The exact singlet-loop response and protected auxiliary gap

Use frame freedom to isolate the loop variable

The Hamiltonian is H=ΔQ−tΣ_j=0³(T_j+T_j†) on the 4m²-dimensional carrier–dual space, with P=|s〉〈s| and Q=I−P. A spanning-tree frame choice sets the first three forward links to identity. The remaining link is the based holonomy W=U_0,3U_3,2U_2,1U_1,0. A simultaneous root-carrier and dual basis change diagonalizes W without changing the singlet.

Write the eigenvalues of W as e^iθa, and let K=tΣ(T_j+T_j†). Then H=ΔI−(K+ΔP). For each eigenangle, the mobile carrier moves on a twisted four-position ring with eigenvalues

Each mobile mode has root probability 1/4. The normalized singlet assigns weight 1/(4m) to each displayed mode. The stationary dual repeats every free mobile eigenvalue m times in the full auxiliary Hilbert space. Those multiplicities are essential for the exact gap statement. §2.2.1, p.9.

Derive the rank-one secular equation

Seek an eigenvalue z of K+ΔP above its free maximum λ*. The equation (K+ΔP)ψ=zψ can be rearranged as ψ=Δ(z−K)⁻¹s〈s,ψ〉. Contracting with s gives 1=ΔR_W(z), where

The ground energy of H is E_m(W)=Δ−z(W). The sum can also be written using one fourth-degree denominator per eigenangle:

For one twisted ring, its characteristic polynomial is D_θ(z)=Π_k(z−λ_ak)=z⁴−4t²z²+2t⁴(1−cosθ). The root diagonal resolvent is D′_θ/(4D_θ), giving the rational form. At special twists apparent cancellations can obscure poles; the spectral sum remains the safest representation. Theorem 2.1, equations (4)–(5), pp.9–10.

The physical branch is unique

For z>λ*, every denominator in the spectral sum is positive and R′_W(z)=−(1/(4m))Σ(z−λ_ak)⁻²<0. The resolvent diverges as z approaches λ* from above because the singlet has nonzero weight on that eigenspace, and it tends to zero as z→∞. Hence ΔR_W(z)=1 has exactly one solution on this branch.

The corresponding eigenvector is proportional to (z−K)⁻¹s. Rank-one eigenvalue interlacing places every other eigenvalue of K+ΔP at or below λ*. Thus its largest eigenvalue is simple and H has a unique ground state. The statement holds at arbitrary real hopping; a small parameter is not needed for this exact finite spectrum. Theorem 2.1 and proof, pp.9–10.

Clearing denominators creates a polynomial equation with roots associated with other spectral branches and possibly apparent factors at cancellations. An arbitrary real root is not the physical solution. Both the analytic proof and a numerical implementation must impose z>λ*. The root’s existence and uniqueness are theorem-level facts; numerical diagonalization is useful verification, not their justification.

Derive the exact gap and its sharp lower bound

Let w*=‖1_{λ*}(K)s‖². Since the top-mode contribution is positive, the secular equation gives 1=ΔR(z)≥Δw*/(z−λ*), hence z−λ*≥Δw*. Each maximizing mobile mode contributes 1/(4m), so w*≥1/(4m). Rank-one interlacing first gives gapH≥z−λ*.

For m≥2, the top free eigenspace has dimension at least two because of the stationary-dual multiplicity. A vector in it orthogonal to s remains an eigenvector of K+ΔP with eigenvalue λ*. The first excited level of H is therefore exactly Δ−λ*, and

The last uniform lower bound is sharp: choose eigenangles with a unique maximizing mobile mode and let |t|/Δ grow. All other free modes separate from the maximum, leaving z−λ*→Δ/(4m). For m=1 the top eigenspace need not contain an orthogonal vector, so the theorem does not assert the same gap equality. At t=0, H=ΔQ has E=0 and gapΔ directly. Theorem 2.1, pp.9–10.

Trivial holonomy is the strict energy minimum

At W=I all blocks have the same free ring. The physical root z₀ is independent of m and obeys

At t=0 use z₀=Δ. For nonzero t and m≥2, the identity auxiliary gap is z₀−2|t|≥Δ/4, stronger than the general Δ/(4m) bound. There are m maximizing mobile modes at identity, so their combined singlet weight is 1/4. The bound is approached as |t|/Δ becomes large.

To compare a general W with I, evaluate its rational resolvent at z₀. The numerator z₀(z₀²−2t²) is positive and independent of angle. Each denominator equals its identity value plus 2t⁴(1−cosθ_a)≥0. Thus R_W(z₀)≤R_I(z₀), with equality only when every eigenvalue is one. Monotonicity implies z(W)≤z₀ and E(W)≥E(I), strictly unless W=I. Corollary 2.2, p.10.

A preferred frozen holonomy is an energy result within this model. It is not by itself a vacuum theorem for a quantum many-link Hamiltonian, whose electric motion and state space add independent structure.

Solve in dimensionless units with a safe bracket

Set ε=t/Δ, ζ=z/Δ, and ℓ_ak=2εcos[(θ_a+2πk)/4]. The dimensionless equation is (1/(4m))Σ(ζ−ℓ_ak)⁻¹=1, and the readouts are E/Δ=1−ζ and gap/Δ=ζ−maxℓ_ak. Only the hopping-to-binding ratio enters the normalized spectrum.

A safe bisection bracket is [maxℓ+1/(4m), maxℓ+1]. At the lower endpoint the top-mode contribution makes the resolvent at least one; at the upper endpoint every denominator is at least one, so the average is at most one. No evaluation at a pole is required. If the midpoint resolvent exceeds one, move the lower endpoint up; otherwise move the upper endpoint down. Handle ε=0 exactly.

Compute the identity root with the same procedure and subtract its energy to obtain ΔE=E(W)−E(I). This avoids solving a cleared polynomial and fixes the normalization at every hopping strength. Negative hopping is allowed; position parity relates t and −t, so the energy is even in t. The eigenangle list may be wrapped modulo 2π without changing the free multiset. These numerical choices follow directly from the proved spectral formula. Theorem 2.1 and the parity argument in Theorem 2.5, pp.9–12.

Compare exact and weak-hopping responses

As a reproducible example, take m=3, Δ=1, and W=diag(−1,1,−1), the φ=π member of diag(e^−iφ,1,e^iφ). It has determinant one and m−ReTrW=4. The table is calculated by bisection of the exact source equation; the leading response is 8t⁴/3.

The fourth-order approximation becomes poor as hopping grows, while the exact ground band remains isolated. The theorem’s explicit perturbative remainder applies only for |t|/Δ<1/4, so the row at 0.25 is already at the boundary and the larger rows are comparisons with a formal leading expression. They do not use that error theorem outside its domain. Theorems 2.1 and 2.5, pp.9–12.

At identity for t=0.1, E(I)=−0.0199923164410 and the exact gap is 0.819992316441. These values provide useful checkpoints for an independent implementation.

Recover the ground state’s singlet weight

The resolvent also determines the normalized ground vector. Since ‖(z−K)⁻¹s‖²=−R′_W(z), it is ψ₀=(z−K)⁻¹s/√[−R′_W(z)]. Its probability of the root-singlet projection is therefore

This expression is a direct consequence of the eigenvector formula in Theorem 2.1. It quantifies how the bound ground state mixes the root singlet with virtual excursions. It equals one at t=0 and is bounded by one because it is a normalized-state probability. Differentiating the ground energy with respect to Δ gives the complementary penalty expectation 〈Q〉=1−|〈s,ψ₀〉|².

This ground-state overlap differs from the finite-time survival experiment |〈s,e^−iτH/ℏs〉|², which starts in the bare singlet and lets all eigenmodes interfere. A ground-band approximation, a prepared dressed state, and an instantaneous completion test need their own preparation descriptions. Theorem 2.1, p.9; Theorem 20.4, pp.61–62.

Keep the auxiliary gap distinct from a continuum mass gap

The exact bound concerns a finite auxiliary Hamiltonian at each fixed unitary-link configuration. It is uniform in hopping and holonomy for fixed positive Δ and rank m, which is a strong model-level statement. It does not give uniform control after an unbounded electric Laplacian is included, nor does it identify the first excitation of an infinite-volume gauge field theory.

The comprehensive manuscript treats bounded electric dispersion separately through an effective-block theorem with a norm remainder and dressed evolution. Removing the electric cap requires high-representation tail estimates; increasing volume and taking a continuum limit require further uniform bounds and a matching construction. None follows from the rank-one resolvent alone. §2.3.2 and §28.3, pp.16–17 and 78.

The exact energy can nevertheless support a focused test of the microscopic model. Measure or calculate its small-angle Hessian in central and traceless directions, compare it with the exact coefficient formula, and identify which extra terms would cause a discrepancy. This is a test of a specified Hamiltonian class, with a clear separation from the open question of physical continuum gauge dynamics.

Continue reading

• The microscopic Hamiltonian and virtual paths
• Exact small-curvature stiffness
• Which mass-gap claims remain open?
• Controlled effective dynamics
• The full SU(3) holonomy domain: alcoves, traces and exact response
• Falsifying a finite model: controls, calibration and predictions kept separate