Microscopic theory for nematic fractional quantum Hall effect

We analyse various properties of the nematic fractional quantum Hall effect (FQHE) in the thermodynamic limit, and present necessary conditions required of the microscopic Hamiltonians for the nematic FQHE to be robust. Analytical expressions for the degenerate ground state manifold, ground state energies, and gapless Goldstone modes are given in compact forms with the input interaction and the corresponding ground state structure factors. We relate the long wavelength limit of the neutral excitations (serving as the nematic FQHE ground state from spontaneous symmetry breaking) to the guiding center metric deformation, and show explicitly the family of trial wavefunctions for the Goldstone modes with spatially varying nematic order. We show for short range interactions, the dynamics of the nematic FQH is completely determined by the long wavelength part of the ground state structure factor. This is the only part that requires numerical studies in the future work, which is potentially more tractable than the conventional numerical approaches.


I. INTRODUCTION
For condensed matter systems with non-trivial topological orders, the robustness of the topological properties at low temperature usually requires the ground state to have a finite energy gap to all excitations in the thermodynamic limit [1].In general for such systems, the universal topological features dominate the ground state response, and the geometric properties of the system are less important.The incompressible FQH states are such examples where topological orders arise from strong interactions between electrons, without needing protection of any symmetry.There are also examples of compressible FQH states with no plateau formation for the Hall conductivity, with anisotropic stripe or bubble phases that are gapless and spontaneously break the rotational/translational symmetry [2][3][4][5][6][7][8][9][10].An interesting exception is the nematic fractional quantum Hall (FQH) effect, which was recently discovered in experiments [11,12].Here we have examples where topological orders and non-trivial geometric effects coexist: there is an anisotropic longitudinal resistivity enhanced by low temperature, and at the same time with a robust plateau for Hall conductivity.
It is generally believed that the robustness of the Hall conductivity plateau in nematic FQH is due to the finite charge gap, while the anisotropic longitudinal resistivity is a result of the neutral excitations in the long wavelength limit becoming gapless [13].Such neutral excitations form a degenerate ground state manifold.They are thus prone to spontaneous symmetry breaking.It is well known that the neutral excitations in the long wavelength limit is a quadrupole excitation that breaks rotational symmetry, potentially leading to anisotropic transport [14,15].Several field theoretical studies of the nematic FQH have been carried out, either by assuming that the neutral excitations go soft in the long wavelength limit [13], or by adding an attractive quadrupolar interaction [16,17].These theories capture the topological order, the nematic order from spontaneous symmetry breaking, as well as the charge gap in a phenomenological manner.
Microscopic theories are needed to better understand the assumptions used in the field theoretical approaches.For the nematic FQH most studies so far are based on numerical computations.Finite system analysis has established that the single mode approximation (SMA) is exact for the neutral excitations in the long wavelength limit [14,28].Both Jack polynomial type representations, the composite fermion picture and the first quantised form of the neutral excitations are also constructed to shed more insights on the nature of such many-body states [14,[18][19][20].Numerical studies have tentatively shown that short range interactions can lead to instability of the intrinsic guiding center metric, and such "squeezed" Laughlin states can harbour uniform nematic order [15].It is, however, difficult to show microscopically how assumptions in the nematic FQH field theory can arise from bare interactions between electrons with numerical studies.In particular, important physics happening at long wavelength limit is inaccessible given the relatively small system sizes that can be computed numerically.
In this paper, we compute analytically the conditions for the long wavelength limit (small q) of the neutral excitation to go soft in the thermodynamic limit.Using the Laguerre polynomials as the basis, variational energy of the neutral excitations at small q is controlled by two universal, tridiagonal characteristic matrix Γ (1) , Γ (2) that can be computed exactly and are independent of microscopic details.The SMA at small q becomes exact eigenstates when it is degenerate with the ground state, and we can identify it with the guiding center metric deformation of the ground state.Thus the onset of the nematic fractional FQH can be understood as the case when the shear modulus of the gapped ground state of the quantum fluid vanishes [22].
We also identify trial wavefunctions for the Goldstone mode from spontaneous symmetry breaking, where the spatially varying "nematic wave" can be shown explicitly.While Γ (1) controls the neutral excitation gap, the dispersion of the Goldstone mode is controlled by Γ (2) .The tridiagonal nature of Γ (1) and Γ (2) implies the dynamics of the nematic FQH only depends on the long wavelength part of the ground state structure factor, if the interaction is short-ranged.The analysis here can much simplify the numerical computation of the nematic phase and its finite size scaling.The results presented here are applicable to FQH phases at any filling factor.The necessary analytic conditions for the robustness of the nematic FQH phases are also illustrated with numerical calculations focused on the Laughlin state at filling factor ν = 1/3.This paper will be organised as follows: In Sec.II we compute the long wavelength energy gap of the neutral excitations from the SMA in the thermodynamic limit, and show that it is determined by the universal characteristic matrix Γ (1) .We term such neutral excitations in the long wavelength limit as the quadrupole excitations.In Sec.III we show the quadrupole excitations can be identified as a uniform area-preserving deformation of the ground state, both from the wavefunction and the energetics perspectives.Thus the quadrupole excitations harbour uniform nematic order [15].In Sec.IV we derive the expression of the spatially varying nematic order from the trial wavefunctions of the gapless Goldstone mode in the nematic FQH phase.We also show the quadratic dispersion of the Goldstone mode is controlled by another universal characteristic matrix Γ (2) .In Sec.V we analytically investigate several families of short range microscopic models, and derive conditions for the nematic FQH to be viable.In Sec.VI we carry out preliminary numerical analysis focusing on the Laughlin phase at filling factor ν = 1/3, corroborating with the analytical results to show tentative evidence of nematic FQH when the two-body interaction is a family of linear combinations of the V 1 , V 3 and V 5 pseudopotentials derived from eigenstates of Γ (1) .In Sec.VII we discuss about the contrasting natures of the quadrupole and dipole neutral excitations, showing further numerical evidence that the latter gives the charge gap that could be much larger than the quadrupole gap.This is the necessary ingredient for the coexistence of the nematicity and Hall plateau.In Sec.VIII we summarize the results of this paper and discuss about the future works.

II. SMA IN THE LONG WAVELENGTH LIMIT
Let us start with a two-body Hamiltonian in a single Landau level (LL) as follows: where ρq = i e iqa Ra i is the guiding center density operator, and Ra i are the guiding center coordinates with only matrix elements between states in the same Landau level.It also satisfies the commutation relation , where l B is the magnetic length.The number of electrons is given by N e and we set l B = 1.Assuming at a fixed filling factor, Eq.( 1) is incompressible with both neutral and charged quasielectron gaps, with ground state |ψ 0 and energy E 0 .Defining the regularised guiding center density as δ ρq = ρq − ψ 0 |ρ q |ψ 0 = ρq − Neδ(q) 2πq with q = |q|, the GMP algebra [21] is given by: The regularised ground state structure factor is defined as S q = ψ 0 |δ ρq δ ρ−q |ψ 0 and we also have the following relationship for fermions [23,24]: We now start with the family of SMA trial wavefunctions |ψ q = δ ρq |ψ 0 , which are orthogonal to the ground state with variational energies E q .The variational energy gap is thus given by [32]: V q 2 sin 1 2 q × q 2 × (s q +q + s q −q − 2s q ) (5) Here we assume rotational invariance.In the long wavelength limit, the ground state structure factor is given by lim |q|→0 S q = ηq 4 , where η = N e κ/2 and κ is bounded below by the Hall viscosity of the ground state [22].By expanding Eq.( 5) to the leading order in q, and using Eq.(3), we have the following expression: Here q 1 = |q | 2 , q 2 = |q | 2 , and 0 F 1 (a, x) is the regularised hypergeometric function [32].For very short range interactions (e.g. with V 1 pseudopotential [1]), δE q→0 > 0 and is buried in the continuum of multi-roton excitations.If V q in Eq.( 1) can be tuned such that δE q→0 → 0, then |ψ q becomes an exact eigenstate, degenerate with |ψ 0 , given that there is no level crossing from V 1 → V q .
To evaluate the numerator in Eq.( 6), we first note that due to the property of the structure factor in Eq.( 3), s q is a linear combination of Laguerre polynomials L m q 2 with odd m.Expanding V q in the same basis of Laguerre polynomials, we have the following: Using the HardyHille formula, Eq.( 9) can be further simplified to give: where both m, n are odd integers, and Γ mn (1) is a tridiagonal matrix.It is then useful to treat V q , s q as vectors c, d respectively, in the basis of Laguerre polynomials, where d is completely from the ground state.The dot product c•d gives the ground state energy E 0 , and the variational energy gap is given by the inner product: Note that Γ (1) is a well-defined mathematical function given by Eq.( 10), while the only physical input to the Hamiltonian is given by c.There is a one-to-one mapping of d from c, with the ground state of Eq.( 1).For short range interactions with c m = 0 for m > m 0 , we only need to consider d n with n ≤ m 0 + 2. A more detailed analysis will be presented in Sec.V.

III. NEMATIC ORDER FOR THE NEUTRAL EXCITATIONS
We now explore the nematic order of the neutral excitations in the long wavelength limit by connecting them to the anisotropic ground state from deforming the guiding center metric of |ψ 0 .The area-preserving deformation generators can be defined as Λab = 1 4 i { Ra i , Rb i } with the following closed algebra [22]: The family of anisotropic ground states can thus be defined as |ψ α = e iα ab Λab |ψ 0 , with α ab as a symmetric matrix.The Bogoliubov transformation of the guiding center coordinates is given by R a = λ a b Rb = e −iα cd Λcd Ra e iα cd Λcd , thus |ψ α is the ground state of Eq.( 1) with the transformation in V q : q a → λ −1 b a q b , or q 2 → g ab q a q b , where g ab is a unimodular metric parametrised as follows: For rotationally invariant |ψ 0 , the variational energy of |ψ α only depends on θ, which parameterises the squeezing of the metric, as follows [32]: Comparing Eq.( 11) and Eq.( 14), we can see the variational energy of the neutral excitations in the long wavelength limit is related to the shear modulous c, d Γ (1) of the ground state.Thus for small |q| and θ, |ψ q and |ψ α approximately have the same energy with θ = 1/ (2η).
In general |ψ q→0 and |ψ α,θ→0 do not have to be related to each other even when they have the same variational energy.However when c, d Γ (1) → 0, they will belong to the same manifold of degenerate ground states.Denoting |ψ θ = |ψ α,φ=θq , where θ q is the angle of the momentum, we can identify the following at small q based on inversion symmetry: This is the ground state of the nematic FQH after spontaneous symmetry breaking, and finite size numerical analysis indicates that |ψ θ could have uniform nematic order [15].Thus in the long wavelength limit, |ψ ± q is equivalent to the guiding center metric deformation of the ground state at q = 0, if the shear modulus c, d Γ (1) vanishes.This leads to the development of the nematic order for the neutral excitations in this limit.

IV. GAPLESS GOLDSTONE MODE
The long wavelength spatial modulation of the nematic order should gives rise to the gapless Goldstone mode.To identify these states let us first define the operator of the local nematic order, which is a slightly modified version from [15]: where θ is the angle of d, and δ ρ (r) is the Fourier component of δ ρq .For a translationally invariant state |ψ 0 , the nematic order is independent of r, and we have: where s d is defined by Eq.(3).Eq.( 17) is clearly zero if |ψ 0 is rotationally invariant, and non-zero if the structure factor has a quadrupole symmetry.For the nematic ground state established in Eq.( 15), simple algebra gives us: where N (1) are two non-universal functions of q that can be computed analytically [32].Thus at least when q is small enough, |ψ ± q is the Goldstone mode with spatially varying nematic order given by the second term in Eq. (18).
To look at the dispersion of the Goldstone mode, we can expand Eq.( 5) to the next order.When Eq.( 11) vanishes at the nematic FQH phase, we have [32]: It is important to note that the dispersion of the nematic Goldstone mode is quadratic.The necessary condition for the nematic FQH phase is thus c, d and Γ (2) are universal tridiagonal matrices independent of the microscopic details of the Hamiltonians.

V. MINIMAL MODEL FOR NEMATIC FQH
To understand the dynamics of the nematic FQH phase from microscopic Hamiltonians analytically as much as possible, we start with spectrum of Γ (1) , which is real given that the matrix is symmetric.The eigenvalues λ 1 and corresponding eigenvectors c λ1 satisfy the following relationship: where k is a non-negative odd integer and c λ1 −1 = 0.In particular, if the microscopic two-body interaction is c = c λ1 , we then have lim q→0 δE q = (λ 1 E 0 ) / (256η), where E 0 = c • d is the ground state energy in the q = 0 sector.It is easy to check that λ 1 = 0 gives c 0 k = const, which is not relevant for realistic interactions.
We now focus on a special family of interactions with c such that c k = c λ1 k for k ≤ k 0 , and c k = 0 for k > k 0 .For the more realistic case where c k decreases with k, we need to have λ 1 < 0. Simple algebra leads to: Thus the variational energy gap requires three inputs from numerical computations: d k0 , d k0+2 and the ground state energy E 0 .This relationship is valid at any filling factor in the thermodynamic limit.
A. k0 = 1 Without loss of generality, we always set c 1 = c λ1 1 = 1.The simplest case is for k 0 = 1.For the Laughlin state at filling factor ν = 1/3, it is the model Hamiltonian leading to E 0 = d 1 = 0.This gives us δE q = 3 128η d 3 − 15 128η d 3 q 2 +O(q 4 ), from Eq.( 22) and Eq.( 19).The neutral mode is gapped with a negative dispersion at q → 0, as it should be.This is also true for filling factor ν ≤ 1/3.More precisely, let N e , N o be the number of electrons and number of orbitals respectively on the sphere or disk geometry, For N o < 3N e − 2, d 1 does not vanish, and the variational energy gap is given by: Thus for very short-range interactions (in the neighbourhood of pure V 1 ), the necessary condition for a gapped translationally invariant ground state is for The global ground state will no longer be translationally invariant with d 1 ≥ d 3 , and spontaneous symmetry breaking will generally occur.For d 1 > 5d 3 /3, the dispersion of the neutral excitation is positive, indicating a possibility of the charged gap and quantised Hall conductivity.We will explore these possibilities in Sec.VI.
For the case of k 0 = 3, the model Hamiltonian is a linear combination of the V 1 , V 3 pseudopotentials (with coefficients c 1 , c 3 ).This can be fully tuned by λ 1 , with c 3 = 1 + λ 1 /6.The physically relevant regime is thus for c 3 > 0 and λ 1 > −6.Let the eigenvectors of Γ (2) be c λ2 with eigenvalue λ 2 , the following expression can be obtained with some algebraic manipulation: Since we are only interested in the signs of each term, we ignore the denominator in Eq. (24).We also have λ the condition for the first line of Eq.( 24) to be zero, and the coefficient of the second line to be positive, leads to a narrow range in the parameter space of c 3 and d 5 /d 3 as shown in Fig. (3).Note that d 5 /d 3 is not an independent parameter.It is fully dependent on c and the filling factor, and can in principle be obtained in numerics by finite size scaling.

C. k0 = 5
We now look at model Hamiltonians with pseudopotential combinations of V 1 , V 3 and V 5 .For simplicity we will only look at the case of c 1 = c λ1 1 , c 3 = c λ1 3 , c 5 = c λ1 5 .While this does not cover all possible cases, it gives much insight into the behaviours of such model Hamiltonians.
FIG. 1: The range of parameters where the nematic FQH is possible at different values of d3/d5, as given by Eq.( 25).The shaded area is the range of c3 as given by the left axis.The line plot is the maximum allowed value of d7/d5 at different value of d3/d5, as given by the right axis.The heat map gives the maximum allowed value of d7/d5 for different values of c3 and d3/d5.
Similar to the case of k 0 = 3, we can obtain the following relationship: Here λ 1 , λ 2 , c λ1 5 , c λ2 5 are defined the same way as in Eq.( 24), while c λ1

VI. NUMERICAL STUDIES
All results in Sec.V are valid in the thermodynamic limit, and are applicable at any filling factor.In this section, we perform some preliminary numerical analysis at filling factor ν = 1/3, about possible microscopic models for the nematic FQHE.All numerical computations in this work are done with the spherical geometry [26], and we analyse the ground state wavefunctions and energy spectra for reasonably large system sizes.While the comparison between finite size scaling of numerical results and the analytical results in Sec.V can never be conclusive, the results here nevertheless illustrate the usefulness and limitations of finite size numerical calculations.The neutral gap of δE q→0 in this section is not computed from the energy spectrum.Instead we use Eq.(11) to evaluate the energy gap numerically from the ground state in the L = 0 sector alone.Not only is this a simpler calculation technically, it also has smaller finite size effect.This is because Γ (1) is calculated from the thermodynamic limit and the only size dependent quantity is d.In addition, it allows us to compute the energy gap in the limit q → 0, which is inaccessible from the full spectra of the finite systems.
For the model Hamiltonian consisting of only V 1 pseudopotential (i.e.c 1 = 1, c i>1 = 0, or the k 0 = 1 model), there are no tuning parameters, and the variational energy gap of the SMA state is completely controlled by d 3 −d 1 (see Eq.( 23)).In Fig. (2), we scan over all possible combinations of N e , N o that are numerically accessible, and compute d 1 , d 3 from the ground state in the L = 0 sector (not necessarily the global ground state).The numerical results show strong evidence that for any FQH phases that can potentially be supported by the k 0 = 1 model (which in particular includes many Abelian Jain states), the SMA states in the long wavelength limit is gapped from the ground state in the L = 0 sector.
An interesting observation is that with the V 1 model Hamiltonian and for all finite size systems we have accessed, the global ground state is in the L = 0 sector if and only if the filling factor and the topologi-cal shift corresponds to the Jain series [28], i.e.N o = (2n + 1) N e /n − n − 1, and their particle-hole conjugates.These Hilbert spaces are highlighted in Fig. (2).For all of these cases we have d 1 /d 3 < 1, indicating gapped neutral excitations as |q| → 0. For other combinations of N e , N o where the global ground state is not in the L = 0 sector, it could be because the neutral excitations go soft even for finite size systems (probably of unknown filling factors), and they could still have a charge gap.However in all cases where N o is reasonably large, d 1 /d 3 < 1 as well.There is thus no numerical evidence of the nematic FQH.For each N e , d 1 /d 3 > 1 only when N o is rather small.This implies as N e increases, we can only have d 1 /d 3 > 1 at rather large filling factors (ν 0.75).
We now move onto k 0 = 3 model Hamiltonians that are linear combinations of V 1 and V 3 pseudopotentials, where without loss of generality we set c 1 = 1.From the general expression of Eq.( 24), the allowed range of c 3 and d 5 /d 3 is given in the shaded area in Fig. (3a), which is computed analytically in the thermodynamic limit.In particular, the nematic FQH phase is not possible in the thermodynamic limit for 0.123 < c 3 < 0.462.Any numerical evidence suggesting otherwise is due to finite size effects.For the Laughlin phase at ν = 1/3, finite size analysis is carried out at different values of c 3 , at which d 5 /d 3 is computed from the global ground state (in the L = 0 sector).They show that it is very unlikely for d 5 /d 3 to be below the maximally allowed value (see Fig.  24)) also seems to be finite, which is consistent (see Fig. (3b)).While the finite size energy spectrum does seem to indicate softening of the neutral mode in the long wavelength limit (see Fig. (3b) inset) for some values of λ 1 , that most likely will not be the case when larger system sizes become accessible numerically.
For k 0 = 5 model Hamiltonians, the addition of V 5 introduces additional variables (c 5 , d 7 ), making thorough numerical investigation difficult.We look at the special case when c comes from the eigenstates of Γ (1) , i.e. c i = c λ1 i for i = 1, 3, 5, and c i>5 = 0. From the analytic expression of Eq.( 25), we have rigorous results on the range of c 3 in different scenario.At filling factor ν = 1/3 numerical computations show it is unlikely for d 3 /d 5 > 2 in the thermodynamic limit for a wide range of c 1 > c 3 > c 5 .For each value of d 3 /d 5 , we can analytically calculate the possible range of c 3 , d 7 /d 5 from Eq.( 25) for the nematic FQH phase.The results are plotted in Fig. (1).In particular, only a small range of c 3 needs to be explored for the potential realisation of the nematic FQH at ν = 1/3.
Since there is a unique relationship between λ 1 and (c 3 , c 5 ), different values of λ 1 are analysed in Fig. (4), by diagonalising the full Hilbert space and extracting d i from the corresponding ground states.In the limit of N e → ∞, d 3 /d 5 seems to fall in between 0.5 and 1.5 (see Fig. (4a)).From Fig. (1) we thus need d 7 ∼ 0.7 (without being too precise), and this also seems quite possible from FIG. 3: a).The shaded region is the possible values of (c3, d5/d3) for the nematic FQH phase, based on the analytical results of Eq.(24) in the thermodynamic limit.The upper end of the vertical dotted lines gives the upper bound of d5/d3 at different values of c3, also given by the horizontal lines in the inset with the same color code.The inset also shows the scaling of d5/d3 for different system sizes at different values of c3.b).The scaling of Eq.(24) for different system sizes and different values of c3.The inset is the energy spectrum for c3 = 0.462, and the low-lying neutral modes are highlighted in red.

Fig.(4b).
For λ 1 < −3.5, the finite size scaling of Eq.( 25) seems to clearly indicate that δE q→0 does not go soft.On the other hand, for λ 1 > −3.2, the finite size effect becomes strong, potentially indicating the divergence of the ground state correlation length and the closing of the neutral gap in the long wavelength limit (see Fig. (4c)).
We thus expect the minimal microscopic model for the nematic FQH at ν = 1/3 consists of a linear combination of V 1 , V 3 , V 5 .The results here applies to zero temperature, where spontaneous symmetry breaking can only happen at δE q→0 = 0.It is possible to have a range of parameters for the nematic FQH phase to be stable, especially if the interaction is allowed to be more long ranged.At finite temperature, nematic FQH phase can be observed as long as the neutral excitation gap is much smaller than the charge gap, and the former is smaller than the thermal energy k B T .Thus in realistic experi- mental setting, the nematic FQH phase could be stable against disorder and small perturbations, as long as the charge gap is the dominant energy scale.

VII. QUADRUPOLE AND DIPOLE EXCITATIONS IN FQH
The robustness of the nematic FQH states not only requires spontaneous symmetry breaking of the soft neutral mode in the long wavelength limit, it also requires the robustness of the charge gap, for the plateau to be present in the Hall conductivity measurement.The charge gap is given by the energy cost of creating a quasielectron or the quasihole excitation.The former should be the dominant energy scale given the incompressibility of the FQH fluid.It is generally difficult to directly explore the charge gap numerically near phase transition or level crossing.This is because it usually involves the comparison of ground state energies of different system sizes [29][30][31], leading to larger finite size effect as compared to, for example, the energy gap in the L = 0 sector from a fixed system size.In this section we will extract information about the charge gap from the neutral excitations, in conjunction with the microscopic models discussed in Sec.VI.
The low-lying neutral excitation for the FQH phase can be viewed as consisting of a pair of quasielectron and quasihole.The separation between them is proportional to the momentum of the excitation.In the long wavelength limit, the quasielectron and quasihole come on top of each other (the separation between them tends to zero), with the vanishing of the dipole moment of the excitation.Thus only the quadrupole moment remains in the long wavelength limit, and such neutral excitation can be characterised as the quadruple excitations [14].
The charge gap is related to the magnetoroton mode at large momenta.This is because at large momenta in the thermodynamic limit, the quasielectron and the quasihole are well-separated with minimal interaction between them.The energy of the neutral excitation is thus equal to the energy cost of creating one quasielectron and one quasihole.It thus gives the charge gap required for the robustness of the topological order.Unlike quadrupole excitations that are not directly accessible in the numerical calculations due to the limit in system sizes, the dipole excitations are readily accessible, though subject to non-trivial finite size effects.For all cases we explored in Sec.V, the dipole excitations at large L clearly seems to be gapped, even though at small L the quadrupole excitations could appear to go soft (see Fig.  4)).Here, we will focus on the neutral excitations in the L = 0 sector, as the low lying excitations in this sector typically consist of two dipole or quadrupole excitations, with roughly double the energy of the single excitation.Such numerical computations give clearer evidence on the nature of low-lying spectra, and the comparison of the energy gaps between the dipole and quadrupole excitations.
The half of the first excitation energy in the L = 0 sector (since it contains two neutral excitations) and the roton minimum energy for various microscopic models are plotted in Fig. (5).The range of parameters also include those suggesting that the quadrupole excitations will likely go soft at small q (as discussed in Sec.V).The physical nature of those excitations can also be revealed by comparing the wavefunctions from the model wavefunctions of two elementary excitations.The root configuration [27] for the two-quadrupole model wavefunction in the L = 0 sector is: where the solid and open circles indicate the locations of quasiparticles (of charge e/3) and quasiholes (of charge −e/3).For model wavefunctions containing two dipole excitations in the L = 0 sector, the quasiholes will be separated as far from the quasielectrons as possible [25].
For example with N e = 12, N o = 34, the root configuration for such a state is given by (see supplementary materials): In Fig. (5) we can clearly see that for small λ 1 , the lowlying excited state consist of two dipole excitations.This suggests that the quadrupole excitation energy is larger than that of the dipole excitations.In the parameter region where we expect the possibility that the quadrupole excitation to go soft, the low-lying excited state consist of two quadrupole excitations, indicating level crossing in the L = 0 sector.This strongly implies that even if the quadrupole excitation cannot become strictly gapless, the quadrupole gap can be substantially smaller than the dipole gap.In experiments at finite temperature, it is thus possible for spontaneous symmetry breaking to occur before the closing of the charge gap.

VIII. CONCLUSIONS
We have computed analytically the dynamical behaviours of the neutral excitations in the long wavelength and thermodynamic limit, which is applicable to any FQH phase with a charge gap.Such excitations are quadrupole excitations, with its gap and dispersion rela-tions captured by two universal tridiagonal matrices that are independent of microscopic details.Both the nematic order and the gapless Goldstone modes from spatially varying nematic order are studied, and the Goldstone mode dispersion is quadratic.Specific criteria for the nematic FQH phase to be robust are also derived, which are necessary (though not sufficient) conditions for the coexistence of the anisotropic transport and the topologically protected Hall conductivity plateau.
We show that the gap of the quadrupole excitation and its dispersion in long wavelength limit can be completely determined from the ground state properties, given the universality of the characteristic matrices in the thermodynamic limit.This provides a new approach in studying the potential transition from isotropic to nematic FQH phases both analytically and numerically at the microscopic level.We focused on the Laughlin phase at filling factor ν = 1/3, and numerical analysis of reasonably large system sizes show evidence that the phase transition is only likely for microscopic Hamiltonians that are linear combinations of at least three leading Haldane pseudopotentials (i.e.V 1 , V 3 , V 5 ).The analytical results can narrow down the parameter range for the short range interactions, allowing us to see tentative evidence of the softening of the neutral excitations in the fermionic systems.By extracting the quadrupole and dipole gap information from the translationally invariant neutral excitations, we also see clear evidence of level crossing between the quadrupole and dipole excitations.It implies the neutral gap in the long wavelength limit can be much smaller than the charge gap for the minimal microscopic models we studied.This serves as the first microscopic evidence for the possibility of the nematic FQH effect.
The characteristic matrices derived in this work shows that the dynamics of the quadrupole excitations has universal aspects that can potentially be useful for constructing effective theories for the FQH effects.From the microscopic perspective, more work is needed to fully understand the competition between the quadrupole gap and dipole gap for different FQH phases.The latter essentially gives the charge gap of the FQH fluid, and needs to be finite for the quantum fluid to be incompressible.Our results tentatively suggests that while short range interaction generally support a finite charge or dipole gap, it can nevertheless lead to softening of the quadrupole gap.For example the k 0 = 5 model Hamiltonians we used in the paper has a much shorter range than the lowest Landau level Coulomb interaction.On the other hand, for very short range interaction (e.g.k 0 = 1 or k 0 = 3 models), the quadrupole gap becomes very large and merge into the multi-roton continuum.The underlying physics of such behaviours is still not well understood.
In our numerical analysis we ignored the q 4 coefficient of the structure factor, which is the denominator of the quadruple gap.This should be justified since it is bounded from below by the Hall viscosity [22].Perturbation from the V 1 model Hamiltonian should only have the possibility of increasing the coefficient (thus reducing the quadrupole gap further).Nevertheless, more detailed numerical analysis is needed to see if including the q 4 coefficient can give clearer finite scaling of various aspects of the quadrupole excitations.The results in this work is also applicable for any filling factors.It is interesting to explore the possibility of the nematic FQH phases in other filling factors, especially for the non-Abelian phases where there are multiple branches of the low-lying neutral modes.= 1 8 ˆd2 qd 2 q 8π 2 V q s q e iq×q q 2 x q 2 y + q 2 y q 2 x + 2q x q x q y q y − i q x q y − q x q y θ 2 + O(θ 3 ) = θ 2 64 Γ mn (1) c m d n + O(θ 3 ) (S18) One should note that the leading order of δE α is θ 2 , so for any rotationally invariant Hamiltonian, the ground state variational energy is minimised when its intrinsic guiding center metric is undeformed.More importantly for small θ, the variational energy gap of |ψ α is in the same form as Eq.(S11), or the variational energy gap of the neutral excitations in the long wavelength limit.Thus for translationally invariant systems, for any momentum eigenstate |ψ q , we have the following relationship: e ir•(q−q ) e id•q1 e i d 2 (q −q) ψ q |δ ρq1 δ ρq−q −q1 |ψ q (S20) With q = q , the integration over the momentum amounts to the Fourier transform of the unregularised guiding center structure factor.The angle integration thus makes the nematic order vanish if |ψ q is rotationally invariant (e.g. for ground state at q = 0).For neutral excitations that break rotational invariance, there could be a uniform nematic order given by: where we have s q, b = ψ q |δ ρ dδ ρ− d|ψ q − s ∞ , da = a b d b , and unregularised part of the structure factor vanishes with the angle integration.Thus the nematic order comes from the nematic properties of the state guiding center structure factor.We are also interested in the case of q = −q, which gives us: e id•(q1−q) ψ q |δ ρq1 δ ρ2q−q1 |ψ −q (S22) Thus for the nematic order of the neutral excitations defined in the main text, |ψ ± q = 1 √ 2Sq (|ψ q ± |ψ −q ), we have the following: q ± cos 2qrN (2)   q (S23) N  e id•(q1−q) ψ q |δ ρq1 δ ρ2q−q1 |ψ −q (S25)

FIG. 2 :
FIG. 2: The value of d1/d3, computed from the ground state of the V1 interaction at different Hilbert spaces (indexed by the number of electrons Ne and number of orbitals No.).The Jain series are highlighted with different colors, where the numbers in the blankets are (Ne, No, ν).The number on top of each Ne sector is the minimum number of No included in the plot; for smaller No not included in this plot we have d1/d3 > 1.
(3a) inset) at all possible values of c 3 .The scaling shows the variational energy gap (first line of Eq.(

FIG. 5
FIG. 5: a) The quadrupole gap (from the half of the first excitation gap in the L = 0 sector) and the dipole gap (from the roton minimum gap) as the function of λ1 for the k0 = 5 model Hamiltonians; b) For the first excited state in the L = 0 sector, we look at its overlap with the model wavefunctions of the two-quadrupole excitations (from Eq.(26)), and the twodipole excitations (from Eq.(27)), as the function of λ1 for the k0 = 5 model Hamiltonians.