${\bf 2k_F}$ Density Wave Instability of Composite Fermi Liquid

We investigate the $2k_F$ density-wave instability of the non-Fermi liquid states by combining exact diagonalization with renormalization group analysis. At half-filled zeroth Landau level, we study the fate of composite Fermi liquid in the presence of the mass anisotropy and mixed Landau level form factors. These two experimentally accessible knobs trigger a phase transition towards a unidirectional charge-density-wave state with a wavevector equals to $2k_F$ of the composite Fermi liquid. Based on exact diagonalization, we identify such transition by examining both the energy spectra and the static structure factor of charge density-density correlations. The renormalization group analysis reveals that gauge fluctuations render the non-Fermi liquid state unstable against density-wave orders, consistent with numerical observations. The possible experimental probes of density-wave instability are also discussed.

Introduction.-Non-Fermiliquid (NFL) is among the most exotic quantum states in condensed matter systems.One class of NFL state is realized at quantum critical points [1][2][3], where the gapless collective modes provide a common route to these NFL states.The discoveries of high-temperature superconductors [4], heavy-fermion materials [5], and Moiré materials such as the twisted bilayer graphene [6] have triggered intensive investigations on the strange metal behavior at quantum critical regime.Instead of appearing at quantum critical point, the NFL state can arise as a stable phase at zero temperature.A prominent example is the two-dimensional (2D) correlated electrons under a strong magnetic field: when the zeroth Landau level (LL) is half-filled, it becomes a fractionalized gapless state [7,8] with large Fermi surface formed by the composite fermions (CFs) [9,10].
In the Halperin-Lee-Read (HLR) description [7] of the composite Fermi liquid (CFL), the CFs strongly interact with dynamical gauge bosons, invalidating the quasiparticle description in Fermi liquid theory.Since the compressible NFL state at half-filled LL is well established both experimentally [11][12][13] and numerically [14,15], it provides a promising platform to explore intriguing properties of NFL states.More importantly, the physical setup also comes with various tuning knobs such as the magnetic field, the geometry, and the number of components including layers, subbands, spins and/or valleys.With these knobs, plenty of states adjacent to CFL are discovered, consequently revealing various instabilities of CFL.For instance, the Cooper instability [16,17] leads to the p + ip paired Moore-Read (MR) state [18][19][20]; the Pomeranchuk instability [21,22] results in nematic quantum Hall states [23,24]; the Stoner instability of CFL gives rise to spin or valley polarizations [25][26][27]; and the instability towards the Halperin 331-state [28,29] in quantum Hall bilayers.
In this letter, we would like to reap yet another natural instability of CFL: the 2k F density-wave instability, which is of equal importance to the previously discovered CFL instabilities and is likely to exhibit distinct physics from the ordinary Fermi liquids [30][31][32].Based on the exact diagonalization (ED) and renormalization group (RG) analysis, we propose one possible mechanism to trigger the density-wave instability of CFL on half filled LLs: tuning the interactions via the mixed LL form factors from an anisotropic CFL state.We numerically demonstrate such instability by examining both the energy spectra and the static structure factors of charge density-density correlations.The underlying mechanism is revealed by employing RG analysis, where the 2k F instability would be dominant over the pairing instability via increasing the gauge fluctuations, which can be achieved by breaking the rotational symmetry.Importantly, the mixed form factor is experimentally accessible in Dirac materials, e.g., in bilayer graphene, by tuning the interlayer electric bias and the magnetic field [33][34][35][36], rendering it possible to examine the 2k F instability in NFL states.
Numerical Setup and Results.-Weconsider 2D electrons on a torus with a strongly perpendicular magnetic field piercing through its surface.Since the kinetic energy is quenched due to the magnetic field, the Hamiltonian only includes the projected Coulomb interaction, which is given by where V (q) is the Fourier transform of the un-projected Coulomb interaction, F (q) denotes the density form factor introduced by projecting the Coulomb interaction, ρ(q) is the guiding center density operators, and A represents the area of 2D plane.Below we will demonstrate one mechanism to trigger the density-wave instability of CFL: tuning the interaction from an anisotropic CFL state.To achieve this, we consider the mixed form factors F (q) = cos 2 ΘF 0 (q m ) + sin 2 ΘF 1 (q m ) to tune the interac-arXiv:2001.03202v1[cond-mat.str-el]9 Jan 2020  tions [33][34][35][36], where F 0,1 (q m ) = exp(−q 2 m /4)L 0,1 [q 2 m /2] are the form factors for n = 0 and n = 1 Galilean LL, respectively.L n (x) is the Laguerre polynomial.The anisotropic CFL can be achieved by introducing mass anisotropy, q 2 m = g ab m q a q b includes the metric g m = diag[ m y /m x , m x /m y ] derived from the band mass tensor.In the isotropic limit (i.e., m y = m x ), the CFL and MR states are stabilized at sin 2 Θ = 0 [7,9] and sin 2 Θ = 1 [18][19][20], respectively.The corresponding pairing instability in this limit, such as tuning sin 2 Θ, has been theoretically confirmed [37][38][39], though the nature of this transition is still controversial [16][17][18][19][20].The mass anisotropy explicitly breaks the spatially rotational symmetry [40][41][42][43][44][45][46][47][48][49], concealing another factor to trigger the instability of CFL.Previous studies have demonstrated the CFL is remarkably robust against mass anisotropy when sin 2 Θ = 0 [48], while the MR state is fragile against mass anisotropy and finally translates to a stripe state [50] when sin 2 Θ = 1 [49].Then it is natural to investigate the possible density-wave instability of CFL by tuning the interactions via sin 2 Θ from an anisotropic CFL state at sin 2 Θ = 0. Below we will detect such possibility by solving the model Eq. 1 using ED [51].
Our numerical results are depicted in the phase diagram shown in Fig. 1(a).In the isotropic limit, we have confirmed the pairing instability of CFL when tuning the interaction via sin 2 Θ, consistent with previous studies.In the presence of mass anisotropy, we find the pairing instability only survives in a small regime in the phase space, instead, the density-wave instability becomes the dominant instability of CFL after the rotational symmetry breaking, which can be triggered more easily with increasing the mass anisotropy [see Fig. 1(a)].
The phase boundaries in Fig. 1(a) are identified from both the energy spectra and the derivatives of groundstate energy.Figure 1(b) shows an example of the energy spectra as a function of sin 2 Θ for N e = 16 system with m y /m x = 8.The CFL state is robust up to sin 2 Θ ≈ 0.64 upon tuning the interaction, which can be further confirmed from the derivatives of the groundstate energy in Fig. 1(c).The energy gap in the spectra of CFL is induced by the shell-filling effect on finite sized system, which can be identified by comparing the quantum number of ground state obtained by ED and the CFL wavefunctions on torus [27,37,[51][52][53].The energy level crossing near sin 2 Θ ≈ 0.32 represents the change of the CFL ground-state momentum sectors, in contrast to the phase transitions around sin 2 Θ ≈ 0.64.We further confirm the nature of these phases by studying the static structure factor N (q) of the density-density correlation, where ρ q = N i=1 e iq•Ri is the Fourier transform of the guiding center density.As shown in Fig. 2(a-b) for sin 2 Θ 0.64, N (q) exhibits strong 2k F scattering feature induced by the scattering among CFs close to the Fermi surface.At sin 2 Θ > 0.64, there are two sharp peaks in N (q) in the same direction, which can be regarded as the hallmark of charge ordering with the wave vector determined by the position of the peaks.Here, N (q) displays stripe feature.
Further increasing sin 2 Θ 0.88, the peaks rotate from (q x , q y ) = (0, ±q * ) to (q x , q y ) = (±q * * , 0) as shown in Fig. 2(c-d).Here, the wave vector ±q * * also can be identified from the low energy spectra of such resulting phase [see Fig. 1(d)], where there is no recognizable gap separating the ground-state manifold from the excited states, instead, the energy spectra displays a conspicuous set of quasi-degenerate states which differ by momentum ∆q and satisfy ∆q = ±q * * .The line connecting the lowest energy states in each momentum sector has a zigzag structure as shown in Fig. 1 (d), which only appears in the energy spectra in one momentum direction, implying a unidirectional charge density waves state.
RG analysis from CFL.-It is natural to put the above transition within the context of the HLR theory [7] and the instabilities of CFL.In the following, we use the patch theory to analyze the competing fluctuations in CFL.Namely, the composite-Fermi surface is approximated by two patches [54,55], S = S f + S a + S int , where and k ≡ d 3 k (2π) 3 , s = ± denotes the two patches, ψ s and a refer to composite fermion and emergent gauge field, respectively.v F and K capture the composite-Fermi velocity and the curvature of the patch, and e is the Yukawa coupling between fermion and gauge boson. is the expansion parameter, = 0 corresponds to the long-range Coulomb interaction [54,55].
The patch theory is an effective description in the range |k x |, k 2 y < Λ (note that k x and k y scale differently).We address the IR properties of the theory by integrating out the high energy mode, Λe −l < k 2 y < Λ to generate RG equations, where l > 0 is the running parameter.There is no renormalization to boson propagator because it is nonlocal.The rational of using a nonlocal bare kinetic term for gauge boson lies in the fact that boson kinetic potential does not receive corrections up to three-loop [56].Taking into account of the fermion selfenergy Σ s (p

RG equation reads (the vertex correction vanishes [57]
) where g ≡ e 2 π 2 v F Λ /2 captures the effective Yukawa coupling.The presence of a nontrivial stable fixed point g * = 2 corresponds to the NFL interacting strongly with gauge field.
Next, we analyze the density-wave instability in the CFL.Because we are interested in the 2k F instability connecting tangential Fermi points, we can consider the scattering processes within the patch theory, namely, In the patch theory, the four-body interaction is irrelevant, which is consistent with the fact that the forwardscattering process does not affect the existence of Fermi surface [58], and the perturbative calculation should be valid.Indicated in Fig. 3(a), the renormalization to the four-body interaction reads Γ (a) where α 0 ≡ Γ(0, 1) ≈ 0.219, and Γ(n, x) ≡ ∞ x dtt n−1 e −t is the incomplete Gamma function.Without gauge fluctuation, the RG equation of dimensionless coupling constant u ≡ which shows that an instability only occurs at finite interaction strength.When u is large enough, i.e., u > 1 2 √ 2α0 , it develops a wave-density instability with the 2k F order parameter φ = ψ † + ψ − .Now we consider the effect of gauge fluctuations.As shown in Figs.
And there is no backreaction from short-ranged interaction to the gauge fluctuation at one-loop order.Thus, in presence of fluctuating gauge bosons, the RG equation becomes In the RG equations, there are four fixed points in (u, g)-plane, including the Gaussian fixed point (0, 0), the density-wave transition point ( We also note that, in the presence of gauge fluctuations, the critical coupling strength of 2k F density-wave transition is significantly reduced.More exotically, when = c , C( c ) = 0, the CFL fixed point and the transition point collide with each other, as shown in Fig. 4(b).The CFL transition fixed point is unstable against 2k F density-wave instability.We would like to point that such fixed point collision is also found in previous literatures [59,60].When > c , the CFL is totally preempted by density-wave orders as shown in Fig. 4(c).These results indicate that NFL fixed point is unstable if the gauge fluctuation is strongly enough.
Discussions.-Thelarge portion of CFL span in the phase diagram Fig. 1 suggests < c .Although it is unclear how the bare interaction strengths, namely, the gauge coupling and the short-ranged interaction, change with the mixed form factors, the RG analysis is able to predict the wavevector of the density wave in the presence of the mass anisotropy.This is because the bare gauge coupling is enhanced by the mass anisotropy through the Fermi velocity.Assuming (u, g) = (u 0 , g 0 ) for the isotropic CFL, we have (u, g) = (u 0 , α1/4 g 0 ) at the patches k = (± √ 2 mx µ, 0) of the anisotropic Fermi surface, where α ≡ mx / my denotes the mass anisotropy of CFs (we will consider α ≥ 1, since the opposite case is equivalent).α is related to the mass anisotropy of electrons α through α = √ α [61].It is easy to see that α1/4 g 0 is the largest bare value in the elliptic Fermi surface, therefore, the above RG analysis predicts that the 2k F instability occurs at 2k F = 2 √ 2 mx µ, which connects the Fermi points with smallest Fermi velocity.This observation is consistent with N (q) in Fig. 2(c) near the transition point.Note that this is a gauge fluctuation induced stripe transition.
Deep inside the charge density wave phase, we also find the switch of stripe orientations, as shown in Figs.2(cd).This phenomenon might be beyond the CFL physics since it is further away from the critical points, however, it can be attributed to the reduction of Hartree energy cost when the stripe orientation coincides with the direction of the smaller mass [49].Moreover, from our ED results in Fig. 1(b-c), the energy level crossing [see Fig. 1(b)] and the sudden jump in the first order derivatives [see Fig. 1(c)] suggest the transition from CFL to charge-density wave might be first order.We should note that it is still under debate whether the 2k F density-wave transition is continuous.While Altshuler et al. [30] argues a first-order transition due to the strong 2k F fluctuation at low energies, a more recent article by Sykora et al. [31] shows a second-order transition is also possible.It will also be an excellent task to investigate the critical phenomena in 2k F transition of NFL, which we leave for future works.
The experimental probe of various instabilities of CFL is still of many challenges and under intensive investigations.Previous studies mainly focus on detecting the pairing instability between the MR and CFL state, which has been proposed by tuning the subband level crossings [62,63] or applying hydrostatic pressure [64][65][66] in GaAs quantum wells, or by tuning either the perpendicular magnetic field or the interlayer electric bias in bilayer graphene [34][35][36].In particular, the hydrostatic pressure experiments [64][65][66] have found that tuning the pressure through P c1 would trigger the transition from MR to an anisotropic compressible phase, which is consistent with either a stripe phase [37,49,50] or nematic phase [22] .Interestingly, further increasing the pressure to P c2 leads to a transition to an isotropic compressible phase, which might be relevant to the density-wave instability, particularly considering that the pressure is believed to change the LL mixing parameters [66].However, we should also note the pressure-driven platform is hard to be captured by an ideal Hamiltonian microscopically, which would be an interesting direction for future study but lies of out the scope of this work.Moreover, the mixed form factor considered in this work could be realized and tunable in bilayer graphene by the interlayer electric bias and magnetic field [33][34][35][36], then breaking the rotational symmetry is potentially to probe the density-wave instability of CFL.The mass anisotropy exists in AlAs quantum wells [67,68] in nature or could be introduced by applying in-plane field [69] or uniaxial strain [70,71], then to realize the density-wave instability on top of an anisotropic CFL is also a promising direction to pursue experimentally.The RG calculation is controllable in the large-N and small ∼ 1 N expansion [54].The patch theory is an effective description in the range |k x |, k 2 y < Λ.In the following, we integrate out the high energy mode, √ Λe −l < |k y | < √ Λ to generate RG equations, where l > 0 is the running parameter.There is no renormalization to boson propagator because it is nonlocal.The rational of using a nonlocal bare kinetic term for gauge boson lies in the fact that boson kinetic potential does not receive corrections up to three-loop [56].The fermion self-energy is (Fig. 1 π 2 v F Λ /2 that captures the effective Yukawa coupling, we have following RG equations, The presence of a nontrivial stable fixed point g * = 2 corresponds to the non-fermi liquid (NFL) interacting strongly with gauge field.

B. Cooper instability and 2kF density-wave instability
Despite the long-range interactions between composite fermion mediated by the gauge field, there are local interactions between the composite fermion that might generate pairing or stripe instability.Thanks to the Pauli exclusion principle of fermions, among infinite channels of four-fermion interaction only BCS and forward-scattering channel survive in the low energy [58].For simplicity, we will send N = 1 in the following and consider four-fermion interactions.We first consider the BCS Hamiltonian for the nondegenerate fermi surface,

Fig. 1 .
Fig. 1. (Color online) The phase diagram and the energy spectra.Depending on the mass anisotropy my/mx, we identify the pairing instability and density-wave instability of CFL when tuning the interaction via sin 2 Θ, the corresponding phase diagram is shown in panel (a).For a fixed mass ratio, e.g., my/mx = 8 in panel (b-d), the phase boundary is consistently identified from the evolution of energy spectra with sin 2 Θ (b) and the derivatives of the ground-state energy (c).In the charge density wave phase, the energy spectra along momentum Kx exhibits the quasidegenerate states that differ by a momentum ∆q (d).Here, we consider half-filled Landau level with Ne = 16 electrons.

Fig. 2 .
Fig. 2. (Color online)The static structure factors N (q).The nature of different phases in Fig.1(b-c) can be identified from the static structure factor N (q) of the density-density correlation.Panels (a-d) show N (q) in the CFL phase with sin 2 Θ = 0.16 (a) and sin 2 Θ = 0.48 (b) , as well as N (q) in the charge density wave phase with sin 2 Θ = 0.68 (c) and sin 2 Θ = 0.96 (d).Here, we consider half-filled Landau level with Ne = 16 electrons and mass ratio my/mx = 8.

FIG. 3 .
FIG. 3. (Color online)The corrections to short-ranged fourfermion interactions within the patch theory.Panel (a) denotes correction from the four-fermion interaction, and panels (b-c) denote corrections from the gauge fluctuations.

FIG. 4 .
FIG. 4. (Color online) The RG flow diagrams of (u, g) at different .The blue points show the Gaussian fixed point and stripe transition point without gauge fluctuations.Red points show the NFL fixed point and stripe transition point in the presence of gauge fluctuations.Fig. 4(a) shows four fixed points when < c.Figs.4(b), 4(c) show the RG flow for = c, > c, respectively.The dashed line indicates the trajectory of two nontrivial fixed points in the presence of gauge fluctuations.After their collision, the fixed points become imaginary values, and disappeare from the flow diagram.

1 2 √, 2 , 3 √ 2
2α0 , 0) in the absence of gauge bosons, and two new fixed points emerged from the interplay between gauge fluctuations and shortranged interactions: F P CFL = where C( ) = (81 + 144α 0 − 1088α 2 0 ) 2 − 12(9 + 8α 0 ) + 36 is a quadratic function in .When 0 < < c , C( ) > 0, all of the four fixed points are physically accessible, and F P CFL (F P T ) corresponds to the CFL fixed point (density-wave transition point).Here c ≡ 6(9−8(number [57].When < c the blue points in Fig.4(a) points correspond to Gaussian and density-wave transition point without gauge fluctuation, while the red points correspond to F P CFL and F P T .