Quantum Paracrystalline Shear Modes of the Electron Liquid

Unlike classical fluids, a quantum Fermi liquid can support a long-lived and propagating shear sound wave, reminiscent of the transverse sound in crystals, despite lacking any form of long-range crystalline order. This mode is expected to be present in moderately interacting metals where the quasiparticle mass is renormalized to be more than twice the bare mass in two-dimensions, but it has remained undetectedbecause it is hard to excite since it does not involve charge density fluctuations, in contrast to the conventional plasma mode. In this work we propose a strategy to excite and detect this unconventional mode in clean metallic channels. We show that the shear sound is responsible for the appearance of sharp dips in the AC conductance of narrow channels at resonant frequencies matching its dispersion. The liquid resonates while minimizing its dissipation in an analogous fashion to a sliding crystal. Ultra-clean 2D materials that can be tuned towards the Wigner crystallization transition such as Silicon MOSFETs, MgZnO/ZnO, p-GaAs and AlAs quantum wells are promising platforms to experimentally discover the shear sound.

Unlike classical fluids, a quantum Fermi liquid can support a long-lived and propagating shear sound wave, reminiscent of the transverse sound in crystals, despite lacking any form of long-range crystalline order. This mode is expected to be present in moderately interacting metals where the quasiparticle mass is renormalized to be more than twice the bare mass in two-dimensions, but it has remained undetectedbecause it is hard to excite since it does not involve charge density fluctuations, in contrast to the conventional plasma mode. In this work we propose a strategy to excite and detect this unconventional mode in clean metallic channels. We show that the shear sound is responsible for the appearance of sharp dips in the AC conductance of narrow channels at resonant frequencies matching its dispersion. The liquid resonates while minimizing its dissipation in an analogous fashion to a sliding crystal. Ultra-clean 2D materials that can be tuned towards the Wigner crystallization transition such as Silicon MOSFETs, MgZnO/ZnO, p-GaAs and AlAs quantum wells are promising platforms to experimentally discover the shear sound.
Ordinary classical fluids only display one kind of sound waves that correspond to longitudinal compressional oscillations of the fluid 1 . On the other hand classical solids display transverse waves as well, which originate from their finite restoring force to shear deformations 2 . Quantum Fermi fluids can dramatically differ from this paradigm by displaying long-lived and propagating collective shear sound waves while lacking any form of static crystalline order 3-6 .
To this date there is no report of the observation of these shear sound waves of electrons in metals, and a pioneering attempt to detect them in 3 He 7 remained inconclusive 8 . However, the appearance of these modes requires only a moderate interaction strength, in the sense that they are expected to become sharp when the quasiparticle mass becomes approximately twice and three times the transport mass in two-and three-dimensions respectively 5 . Therefore it is possible that these elusive collective modes are actually present in a variety of electron liquids but they have remained undetected so far because their transverse nature makes them unresponsive to charge-sensitive probes.
In this letter, we demonstrate that shear modes leave clear fingerprints in the conductivity of clean metallic channels. Our idealized setup is depicted in Fig. 1a, where a uniform ac electric field generates an alternating current along the y direction. In a clean channel, the current can only be damped at the boundary. This is illustrated by the current profile shown in Fig. 1b-c, which is suppressed at the boundaries due to friction. The current magnitude varies in a direction transverse to the electron flow signaling the excitation of shear modes.
The central result of our work is summarized in Fig. 1d, which shows the conductance of the strip as a function of frequency. When scattering due to impurities or electron-electron collisions is weak, the conductance exhibits sharp dips at frequencies ω = nω 0 , where ω 0 is the shear sound frequency at momentum 2π/W determined by the width W of the channel. In fact, when friction only occurs at the boundary (blue curve), the conductivity vanishes on resonance and the liquid responds in a dissipationless fashion. As we will show, this is a characteristic transverse response of a sliding crystal which is only subjected to friction at the boundaries. Therefore these resonances reveal a type of crystallinity that appears in Fermi liquids when probed dynamically. Such remarkable collective behavior could be observed in ultra-clean samples such as those recently employed to observe the hydrodynamic electronic flow 9-12 but in the low-temperature quantum regime where the classical hydrodynamic description breaks down.
The conductivity dips shown in Fig.1d are unique signatures of the shear sound that would be absent in weakly interacting metals where this mode does not exist (black curve). Likewise, the dips are washed out once scattering in the bulk becomes comparable to the boundary friction (dashed curve). This is a consequence of a reduced shear force when the force difference between the interior and the boundary is small.
At low temperatures metals enter the quantum Landau Fermi liquid (LFL) regime. A Fermi liquid can be thought of as having an infinite number of slow degrees of freedom that describe the relaxation of the shape of the Fermi surface. Unlike superfluids or ordinary classical liquids, the low energy excitations of LFLs cannot be captured completely by a description in terms of a finite number of dynamical fields such as density and current. We will focus on 2D systems but many of our conclusions carry over to the 3D case.
We begin by stating a central finding of our study: even in the presence of collisions, 2D Fermi liquids display a sharp propagating transverse sound mode with speed v s = v F (1 + F 1 )/2 √ F 1 , for Landau parameter F 1 > 1, and for wave-vectors q q * , with q where v F is the Fermi velocity and Γ 1 , Γ 2 are the momentum relaxing and preserving collision rates respectively. We will now derive these results within the Landau theory of Fermi liquids. 1. a, Experimental setup to detect shear sound. The blue region illustrates the out-of-phase (imaginary) current profile in the channel. b, Out-of-phase (imaginary) and c, in-phase (real) current profiles for driving frequency on-and off-resonance with the shear sound frequency ω shear . d, Real part of the transverse conductivity in units of DΓ eff /ω 2 when the shear sound is present (solid blue line) and absent (solid black line) in the limit of boundary dominated scattering (boundary scattering parameter b = 0.1(2πvF)), where D = ne 2 /m is the Drude weight and Γ eff is an effective scattering rate 13 . For finite bulk scattering (Γ1 = 0.1vFq0) the resonant zeroes at the shear sound harmonics (solid blue line) evolve into smooth dips (dashed blue line).
In LFL theory the shape of the Fermi surface becomes a dynamical object and small deviations of the radius p F (r, θ) from the equilibrium shape obey the linearized Landau kinetic equation (LKE) 3 : Here v p = v Fp is the velocity normal to the Fermi surface at angle θ, f (θ −θ ′ ) is the Landau function including short-range and Coulomb interactions, E is the applied electric field and I are collision terms. There are two kinds of collisions terms: those which relax momentum, such as electron-impurity collisions, and those that preserve momentum, originating from electron-electron collisions, which can be modeled as 14-16 : Here P m [p F ] projects the Fermi radius onto the m-th harmonic e imθ . There are two types of solutions to the LKE: incoherent and collective modes. The incoherent modes are sharply localized angular deformations of the Fermi surface 3,5 that form the particle-hole continuum with a dispersion of the form: Collective modes, however, are angularly delocalized deformations of the Fermi surface 3,5 . When the system has a microscopic mirror symmetry and the wave-vectors of the modes lie along the mirror invariant line, the modes can be separated into odd (transverse) and even (longitudinal) under the mirror operation 3,5 . The well-known plasma mode of metals is a longitudinal mode, whereas, the shear sound is a transverse mode.
To illustrate the features of the shear sound, we consider a simplified model in which all the n > 1 angular moments of the Landau interaction function vanish, F n>1 = (dθ/2π)f (θ) cos(nθ) = 0. The F 1 parameter controls the ratio of the quasiparticle mass (m * ) to the Drude mass (m) of a Fermi liquid, m * = (1 + F 1 )m. The Drude mass would equal the non-interacting mass (m 0 ) in Galilean invariant systems 17-20 .
Our key results are expected to remain valid in the presence of other Landau parameters whenever the shear sound mode remains the only sharp collective mode in the transverse sector 5 . For this model, a LFL with F 1 > 1 would feature a propagating shear sound mode with dispersion: This mode exists for q > q 2 , whereas for q < q 2 one encounters diffusive collective modes as depicted in Fig. 2a and detailed in the Supplementary Material 13 . Therefore, the shear sound is expected to become a sharp collective mode in moderately interacting Fermi liquids (F 1 > 1) for q > q * , with In the q 2 ≪ q ≪ p F limit, the shear sound velocity asymptotes to its undamped value v s 5 . On the other a b FIG. 2. a, The dispersive shear sound (blue solid curve) exists only for moderately interacting Fermi liquids (F1 > 1) and relaxes at a lower rate than that of the incoherent particle-hole excitations (green wedge), Γs = Γ1 +vsq2 < Γ1 + Γ2. b, The dispersive shear sound is absent when interactions are too weak (F1 < 1). Red and blue dashed curves indicate the dispersion of decaying collective shear modes.
hand, for a weakly interacting LFL with |F 1 | < 1, only a single, purely decaying collective mode exists as depicted in Fig. 2b, with dispersion: This decaying mode exists for 0 ≤ q ≤ Q, where its relaxation increases with q from Γ 1 to Γ 1 + Γ 2 , the value of incoherent particle-hole excitations, as depicted in Fig. 2b. Notice that this transverse mode becomes strictly diffusive only in the limit of vanishing momentum relaxing collisions Γ 1 → 0, and exists only for a non-vanishing rate of momentum preserving collisions Γ 2 > 0. Therefore, at such small wave-vectors the weakly interacting Fermi liquid (|F 1 | < 1) behaves like a classical fluid, where the slow diffusive relaxation of transverse currents is a consequence of the local conservation of momentum 1 . When F 1 < −1, one finds instead exponentially growing modes associated with a Pomeranchuk instability 6,13,21 .
To include boundary effects, we adopt the minimal but realistic model proposed in Ref. 15, which combines specular boundary conditions with boundary friction modeled as an enhancement of the momentum relaxing collisions at the boundary of the form where x ∈ (−W/2, W/2), y ∈ (−∞, ∞). As demonstrated in Ref. 15 this model captures the hydrodynamic, diffusive, and ballistic regimes of metals and their crossovers. For related studies see Refs. 14-16,22-24. We have found an exact analytic solution of the LKE Eq. (1) for this model with finite Landau parameters {F 0 , F 1 } in addition to all of the above ingredients which we present in the following (see Supplementary Material 13 for details). Because translation symmetry along x is broken by the presence of the boundaries, the conductivity that determines the current along the channel, j y (x, t), in response to a driving electric field along the channel, E y (x, t), is a function of two-wave vectors: The conductivity can be expressed as the sum of a bulk (bk) contribution: and a boundary (bd) contribution: where q is the momentum along x, q 0 = 2π/W , m is the transport mass,σ bk y (ω) = n∈Z σ bk y (nq 0 , ω) is the transverse conductivity measuring the bulk response to a periodic array of delta-function perturbations, andσ bd y = ne 2 W/mb parametrizes boundary scattering. The total conductivity for a uniform driving field is obtained by taking the q, q ′ → 0 limit of the above expressions, where σ D (ω) = ne 2 /m(iω + Γ 1 ) is the frequencydependent Drude conductivity. The expression in Eq. (14) can be understood as the self-consistent response of the LFL to both an externally applied electric force and the boundary friction. In a single equation, our solution encompasses the effects of disorder, interactions as well as boundary scattering, controlled respectively by the parameters Γ 1,2 , F 1 and b/W , and therefore captures the hydrodynamic, diffusive, ballistic, and LFL regimes on equal footing. Notice that F 0 is absent in our expressions because of the absence of density fluctuations for driving electric fields parallel to the channel.
The conductivity in Eq. (14) is shown for a metal with (F 1 = 3.0) and without (F 1 = 0.5) shear sound in Fig. 1d. In the former case, there are sharp dips at the shear sound energy, ω = Re ω s , evaluated at integer multiples of q 0 . In Fig. 1b-c, we see that the resonant current becomes purely imaginary, i.e., it is out of phase with the applied field. Therefore, in the limit of boundary dominated scattering, metals with shear sound display a dissipationless response at the resonant frequencies of this mode. As we will see, this is analogous to the response of a sliding crystal which is subject to friction only at the boundaries.
These conductivity minima acquire finite values in the presence of weak bulk scattering. The electronelectron collision rate is expected to scale as Γ 2 = (E F /2π) (k B T /E F ) 2 25 up to logarithmic corrections, and therefore, can be easily suppressed by cooling the metal well below the Fermi temperature. The electron-impurity collision rate is limited at low temperatures by the bulk elastic mean free path, λ = v F /Γ 1 . We estimate that the shear sound dips would be visible in metals with λ 5W at low temperatures. Furthermore, samples with enhanced boundary scattering relative to bulk scattering should lead to more pronounced conductivity dips.
Dissipationless resonant driving is a hallmark of a sliding crystal. To illustrate this, we consider a toy model of a two-dimensional Debye tetragonal crystal 26 confined in a channel with boundary friction aligned with one of its crystal axes (see Fig. 3b). The crystal slides in response to an alternating force along the channel, experiencing friction at the edges analogous to the boundary scattering in the LFL. Figure 3a shows the conductivity, i.e., the average velocity of atoms divided by the external force, of such a sliding crystal. In the absence of bulk friction, the real part of the conductivity exhibits zeros at frequencies corresponding to the harmonics of the transverse phonon of the crystal at wavelength W . Resonantly driving the system at these frequencies creates a current profile that is out of phase with the drive-the crystal pins at the boundary and self-consistently avoids energy dissipation in an analogous fashion to the Fermi liquid with shear sound (see Fig. 3c-d).
When probed optically, the sliding crystal therefore does not exhibit the resonant absorption typical of a crystal with pinned boundaries. The latter scenario can be described as a limiting case of the sliding crystal at infinite boundary friction. Indeed, when the boundary dissipation increases, the dips broaden, ultimately giving rise to resonant peaks at half-integer multiples of the fundamental frequency once the dissipative boundary force exceeds the shear restoring force of the crystal 13 . Such peaks do not have a counterpart in the case of the LFL, where off-resonant pinning at the boundary is prevented by scattering to the incoherent particle hole-continuum. Consequently, the conductivity dips signaling the shear sound in the LFL remain narrow even in the limit of arbitrarily strong boundary scattering 13 .
As we have shown, moderately interacting metals display a sharp shear sound collective mode which exists even in the presence of weak impurity and electronelectron collisions. This mode leaves clear fingerprints in clean metallic channels at low temperatures in the form of sharp resonant dips in the conductivity at frequencies controlled by the shear sound dispersion in Eq. (4), and that resemble the transverse sound resonance of a sliding crystal, despite the metal lacking any form of long-range crystalline order. There already exists various ultra-clean materials that feature a strongly interacting metallic state before a metal insulator transition which are therefore ideal platforms to discover the shear

SUPPLEMENTARY MATERIAL I. SOLVING THE LINEARIZED KINETIC EQUATION WITH BULK AND BOUNDARY RELAXATION
Using an ansatz of the form p(x, θ, t) = p(q, θ)e i(ωt−qx) , the linearized kinetic equation (LKE, main text Eq. (1)) becomes: (iω − iv F q cos θ + Γ 1 + Γ 2 ) p(q, θ) = eE q sin θ + (iF 0 v F q cos θ + Γ 1 + Γ 2 ) P 0 (q) where p l>0 (q, θ) = P l [p] + P −l [p] and P 0 (q) = P 0 [p] with P l [p] = e ilθ dθ 2π e −ilθ p(q, θ). The Fermi radius then satisfies the expression: To exploit the symmetry of the solutions, we express the Fermi radius explicitly in terms of even and odd Chebyshev polynomials, The general solution is then solved component by component by projecting Eq. (16) onto the various Cheybyshev sectors using Eq (20). This gives rise to a set of coupled self-consistency equations between P 0 (q) and P ± 1 (q): The solutions of which can then be used to obtain the other P ± l≥2 (q) components. Projecting to the Chebyshev sectors reveals the decoupling between the even (+) and odd (−) sectors.
In this work, we are particularly interested in probing the shear collective mode and therefore restrict the following discussion to the solutions of the odd sector, P 0 (q) = P + l (q) = 0. Rearranging Eq. (23) leads us to the self-consistent solution To proceed, we expand the cosine above and introduce the quantities C − 1 = q ′ cos πq ′ q0 P − 1 (q ′ ) and S − 1 = q ′ sin πq ′ q0 P − 1 (q ′ ). Since q ′ in Eq. (25) are integer multiples of q 0 = 2π W , the discretization wave-vector set by the channel width W , we have S − 1 = sin πq q0 = 0. Performing q cos πq q0 on both sides of Eq. (25) leads to the solution: . (28) Substituting this expression back into Eq. (25) gives us the self-consistent solution: Restoring the explicit ω-dependence of the various quantities, the transverse current density is given by: as per Eqs.(9)-(14) in the main text, where we explicitly separate into the bulk and boundary contributions. In going to the second line, we used n = p 2 F 4π , the electron density for a circular Fermi surface. For the specific case of a uniform electric field, E y (q, ω) = E y (ω)δ q,0 , the average transverse conductivity is i.e. Eq.(14) in the main text. Here we derive the frequency-dependent Drude conductivity explicitly: Finally, the spatial current profiles can be obtained via the inverse Fourier transform: In this section we outline the derivation of the generic complex dispersion relation for the shear sound in the infinite system (b = 0) in the absence of external fields, but with finite Landau parameters and scattering. The dispersive and purely decaying solutions, as well as the Pomeranchuk instability discussed in the main text can be obtained from the different regions in parameter space for which the solution is valid.
Primarily, we solve Eq. (23) with b = 0, or equivalently, for the zeroes of τ −1 (q, ω), The integrals Ω ij (q, ω) = Ω ij (ζ) are functions of the single variable ζ = s − i(Γ 1 +Γ 2 ), where we restrict to the casẽ Γ 1,2 > 0. We extend s into the complex plane, s → s = s + iγ, and evaluate these integrals by a change of variables z = e iθ , where C denotes the unit circle. The above results are valid for |z + | = |z − | = 1, corresponding to solutions outside the particle-hole continuum. Substituting them into Eq. (39) and self-consistently solving for ω, we arrive at the following solutions which are only valid for the following regions in parameter space with q ≥ 0: • ω + solution only exists for |F 1 | > 1 and for the following wave-vectors: • ω − solution exists for all F 1 and for the following wave-vectors: These results are summarized in Fig. 4, highlighting the three regimes of F 1 values with qualitatively distinct solutions. Particle-hole continuum is given by the green region.

III. RESONANT CONDUCTIVITY ZEROES OF THE SLIDING CRYSTAL
In this section we discuss in greater detail the sliding crystal toy model to illustrate the phenomenon of resonant conductivity dips.
Consider the propagation of elastic waves in a two-dimensional crystal tetragonal crystal. For a wavevector pointing along one of the principal axes of the crystal, entire lines of atoms move in phase with displacements either parallel (longitudinal) or perpendicular (transverse) to the direction of the wavevector. For simplicity, we assume that a line j with displacement u j experiences a restoring force F res j only from its adjacent lines j ± 1 that is linearly proportional to the difference of their displacements 26 , Here κ is the force constant.
Let us now confine the tetragonal crystal in a channel (see Fig. 3b of the main text). For an N -atom wide channel, the two columns of atoms at the edges x ±N/2 experience boundary friction γ b due to the roughness of the channel. In addition, we introduce a homogeneous bulk friction γ, and further drive the system uniformly alongŷ parallel to the channel with an external force F j ∝ e iωt . By symmetry, only the transverse waves are excited by the drive so that we write explicitly u j = y j .
To highlight the qualitative aspects of the system, we outline below the solution for the case of periodic boundary conditions along x so that x j = x j+N . Setting the particles' masses to unity, we have the following equation of motion: Using the ansatz y j ∝ e iωt , the equation of motion in Fourier space reads where F q = j e −i 2πq N j F j and Y q = j e −i 2πq N j y j are the respective Fourier components of F j and y j . We rewrite the above in a form similar to Eq. (25), and proceed to solve it in a similar fashion outlined earlier to find For a spatially uniform external force F j = F , F q = F N δ q,0 , an analogous average transverse conductivity relating the average transverse velocity density to the external force can be defined, with the quantities for the toy model analogous to the Fermi liquid, σ bk toy (ω) = q σ bk toy (q, ω) = q iωη toy (q, ω), The collective modes of this toy model analogous to the shear sound is simply the the transverse phonon, whose positive frequency dispersion is given by the poles of lim γ→0 η toy (q, ω), where ω ph denotes the fundamental frequency of the transverse phonon. Rewriting the bulk conductivity, we find whose real component is This is none other than the relaxation broadened Lorenzian peak centered at the resonant frequencies ±ω γ (q) of the (damped) transverse phonon whose peak value diverges as 1 γ . The average conductivity of the channel itself however does not necessarily follow the behavior of the bulk conductivity depending on the value of the boundary friction γ b . To see this, we rewrite the real part of Eq. (54) to leading order in γ ω , The quantityσ bk toy (ω) is therefore a series of Lorentzian broadened resonant peaks centered about ω γ (q ∈ Z) (a Dirac comb in the γ → 0 limit). Consequently, there are three qualitatively distinct regimes: • No resonance features: γ ≫ γ b N . In this case β −1 toy (ω) ≃ N γ b so that in general, σ ′ toy (ω) ≃ 1 ω 2 γ.
In this case, the contribution from the resonant peaks become comparable and the typical conductivity becomes roughly constant. At a frequency midway between two adjacent resonant frequencies ω 0 (q) and ω 0 (q + 1) , their imaginary parts cancel. A narrow peak develops, whose height and inverse width is approximately given by The second regime corresponds to strong pinning of the crystal at the boundary. The average channel conductivity recovers the resonant peak structure of the bulk conductivity but at shifted frequencies, These are none other than the odd harmonics of the transverse phonon whose fundamental wavelength is now twice the channel width (c.f. particle in a box). Only the odd harmonics are excited by the spatially uniform driving field due to the symmetry about x = 0. The crossover between the two regimes is illustrated in Fig. 6a in Sec. IV.
once the experimental resolution and the energy broadening due to disorder and electron collisions is smaller than both the effective boundary scattering rate b/W and the typical energy of excitations v F /W . Similar estimates hold for the resonances at higher harmonics of the shear-sound frequency, ω = lω shear . In practice, samples with a mean free path of λ > 5W should be sufficient to support well developed dips as shown in Fig. 5b and c. While the widths of the dips are determined by the boundary scattering rate b/W , they saturate to a finite value ∼ v F /W in the limit of strong boundary scattering, in which case the widths are typically of the order of the distance between neighboring dips ∼ v F /W . This is in contrast to the toy model of a sliding crystal, where the width of the dips continue to grow with increasing boundary scattering until they evolve into conductivity peaks located at the mid point between two resonances. This difference is illustrated by a comparison of the Fermi liquid and the toy model in Fig. 6. In this figure (also in Fig. 5b and Fig. 5c and in main text Fig. 1d and Fig. 3a), the conductivities are respectively normalized by for the LFL, with D denoting the Drude weight, and for the toy model. The dimensionless friction parameterb for the LFL is defined relative to v F q 0 , while its analogue for the toy modelb is defined relative to the fundamental transverse phonon frequency. The parameters Γ eff and γ eff characterize respectively the effective scattering rates in the LFL and the toy model. The saturation of the dip width in the case of the Fermi liquid arises from the additional contribution to the conductivity due to particle-hole excitations. Unlike the toy model, it obstructs the complete formation of the resonant peaks. Indeed, in the somewhat artificial limit of F 1 → ∞, the contribution of the particle-hole continuum is diminished and the Fermi liquid becomes more similar to the toy model.