navstokgap

The strong-coupling gap of the Kogut–Susskind Hamiltonian is uniform in the volume

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Statement T2 of the obligations map holds in Hamiltonian form: for every compact connected gauge group \(G\) and every lattice size, the Kogut–Susskind Hamiltonian at coupling \(g\ge g_0(G)\) has a non-degenerate ground state and a gap \[\Delta_{a,L}\ \ge\ \gamma\,\frac{g^2}{2}\,C_2(R_{\min})\,\frac{\hbar c}{a},\] with \(\gamma>0\) independent of the lattice size, and the physical (gauge-invariant) gap is at least as large. The proof is a hypothesis check against Yarotsky’s Theorem 1 on relatively bounded perturbations of classical lattice systems (Commun. Math. Phys. 261 (2006) 799, Sections 1 and Theorem 1 read at passage level): the electric term is a classical Hamiltonian in the Peter–Weyl partition with a product ground state and unit local gap after normalization, and the magnetic term is a bounded, translation-invariant, finite-range perturbation whose normalized size is \(\beta(g)=48N^2/[g^4(N^2-1)]\) for \(SU(N)\), so the theorem’s smallness condition \(\beta\le\beta_*(3,\Lambda_0)\) is the strong-coupling condition \(g^4\ge48N^2/[(N^2-1)\beta_*]\). The same theorem gives the infinite-volume ground state and exponential clustering at these couplings. The constants \(\beta_*\) and \(\gamma\) are existential in Yarotsky’s statement, so \(g_0\) is a well-defined but not numerically explicit threshold. The result does not distinguish abelian from non-abelian groups; that distinction is T2\('\), and the rest of the problem is the descent of \(g_0\) to zero along the scaling curve. No claim is promoted.

1. The Hamiltonian in Yarotsky’s normalization

Notation as in the obligations map §1: periodic cubic lattice \(\Lambda=(\mathbb Z/N_s\mathbb Z)^3\), links \(\mathcal E\), plaquettes \(\mathcal P\), \(\mathcal H=L^2(G^{\mathcal E})\), and \[H=\frac{\hbar c}{a}\Big[\frac{g^2}{2}\sum_{\ell}(-\Delta_\ell) +\frac{2}{g^2}\sum_{p}\big(N-\operatorname{Re}\operatorname{tr}U_p\big)\Big].\] Let \(C_2\equiv C_2(R_{\min})\) be the smallest nonzero Casimir eigenvalue of \(-\Delta\) on \(L^2(G)\); for \(SU(N)\) with \(\operatorname{tr}T^aT^b=\tfrac12\delta^{ab}\), \(C_2=(N^2-1)/(2N)\). Divide by the unit \(\varepsilon_0=(g^2/2)C_2\,\hbar c/a\): \[\frac{H}{\varepsilon_0}=\sum_{x\in\Lambda}h_x+\sum_{x\in\Lambda}\phi_x,\qquad h_x=\frac1{C_2}\sum_{i=1}^3(-\Delta_{(x,x+e_i)}),\qquad \phi_x=\frac{4}{g^4C_2}\sum_{i<j}\big(N-\operatorname{Re}\operatorname{tr}U_{p(x,i,j)}\big),\] where \(p(x,i,j)\) is the plaquette with corner \(x\) spanned by \(e_i,e_j\). Every link belongs to exactly one \(h_x\) and every plaquette to exactly one \(\phi_x\), so the two sums reproduce \(H/\varepsilon_0\).

2. Hypothesis check

Yarotsky’s setting (Section 1 of the paper): a lattice \(\mathbb Z^\nu\), site spaces \(\mathcal H_x\) possibly infinite-dimensional with a preferred vector \(\Omega_x\), translation-invariant interactions of finite range \(\Lambda_0\), a classical part \(H_0=\sum_xh_x\) where \(h_x\) is a function of a product partition of unity containing the projection onto \(\Omega_x\) and satisfies \(h_x\Omega=0\), \(h_x\ge1\) on the orthogonal complement of the product vector, and a perturbation \(\sum_x\phi_x\) with each \(\phi_x\) a symmetric form on \(\mathcal H_{\Lambda_0+x}\) obeying \(|\phi_x(v,v)|\le\alpha\|h_x^{1/2}v\|^2+\beta\|v\|^2\) with \(\alpha<1\). Theorem 1 there: there are \(\alpha_*,\beta_*>0\) depending only on \(\nu\) and \(\Lambda_0\) such that if \(\alpha\le\alpha_*\) and \(\beta\le\beta_*\), then for periodic cubic volumes \(H_\Lambda\) has a non-degenerate ground state with a gap \(\gamma>0\) independent of \(\Lambda\), the ground states have a weak\(^*\) thermodynamic limit, and that limit has exponentially decaying correlations.

  1. Lattice and site spaces. \(\nu=3\); \(\mathcal H_x=L^2(G)^{\otimes3}\), the three links leaving \(x\); \(\Omega_x\) the constant function. Then \(\mathcal H_\Lambda=\bigotimes_x\mathcal H_x=L^2(G^{\mathcal E})\) and \(\Omega_{\Lambda,0}\) is the constant function on \(G^{\mathcal E}\).
  2. Partition and classicality. On each link, Peter–Weyl gives the orthogonal partition of unity by isotypic projections \(P_R\), \(R\) ranging over irreducible representations, with \(P_{\rm triv}\) the projection onto constants. \(-\Delta_\ell=\sum_RC_2(R)P_R\) is a function of this partition, so \(h_x\) is a function of the product partition on \(\mathcal H_x\), and \(h_x\) is self-adjoint and unbounded, as allowed.
  3. Local ground state and gap. \(h_x\Omega_x=0\). On the orthogonal complement of \(\Omega_x\) in \(\mathcal H_x\) at least one of the three links carries a nontrivial representation, so \(h_x\ge C_2(R_{\min})/C_2=1\) there: condition (1) of the paper holds with the range \(\Lambda_0=\{0\}\) for \(h_x\), which may be enlarged to the common range below.
  4. Perturbation. \(\phi_x\) is multiplication by a continuous function on the twelve links of the three plaquettes at \(x\), whose sites lie in \(\Lambda_0+x\) with \(\Lambda_0=\{0,1\}^3\); it is symmetric, bounded, and \(0\le N-\operatorname{Re}\operatorname{tr}U_p\le2N\) gives \[|\phi_x(v,v)|\le\beta\|v\|^2,\qquad \beta=\frac{4}{g^4C_2}\cdot3\cdot2N=\frac{24N}{g^4C_2} =\frac{48N^2}{g^4(N^2-1)}\ \ (SU(N)),\] so condition (2) holds with \(\alpha=0\) and this \(\beta\).
  5. Translation invariance and periodicity. \(h_x\) and \(\phi_x\) are translates of \(h_0\) and \(\phi_0\), and the lattice is a periodic cube, the case in which the paper states Theorem 1.

Theorem (T2, Hamiltonian form). Let \(\beta_*=\beta_*(3,\{0,1\}^3)\) and \(\gamma\) be the constants of Yarotsky’s Theorem 1 for this range. For every \(g\) with \(48N^2/[g^4(N^2-1)]\le\beta_*\), that is \(g\ge g_0(N)=\big(48N^2/[(N^2-1)\beta_*]\big)^{1/4}\), and every \(N_s\ge2\): \(H\) has a non-degenerate ground state, and \[\Delta_{a,L}\ \ge\ \gamma\,\varepsilon_0=\gamma\,\frac{g^2}{2}\,\frac{N^2-1}{2N}\,\frac{\hbar c}{a}.\] The physical gap obeys \(\Delta^{\rm phys}_{a,L}\ge\Delta_{a,L}\) by Theorem 1(3) of the obligations map. The ground states have a thermodynamic limit \(\omega\) with exponentially decaying correlations of local observables.

Proof. Items 1–5 are the hypotheses of Yarotsky’s Theorem 1 with \(\alpha=0\le\alpha_*\) and \(\beta\le\beta_*\); its conclusions 1–3 are the statements, after multiplying the normalized gap \(\gamma\) by \(\varepsilon_0\). The inheritance by the physical sector is the positivity argument already written for T1. \(\square\)

For a general compact connected \(G\) the same holds with \(C_2=C_2(R_{\min})\) and the bound \(N-\operatorname{Re}\operatorname{tr}U_p\le2\dim\) in place of \(2N\); the threshold is \(g_0^4=24\dim/(C_2\beta_*)\).

3. What the theorem does and does not give

4. Consequence for STATE

T2 is closed in Hamiltonian form as a corollary of an established theorem, with the threshold \(g_0(N)=(48N^2/[(N^2-1)\beta_*])^{1/4}\). The obligations table moves T2 to “established”; the open items are T2\('\), T3 and T4. The next theorem-sized target on the lattice route is a weak-coupling counterpart: a variational upper bound and, if possible, a lower bound on the Kogut–Susskind gap at small \(g\) on a lattice whose side is small against \(\hbar c/\Lambda\), using the constant-mode ground state of C133 as the trial state, to connect the lattice gap \(\Delta_{a,L}\) with the continuum small-volume corner and to locate the regime where neither expansion applies.