one count ∂ t as two derivatives, ∂ t ∼ ∂ 2 i .This counting is natural as the Tkachenko mode, which is the only low-energy degree of freedom, has a quadratic dispersion.Keeping at each power of ϕ only terms with the minimal number of derivatives, we have Effective Lagrangian.-We can now write down the Lagrangian of the Tkachenko mode, keeping at each power of ϕ terms with the minimal number of derivatives, counting each occurrence of ∂ t as two derivatives.This Lagrangian would allow one to compute the rate of any scattering process to leading order over the momenta of the particles involved, similarly to the nonlinear Lagrangians for superfluids [50] or solids [44,45].In the SM [9] we explicitly derive a nonlinear effective theory of the Tkachenko field ϕ from the leading-order effective theory of a vortex lattice introduced in Refs.[51,52].
The most general Lagrangian consistent with the U (1) and magnetic translation symmetries is a function of the invariant structures defined above: The form of the Lagrangian can be restricted further by imposing additional symmetries.In particular, assuming the vortex lattice is a triangular lattice, one should expect the C 6 group of rotations by angles multiple of 2π 6 .Introducing the complex coordinate z = x + iy, the rotationally invariant structures are now A system of particles in a magnetic field has an antiunitary RT symmetry that combines spatial reflection (R) and time reversal (T ): Under this symmetry, ϕ → −ϕ, which can be seen from its connection to the displacement u a in Eq. ( 9).Among the C 6 invariants in Eq. ( 20), Re(D zz ϕ) 3 is odd, while the rest are even.Thus the most general effective Lagrangian is a function of four arguments, The Girvin-MacDonald-Platzman (GMP) algebra.-TheNCFT construction realizes a key feature of the LLL-the GMP algebra [53].Indeed, upon canonical quantization, the particle number density n = −δS/δ(D 0 ϕ) realizes the NC U(1) gauge transformation, i.e., where δ λ O is the infinitesimal change of O under the gauge transformation, under which e iϕ → e iλ ⋆ e iϕ .But the gauge transformations do not commute: [δ α , δ β ] = δ {{α, β}} .From this, one derives the GMP algebra satisfied by n(x).This is confirmed by explicit calculation in the SM [9].Quadratic Lagrangian.-Theonly terms that contribute to the quadratic Lagrangian are (D 0 ϕ) 2 and (D ij ϕ) 2 .Modulo a total derivative, the quadratic Lagrangian is that of the quantum Lifshitz model [12] which corresponds to a quadratic dispersion relation ω ∼ q 2 , see also the SM [9] for the explicit expression for the coefficients c 0 and c 1 .This quadratic dispersion relation is protected by the magnetic translation symmetry [8].
From Eq. ( 24), one easily reproduces the power-law behavior of the correlation function of the superfluid order parameter at large distances, first found in Ref. [24] (see also Refs.[35,37,54]).Decay width of the Tkachenko mode.-Thequadratic dispersion relation of the Tkachenko mode allows a decay of one Tkachenko quantum into two quanta.To find the rate of such decay, we need to determine the interaction vertices cubic in the field ϕ.It is easy to see that, even as cubic terms appear when one expands the "quadratic" terms (D 0 ϕ) 2 and (D ij ϕ) 2 to cubic order in ϕ, these terms are total derivatives.The real cubic interaction appears from the following terms in the Lagrangian: (D 0 ϕ) 3 , D 0 ϕ, (D ij ϕ) 2 , and Im(D zz ϕ) 3 .Up to a total derivative, the cubic Lagrangian has the form From this, one easily finds the energy dependence of the decay width of the Tkachenko mode.All the cubic interaction terms scale the same way in the scaling scheme with ∂ 0 ∼ ∂ 2 i .In this scheme, ϕ is dimensionless and the g's have dimension p −2 ∼ E −1 .The decay width Γ is proportional to g 2 , and to have the correct dimension, Γ should scale as ∼ g 2 E 3 .This can be confirmed by writing down the decay rate of the Tkachenko mode: Estimating the integral with p ∼ q, M ∼ gq 6 , we get Γ q ∼ g 2 q 6 ∼ g 2 E 3 .The presence of an anisotropic cubic vertex means that the decay rate depends on the direction of the momentum of the decaying particle.At small but finite temperature T , the U (1) condensate phase disappears, but the order parameter of translation symmetry breaking ∂ i ϕ has a logarithmic correlation at long distances [55] (see also Refs.[35,37]).Below the Berezinskii-Kosterlitz-Thouless phase transition where the lattice melts, the Tkachenko mode should still exist.The 1 → 2 decay rate ( 26) is modified for modes with energy much less than T by the factor (1+f p +f q−p ) where f p and f q−p are the occupation numbers in the final state.For E ≪ T , this factor is of order T /E, hence the 1 → 2 rate is now g 2 T E 2 for E ≪ T .However, the dominant contribution to the width is now a different process: the "Landau damping" process, i.e., the absorption of the soft Tkachenko quantum by a hard thermal Tkachenko photon in the medium: The width of the Tkachenko mode due to this process is g 2 (T E) 3/2 , which means that the Tkachenko mode is still a well-defined resonance.
The estimate above assumes that the hard Tkachenko quanta participating in the scattering process has no width and is valid only when the energy of the Tkachenko mode under consideration is larger than the width of a typical thermal mode, which is, by dimensional analysis, g 2 T 3 .Thus the estimate Γ(E) ∼ g 2 (T E) 3/2 is valid in the interval g 2 T 3 ≪ E ≪ T .The regime E ≪ g 2 T 3 is the hydrodynamic regime, the analysis of which we defer to future work.
We note that our formulas for the width of the Tkachenko mode, both at zero and nonzero temperature, are in conflict with a previous result obtained from a microscopic calculation [13].For bosons on the LLL, the authors of Ref. [13] found that at zero temperature ratio of the width to the energy of the Tkachenko mode is a constant independent of the energy (which depends only on the filling factor), and at nonzero temperature the mode is overdamped.The results are untypical for a NGB, and we cannot reconcile them with the symmetries of the system.This discrepancy needs to be investigated further.
Conclusion.-In this Letter, we have provided a new interpretation of the Tkachenko mode in a rotating superfluid: it is a noncommutative Nambu-Goldstone boson that arises from the breaking of U (1) and translation symmetries.Noncommutative field theory provides a convenient way to impose the invariance of the theory with respect to U (1) and magnetic translations, and the resulting theory gives us a prediction for the decay width of the Tkachenko mode at low momentum.
-Supplementary Material -Noncommutative Field Theory of the Tkachenko Mode: Symmetries and Decay Rate Yi-Hsien Du, Sergej Moroz, Dung Xuan Nguyen, and Dam Thanh Son

REDUNDANCIES OF SPONTANEOUS BROKEN SYMMETRIES
We believe that the most transparent understanding of spontaneous symmetry breaking and the Goldstone boson counting for a two-dimensional superfluid vortex crystal was obtained by Watanabe and Murayama in Ref. [8].Here we summarize their explanation adopted to the lowest Landau level regime.
Emergence of a vortex lattice ground state in a rotating superfluid breaks spontaneously global particle number U (1) symmetry, magnetic translation symmetry, and magnetic rotation symmetry.Notwithstanding, the generators of all these symmetries are linearly related to each other.In particular, the momentum density T 0i is given by where j i and n are the boson current and particle densities, respectively, and B is the effective magnetic field originating from the rotation [57].In the massless regime m → 0, where only the lowest Landau level states survive, we can ignore the first term in Eq. (S1), so the momentum density operator and the particle density operator are proportional to each other.Furthermore, we can define the angular momentum density as J = ϵ ij x i T 0j which is also simply related to the boson density operator J = B⃗ x 2 n.As a result, the densities of all symmetries that are spontaneously broken are not independent, but are linearly related to each other.Therefore we only have a single Goldstone boson, which is the Tkachenko mode.

Linearized effective Lagrangian
Our departure point is the low-energy linearized effective theory of a two-dimensional superfluid vortex lattice introduced in Refs.[51,52].We consider a system of bosons with density n 0 placed in a constant magnetic field B. This magnetic field may be effectively created by rotating the system with angular frequency B/(2m), at the same time putting it in a harmonic trap with the trap frequency fine-tuned to cancel the centrifugal force.The lattice is parametrized by the displacement field u i , i = x, y, while the superfluid is characterized by the dual u(1) gauge field a µ .The Tkachenko mode emerges as the result of the mixing between the of elastic waves on the vortex lattice and the superfluid fluctuations.We start from the leading-order (LO) quadratic Lagrangian linearized around the vortex crystal ground state [58] el (∂u).(S2) The formula for the particle number spacetime current in terms of the gauge field, j µ = δS/δA µ = 1 2π ϵ µνρ ∂ ν a ρ , relates the particle number density n with the magnetic field b and the particle number current with the electric field e i = ∂ t a i − ∂ i a t .The term with one time derivative, proportional to ϵ ij u i uj , encodes the Berry phase that a vortex acquires when moving in a superfluid.This term gives rise to the "Magnus force" acting on the vortex.The elastic energy density E (2) el (∂u) is a function of the linearized strain tensor u ij = 1 2 (∂ i u j + ∂ j u i ).The superfluid internal energy is a function of the superfluid density n = 1 2π b and here is expanded around the ground state value n 0 to quadratic order in fluctuations δb = b − 2πn 0 .Finally, we included the coupling to an external U (1) source A µ which is set to vanish in the ground state [51].
The quadratic Lagrangian (S2) can be easily obtained from the Lagrangian (3) in the main text [59].To this end following [51], we substitute into Eq.( 3) the definition of the vortex current (2), and the definition of X a in terms of u a , see Fig. S1.We then expand the action up to the quadratic order in the field fluctuations and arrive at the linearized action (S2) with the fluctuation of the vector potential source A µ = A µ − Āµ on top of the background Āµ that produces the constant background magnetic field B.
in the second integral.So the displacement velocity is measured with respect to the LLL drift that has the velocity −ϵ ij E j /B.As a result, the action is invariant under Galilean boosts.We thus have the effective action where B = ϵ ij ∂ i A j is a variation of the magnetic field on top of the constant background B. In the presence of a such inhomogenity, the Gauss law is ∂ i u i = −B/B, so the crystal becomes compressible.In momentum space (our conventions: ∂ i → ik i and ∂ t → −iω), this Gauss law can be resolved as u i k = −θϵ i j (ik j ϕ k − A j k ).The way the dimensionless scalar ϕ couples to the U (1) source suggest that in addition to fixing the transverse displacement of the vortex crystal, ϕ also represents the regular part of the superfluid phase of the Bose-Einstein condensate.The latter interpretation was the key point for Watanabe and Murayama, who developed the effective theory of the superfluid vortex crystal in Ref. [8].Now we are ready to write the generalization of the effective theory (S4) in the presence of the U (1) source.The simplest result is obtained, when one considers a special type of the source with vanishing B which thus does not violate the incompressibility condition.To this end, we will set A i = 0.In that case, after resolving the Gauss law, we find the quadratic effective action for the ϕ fluctuation where k = (ω, k).
We will extract now the density susceptibility χ k that (up to a sign) is just the correlation function ⟨n −k n k ⟩.To this end, we first compute the superfuid density substitute into it the solution of the equation of motion for ϕ −k and get Finally, we differentiate the result with respect to A 0 k to get This agrees with the result of Ref. [51].More generally, the linear electromagnetic responses extracted from the effective action for the Tkachenko field ϕ are expected to agree with the massless LLL limit of the results derived in Ref. [51].This amounts to discarding the contribution originating from the Kohn's mode [61].

Hamiltonian formulation
Starting from the Lagrangian (S4), the canonical momentum conjugate to ϕ is which according to Eq. ( S8) is (minus) the superfluid density π ϕ = −n.So ϕ is indeed the superfluid phase field.The Hamiltonian density can now be computed to be Using now the canonical Poisson bracket [ϕ(x), π ϕ (y)] = δ(x − y), we end up with the Hamiltonian equations of motion We observe that the time-evolution of the Tkachenko field ϕ is fixed by the superfluid density, while the time-evolution of the latter is fixed by the fourth-spacial derivative of ϕ.

Constant magnetic field
As argued in the main text, in a constant effective magnetic field B = 2mΩ, the scalar fields X a must satisfy the non-linear constraint To resolve the constraint, introduce an auxiliary Poisson bracket As the consequence of the constraint (S15), the fields X a also satisfy Thus the transformation from x i to X a belongs to the group of canonical transformation and can be generated by a scalar function ϕ Notice that this expression of X a is identical with Eq. ( 7) in the main text.As a superfluid order phase, the scalar generator ϕ is even under 2d parity x ↔ y and odd under time reversal t → −t.
Here we comment on two natural ways how to organize the derivative expansion in terms of ϕ: If we scale ϕ ∼ O(ϵ −1 ) and ∂ i ∼ O(ϵ), higher order non-linearities in the expansion (S19) are systematically suppressed.On the other hand, in order to include all non-linearities on equal footing, we can scale ϕ ∼ O(ϵ −2 ) and ∂ i ∼ O(ϵ).In this way, all non-linear terms in Eq. (S19) are of the same order.

Inhomogeneous magnetic field
It is possible to generalize the above construction to the case, where the effective magnetic field is not constant.The constraint to be resolved is now where the non-linear terms originate from the non-commutativity of the area-preserving diffeomorphisms [69].It is straightforward to check that these transformations realize the canonical w ∞ algebra on the Tkachenko field ϕ, namely [δ α , δ β ]ϕ = δ {α,β} θ ϕ.
One can check that up and including the leading order non-linearity, the action built from the Lagrangian (S36) is invariant under the combination of (S46) and (S47).Indeed, the variation of the first and second terms of (S36) is a total derivative [70], while the elasticity energy density is invariant on its own.Notably, this invariance automatically insures the emergence of the LLL GMP algebra [56] that we derived explicitly above.Moreover, it implies that in the LLL limit the charge current density can be expressed as a derivative of the stress tensor and the charge conservation law has a higher-rank form that arises naturally in higher-rank tensor gauge theories coupled to fractons [26,71].
Beyond leading-order theory (S28) Following the power-counting scheme (S30), one can systematically add sub-leading symmetry-allowed terms to the LO effective theory Lagrangian (S28).In fact, some next-to-leading (NLO) terms have already been investigated before.
In particular, already in Ref. [51], the non-linear NLO term me 2 i /(4πb) (whose form is fixed by Galilean invariance) has been incorporated.Here e i = ∂ t a i − ∂ i a t is the dual electric field that encodes the superfluid current, and m is the mass of the elementary boson.This term allows us to go beyond the LLL approximation and incorporate some effects of higher Landau levels into the low-energy description of the superfluid vortex lattice.In particular, the inclusion of this term gives rise to the finite-frequency Kohn mode in the EFT excitation spectrum, which correspondingly modifies the U (1) linear response [51].
Another NLO term that breaks time-reversal symmetry and has the form e i ∂ i b/(4πb) has been discussed in Ref. [52].Given that this term does not depend on the mass m of the boson, it survives in the LLL limit and incorporates higher-order corrections to the LLL coarse-grained description developed above.In particular, we expect that this term is responsible for leading-order non-linear corrections to the low-momentum GMP algebra (S45).
We expect that the NLO terms discussed above will generate corrections to the decay rate Γ of the Tkachenko mode.Those however will disappear faster in the limit E → 0 than the leading-order result Γ ∼ E 3 that we discovered in this paper.
Notice that either of the NLO terms mentioned here modify the Gauss law constraint (S15) and make the vortex crystal compressible.As a result, the resolution of the constraint by a canonical transformation (S19) is not applicable anymore.Nevertheless, it should be possible to resolve the modified constraint by generalizing the method used in Sec. .