A Fock-space inequality and the Laughlin spectral gap

1 The gap and the model

Laughlin’s wave function gave an explicit description of the fractional quantum Hall state at filling 1/3 [12]. Haldane’s spherical geometry and pseudopotentials identify a particularly simple parent Hamiltonian: every pair pays energy one in relative angular momentum one and zero in the other pair sectors [7]. Its exact Laughlin zero mode is known, but a uniform lower bound on the positive spectrum requires control of all competing many-particle states. We prove this bound for the full interaction on the round sphere, resolving positively the spherical fermionic Laughlin spectral-gap conjecture.

The gap question has accompanied this model since its early development. Haldane and Rezayi’s finite-size study supplied numerical evidence and explicitly distinguished the expected hard-core gap from a formal proof [8]. The density-wave variational picture of Girvin, MacDonald, and Platzman explains important features of the excitation spectrum [6]. A variational excitation energy is an upper bound; the lower bound sought here must hold for every state orthogonal to the zero modes.

1.1 Normalization and main results

For an integer Q\ge1, let U_Q be the spin-Q/2 representation of SU(2), equipped with its invariant inner product. It is the one-particle lowest Landau level on the sphere with Q flux quanta. Equivalently, it is the space of homogeneous polynomials of degree Q in spinor coordinates (u,v). For 2\le N\le Q+1, define

where P_{ij}^{(1)} is the orthogonal projector onto pair spin Q-1 in tensor positions i,j. This is relative angular momentum one. The sum preserves the antisymmetric subspace. Each unordered pair has coefficient one, with no rescaling depending on N or Q.

Set H_{0,Q}=H_{1,Q}=0, and write

Our principal estimate concerns this entire Fock space.

Thus the spectrum is contained in \{0\}\cup[\gamma,\infty). In a particle sector without a zero mode, this statement bounds the energy away from zero; it does not assert a gap between that sector’s two lowest eigenvalues. At Laughlin filling the zero mode is unique, and we obtain the usual ground-state spectral gap.

At fixed magnetic length, the sphere radius is proportional to \sqrt Q, so this is the two-dimensional thermodynamic limit. The thresholds in Theorem 1.1 and Corollary 1.2 are existential. The proof does not give an explicit flux threshold or the endpoint coefficient \gamma_* on finite spheres.

1.2 Prior work

The exact ground-state structure is an established part of the theory. Besides Haldane’s spherical construction, Trugman and Kivelson obtained exact Laughlin ground states for repulsive interactions in a vanishing-range limit [23]. The second-quantized algebra of pseudopotentials was developed systematically by Ortiz, Nussinov, Dukelsky, and Seidel [18]. Chen and Seidel, and Mazaheri, Ortiz, Nussinov, and Seidel, gave algebraic constructions and characterizations of their zero modes [2, 15]. These results identify the zero modes that a spectral estimate must preserve. We include a short self-contained proof of the known spherical zero-mode uniqueness at the Laughlin flux in Lemma 6.2.

For Laughlin states on a cylinder of any fixed radius, Jansen established thermodynamic correlation limits and axial periodicity [10]. Exact ground states of related finite-range orbital chains, including squeezed and matrix-product states, were constructed by Nakamura, Wang, and Bergholtz [17] and by Wang and Nakamura [24]. Nachtergaele, Warzel, and Young proved uniform gaps for a truncated fermionic pseudopotential [16]; Warzel and Young obtained thin-torus bounds that separate bulk behavior from open-chain edge modes [25]. Their invariant-subspace method also treats a truncated bosonic pseudopotential [26]. The fermionic estimates already hold uniformly across particle sectors. The distinction in Theorem 1.1 is the full spherical pair interaction, with its fixed projector normalization and sphere radius growing in the thermodynamic limit. The finite support used in our auxiliary comparison operators does not truncate that interaction.

Recent work gives further structural and spectral information. Lemm, Nachtergaele, Warzel, and Young derive recursions between particle sectors and compare charge and neutral gaps [13]; a uniform bound for the full pseudopotential still requires control of the model-specific ground-space overlap matrices in those criteria. Schraven and Warzel prove entanglement-gap and clustering results for Laughlin states on sufficiently thin cylinders [22]. The entanglement spectrum is distinct from the Hamiltonian spectrum considered here. Density rigidity within the family of fully correlated Laughlin states is another complementary form of incompressibility, established by Lieb, Rougerie, and Yngvason [14]. No density-screening assumption enters our argument.

Rougerie’s survey records the spectral-gap conjecture and its geometric formulations [21, Appendix A]. For N fermions in the planar lowest Landau level, consider the sum over unordered pairs of the orthogonal projectors onto relative polynomial degree one, each with coefficient one and with center-of-mass degree unrestricted. Section 7 proves that its positive spectrum is bounded below by the endpoint \gamma_*, uniformly in particle number N\ge2 and total polynomial degree L. In particular, this proves the fermionic cubic case of the folklore spectral-gap conjecture in [21, Conjecture A.1], whose degree range is L\le3N(N-1)/2.

1.3 Proof strategy

Proving positivity of H^2-\gamma H is a standard route to a gap, underlying finite-size criteria [11] and methods based on positive local decompositions [3, 20]. Our argument uses rotations and angular-momentum decomposition to construct such a comparison. It starts from the normal-ordered expansion of H_Q^2. It has two-, three-, and four-body terms. We compare them with the corresponding terms of a positive operator

where each F_\rho is a specified linear combination of a pair annihilator and operators that annihilate three particles and create one. Here F_\rho(g) denotes rotation by g, and Haar measure has mass one. These auxiliary operators involve finitely many orbital modes; the Hamiltonian itself is never truncated.

Rotational symmetry reduces the comparison to fixed spin blocks. The three-body blocks have an explicit spectrum and a summable tail. For four particles, coupling the interacting pair to the different spin sectors of a second pair can produce several copies of the same total spin. Their four-particle images under exterior multiplication can be linearly dependent. For the retained blocks, seven rows of rational data establish finite inequalities in their large-flux limits. The remaining issue is to transfer those inequalities to finite spheres without an error that grows with the number of particles.

The key device is an exact diagonal change of variables for the pair coefficients. On a fixed set of orbital modes, it converts finite-flux annihilators into their limiting forms. Every perturbation is then bounded by the local pair energy. Rotation averaging turns this into o(1)H_Q on the full Fock space. This relative estimate remains valid when the Gram matrices of these limiting four-particle images have kernels. A second useful observation is to apply Bessel’s inequality to all equivalent spin copies at once; this substantially reduces the four-body comparison error.

The comparisons leave a positive margin:

All errors and convergence thresholds are independent of particle number. Figure 1 isolates the passage responsible for this uniformity. A small norm error on a growing many-particle space would not suffice: it must vanish on the zero modes and be controlled by the pair energy. The fixed-mode comparison gives precisely that control before rotations distribute it over the sphere.

Section 2 derives the normal-ordered square, and Section 3 establishes the spin formulas. Section 4 gives the finite comparisons. Section 5 proves their uniform transfer, and Section 6 controls the tail and proves the spherical results. Section 7 derives the planar consequence. The exact row data, rational formulas, and complete arithmetic procedure for the positivity checks are in Appendix A.

2 Exterior algebra and the square of the interaction

Our first task is to isolate the two-, three-, and four-body terms of H_Q^2. The lifting notation below keeps this calculation independent of particle number.

Throughout the operator argument take Q\ge2; statements involving limits hold for all sufficiently large integer Q. We use the convention that wedges of distinct increasing orthonormal basis vectors have norm one. On

let B(v) be the adjoint of left wedge multiplication by v. Thus B(v) is conjugate-linear in v, and

For an orthonormal basis e_0,\ldots,e_Q of U_Q, write c_j=B(e_j). These operators satisfy the canonical anticommutation relations, and \norm{c_j}\le1.

For a matrix on \bigwedge^k U_Q, define its lift by linear extension of

This map preserves positivity, intertwines rotations, acts as the original matrix on k particles, and vanishes on fewer than k particles. Positivity follows by decomposing a positive matrix into a sum of rank-one positive matrices.

Put a=Q-1 and let V_Q\subset\bigwedge^2 U_Q be the spin-a summand. If v_0,\ldots,v_{2Q-2} is an orthonormal basis of V_Q, set

This pair-annihilator form is standard in the algebraic treatment of pseudopotentials [18, Sections II.B–D]; see also [2, Equation (2)]. We verify its coefficient-one normalization here. Its restriction to \bigwedge^N U_Q is (1). Indeed contraction in two specified tensor positions corresponds to B(v)/\sqrt{\binom N2} under the isometric identification of wedge and antisymmetric tensor spaces; summing over the \binom N2 unordered pairs gives (6).

Define intertwining wedge maps

by W_3(v\otimes u)=v\wedge u and W_4(v\otimes w)=v\wedge w. We suppress the subscript Q when no confusion can result.

A finite normal-ordering calculation in the sense of Wick [27] now determines the square. Only contractions between the two middle pair operators are needed.

3 Spin coefficients and their limits

The three-body comparison will use the exact spin spectrum of the wedge map. The four-body comparison will use only finitely many coupling coefficients and their limits.

The spherical pair coefficients can be expressed through angular-momentum coupling [18, Section II.C and Appendix A]. We derive the particular coefficients and fixed-deficit limits needed here to keep their phases and normalization explicit.

We now fix e_p and v_p to be the normalized lowering bases of U_Q and V_Q, respectively, with v_0=-e_0\wedge e_1. For spin j, lowering from label p to p+1 has coefficient \sqrt{(p+1)(2j-p)}. Write (s)_p=s(s-1)\cdots(s-p+1), with (s)_0=1.

The elementary tensor product rule decomposes spins j_1,j_2 into one copy of each spin j_1+j_2-z, for 0\le z\le\min(2j_1,2j_2). This rule can be seen directly in the polynomial model: at deficit z the raising equation determines a highest vector, and normalized lowering generates its 2(j_1+j_2-z)+1 orthogonal weight vectors. Different highest spins give orthogonal invariant subspaces, and the sum of these dimensions is (2j_1+1)(2j_2+1), exhausting the tensor product. Let R_{3,z} be the corresponding projection in V_Q\otimes U_Q, and put

For Q\ge2, the allowed indices are 0\le z\le Q.

We will repeatedly use a fixed-deficit limit of this same calculation. In a product of spins j_1,j_2, total lowering deficit means T=j_1+j_2-m, where m is the total magnetic quantum number. Coefficients below are with respect to normalized monomials X^pY^{T-p}/\sqrt{p!(T-p)!}, and are zero outside the stated index ranges.

4 Rotational averages of explicit squares

We construct a positive rotational average and compare its three- and four-body terms with those of H_Q^2. This section specifies the finite inequalities needed; Appendix A proves them by exact rational arithmetic.

Haar measure dg on SU(2) is normalized to have total mass one. For an operator A, write A(g)=U(g)AU(g)^\dagger, with the appropriate Fock representation. Schur averaging sends a rank-one operator on an irreducible spin-j space to its trace times I/(2j+1). For several equivalent copies, it acts in the same way on the spin factor and retains the matrix between copies. Explicitly, if the representation is \bigoplus_j(V_j\otimes\mathbb C^{m_j}) and P_j is its spin-j projection, then

Here \operatorname{Tr}_{V_j} is the partial trace over the spin factor. This follows from Schur’s lemma applied to each pair of irreducible copies, with the scalar fixed by its trace; see [1, Section II.C]. In particular, averaging preserves off-diagonal entries between equivalent spin copies.

The positive-square construction belongs to the quadratic-gap approach described in Section 1. Fermionic positivity constraints, normal ordering, and symmetry also appear in the quantum Hall bootstrap of Gao et al. [4], which bounds ground-state energies. Here the explicit squares are compared with H_Q^2-\gamma H_Q to control the positive excitation spectrum.

For each of the seven rows in Appendix A, the integer t_\rho lies in \{0,\ldots,7\}. Set

The row specifies real coefficients \lambda_\rho and \alpha^\rho_{pj} for (p,j)\in\mathcal S_\rho; unlisted entries are zero. Every coefficient sum for row \rho below is over \mathcal S_\rho. Our comparison operator is

Each summand of F_\rho removes two particles and decreases the sum of occupied orbital labels by t_\rho+1: for the second term the decrease is (p+1)+j-(p+j-t_\rho)=t_\rho+1. This common weight allows the two parts to interfere in their square, adjusting the three-body comparison at the two-body cost \lambda_\rho^2B_{t_\rho}^\dagger B_{t_\rho}. Only modes 0,\ldots,8 occur in these auxiliary squares.

For a single row, omit \rho and put i=p+j-t, k=q+l-t. Define

Reordering c_ic_k^\dagger=\delta_{ik}-c_k^\dagger c_i gives the two-, three-, and four-body terms of F^\dagger F:

Let A_2,A_3,A_4 denote these terms, summed over rows and averaged with factor d_a. Thus

4.1 The three-body comparison

Before wedging, (16) is a matrix on V_Q\otimes U_Q. Its nonzero entries preserve T=p+j, and T\le15. Averaging therefore gives

for all sufficiently large Q, where e_z^Q is the trace of the unaveraged matrix in that spin block. By Lemma 3.3, with u=1/\sqrt3, these traces converge to

where s=s^{(1/\sqrt3)} and empty inner sums are zero. The condition i=k in (16) is exactly the equality of the two values of T.

The exact calculation in Appendix A proves

There are finitely many strict inequalities, d_{3,z}/d_a\to3/2, and the omitted blocks z=0,1,3 are annihilated by W_3. Consequently, for all sufficiently large Q,

4.2 The four-body comparison before taking limits

Let

be the wedge map. The second tensor factor decomposes into spins Q-r, with r odd, 1\le r\le Q. In its lowering bases use the sign of Lemma 3.3; at r=1 this is precisely the basis v_p.

A copy of spin 2Q-1-D is obtained by coupling a=Q-1 with Q-r at deficit D-r. Denote its vector at total deficit T by \ket{D,T;r}_Q. Its physical orbital-index sum is 1+T, and it is a highest vector when T=D. For fixed D,T and sufficiently large Q, the allowed copy labels are

Define the real finite-flux coefficients by

and extend them antisymmetrically in j,k. By two applications of Lemma 3.3, they converge to

interpreted as zero if r>j+k. The formula holds for either ordering of j,k, by antisymmetry. It follows by applying Lemma 3.3 first to the two single-particle spins and then to the two pair spins.

Only 1\le D\le23 contribute: every unaveraged term has

Put d_D=4Q-1-2D. In the coupled bases \ket{D,T;r}_Q, the spin-2Q-1-D block is identified with \mathbb C^{d_D}\otimes\mathbb C^{\mathcal R_D}: the first factor uses the (T-D)-th normalized spin descendant, and the second has basis vector indexed by r. Let E_D^Q be the partial trace over the spin factor of the compression to this block of the matrix representing (17) before wedging, summed over rows. Schur averaging then gives, for all sufficiently large Q,

where the direct sum acts as zero on all other spin blocks. For these finitely many D, entrywise convergence gives E_D^Q\to E_D, with

where i=p+j-t_\rho, k=q+l-t_\rho, and T=p+j+k. The exact finite-flux matrix E_D^Q is given by the same formula with each S_r replaced by S_r^Q. Exchanging the two ordered entry pairs exchanges r and s, so both matrices are real symmetric.

The target term \LL_4(W_4W_4^\dagger) is obtained from the r=1 summand, since that summand is V_Q\otimes V_Q. Thus its copy matrix, in the units just used, is (d_D/d_a)\chi, where

For even D, the r=1 coupled vector is antisymmetric under exchange of its two spin-a factors, since its coupling deficit is D-1. Wedging two degree-two vectors is symmetric, so this target column then vanishes after applying \widehat W. The formula with \chi still represents it exactly; we never assume that the wedged copy vectors are independent.

Target spin blocks outside 1\le D\le23 contribute positively and may be discarded in a lower bound.

Set

The star is a limit label, not an adjoint. Let W_D^* be the map with columns w_{Dr}^*. With allowance \eps=3\cdot10^{-6}, the limiting comparison on these four-particle images asks that W_D^*(2\chi-E_D+\eps I)(W_D^*)^\dagger be nonnegative. Lemma 5.1 identifies this with the exact certificate

Appendix A specifies and verifies it entirely by rational arithmetic.

5 A relative error bound for finite spheres

The limiting Gram matrices have kernels, so their positivity cannot be transferred by assuming that their ranges vary continuously. Instead, we compare the annihilators themselves. An exact diagonal change of variables reduces every error to a fixed finite family of pair-generated annihilators. This gives an error relative to H_Q, uniformly on the full Fock space.

Geometry-dependent diagonal changes of occupation-number coordinates already relate pseudopotential annihilators and their zero modes [18, Section II.E]. For a gap estimate, the needed additional statement is quantitative: on a fixed set of modes the change of variables approaches the identity, and its error is controlled by pair energy. The proof below establishes that statement before adding spectator modes and averaging over rotations.

For the local argument, let \FF_{\mathrm{loc}} be the exterior algebra of \mathbb C^{25} with orthonormal basis e_0,\ldots,e_{24}. For Q\ge24 we identify it with the Fock space of the first 25 orbital modes in U_Q. By (9), each limiting pair vector v_p^* with p\le23 lies in \bigwedge^2\mathbb C^{25}. Contraction by these vectors is defined on \FF_{\mathrm{loc}} by the same exterior-algebra convention as before.

The allowance \eps I in the certificate must also be paid for in pair energy. Here it is useful to sum over the orthonormal highest vectors before estimating their wedges. Bessel’s inequality then avoids an extra factor equal to the number of spin copies.

6 The tail estimate and the gap

The finite comparisons are now in place. We first bound all remaining three-body blocks by a summable multiple of H_Q, then identify the zero mode at Laughlin flux.

We now recover the known unique zero mode of Haldane’s spherical parent Hamiltonian [7, p. 606, Equations (6)–(7)]. The elementary divisibility proof also fixes the exact flux at which the kernel is a line.

The following comparison specializes the recursive spectral relations of Lemm, Nachtergaele, Warzel, and Young [13, arXiv v1, Theorem 2.1(I), Lemma 2.3, and (2.10)].

7 A planar corollary

The full-Fock-space estimate also gives a gap on the plane. We use the analytic lowest-Landau-level representation described by Girvin and Jach [5, Section II], with Gaussian scale fixed below. Let \mathcal B be the Bargmann space of entire functions with norm

Multiplication by \pi^{-1/2}e^{-|z|^2/2} identifies this space unitarily with the planar lowest Landau level at magnetic field 2; changing magnetic length is a unitary dilation. For a pair of coordinates set w=(z_1-z_2)/\sqrt2 and Z=(z_1+z_2)/\sqrt2. Let P^{(1)} be the orthogonal projector onto relative degree one, with no restriction on the degree in Z, and put

Thus each pair projector has coefficient one, as on the sphere.

A The finite rational certificate

This appendix specifies the seven rows and all arithmetic needed to verify (21) and (28). Starting from the integer rows in Section A.1, evaluate the rational three-body traces in Section A.2 and the rational four-body matrices in Section A.3. The margin and coefficient-sign tables in those sections record the resulting checks. The formulas below determine every value using only integers and rational numbers.

A.1 Row data

Let P=10^7. The first two numbers of each row are t and \ell=P\lambda. An entry pj:a specifies the integer a_{pj}, with p and j parsed as separate single-digit labels; omitted entries are zero. Continuation lines belong to the preceding row. Define

Every nonzero entry has 0\le p+j-t\le8.

In particular P^2\eta=\sum_\rho\ell_\rho^2=93527408868499.

A.2 Three-body arithmetic

For c\in\{1,2\}, an integer T\ge0, and integers z,p, define

and set it to zero if z or p lies outside [0,T]. Expanding (13) gives

Substitution in (20) and (39) cancels all radicals:

where X_{\rho,T}=\sum_{p+j=T}a^\rho_{pj}\mathsf U(2,z,T,p), with value zero on empty levels. For each z\in\{2,4,5,\ldots,15\}, evaluate (41) and form the exact rational margin m_z=\frac32(3z-1)(-1/2)^z-e_z. Write 10^6m_z=n_z/d_z with integers n_z,d_z and d_z>0. Integer division gives the floors \lfloor n_z/d_z\rfloor listed below:

All are positive, proving (21).

A.3 Four-body arithmetic

Fix 1\le D\le23; all copy indices below lie in \mathcal R_D. Put

The matrices Y,Z are rational. Define

with value zero for r>j+k. Two applications of (13) give

Therefore

where i=p+j-t_\rho, k=q+l-t_\rho, and T=p+j+k. Both entry sums are ordered. To verify the cancellation, multiply the two factors in (42) by the two factors (39); the factorial product under the combined square root is (t_\rho!i!k!)^2u_ru_s/((T-D)!)^2. Using j+k=T-p and l+i=T-q, its power of two has exponent 2+2p+2q-2t_\rho-4T+r+s. Taking the positive square root and removing \sqrt{u_ru_s} gives exactly the displayed factor.

For an increasing quadruple A of nonnegative indices with sum D+1, put f(A)=\prod_{b\in A}b!. If (x,y,j,k) lists its elements, let \sigma(x,y,j,k) be the sign of sorting this list into increasing order. In each summand below put p=x+y-1, and define

Combining (9) and (42), the coefficient of w_{Dr}^* on the Slater wedge A is \sqrt{u_r/f(A)}L_r(A). Thus

All exponents of two in these formulas are integers, since r,s are odd.

Let D_u=\diag(u_r) and D_{\sqrt u}=\diag(\sqrt{u_r}). The matrix in (28) equals D_{\sqrt u}MD_{\sqrt u}, where

Thus it suffices to check M\ge0 using rational arithmetic. This congruence uses only the positive diagonal D_{\sqrt u}; it does not invert G_D or presume that its columns are independent.

To check positivity for each D=1,\ldots,23, form Y and Z from (43) and (45), then form B and M from (46), using \eps=3/10^6. All sums range over the printed rows and the finite index sets specified above. Compute with exact fractions throughout. If d=|\mathcal R_D|, initialize J=I_d. For b=1,\ldots,d, compute in order

This is the Leverrier–Faddeev recurrence applied to -M; see [9]. It gives \det(xI+M)=x^d+c_1x^{d-1}+\cdots+c_d. For example, this recurrence follows by multiplying the polynomial adjugate of xI+M by xI+M and comparing coefficients, together with the derivative-of-determinant trace identity. Exact evaluation of (40), (43), (44), (45), and (47) gives:

The signs list the coefficients after the leading 1, in descending degree. For each computed coefficient, choose a positive denominator and inspect its integer numerator: + denotes a positive numerator and 0 denotes zero. We use the characteristic-coefficient positivity test for symmetric matrices [19, Theorem 5]. Every coefficient is nonnegative, so \det(xI+M)>0 for x>0. Since M is real symmetric, a negative eigenvalue would give a positive root of this polynomial. Therefore M\ge0, proving (28).