Limiting Free Energy in Lattice Maxwell Theory

• The paper derives an explicit closed-form expression for the universal constant K_d, quantifying leading free energy corrections in lattice gauge theories.
• It employs rigorous spectral analysis and Gaussian integration under axial gauge to reduce gauge redundancy and isolate physical degrees of freedom.
• The work illustrates how boundary corrections vanish in the thermodynamic limit, ensuring universal free energy formulations for both Abelian and non-Abelian models.

The limiting free energy of lattice Maxwell theory quantifies the infinite-volume free-energy density of Abelian gauge fields on a $d$-dimensional lattice, and encapsulates the universal, nontrivial contribution to the leading term in the free energy of lattice Yang-Mills theories as the volume and, where appropriate, the lattice cutoff become large. The calculation centers on evaluating Gaussian integrals under a specific gauge-fixing scheme—most notably axial gauge—and extracting the thermodynamic limit while controlling the boundary- and gauge-dependent subleading corrections. The key structural object is a universal constant, $K_d$, which arises as the normalized logarithmic determinant of a discrete Maxwell (Laplacian-like) operator projected onto physical degrees of freedom. This constant appears explicitly in closed form, as derived by finite-dimensional analysis and continuum Riemann-sum limits (Brennecke, 10 Nov 2025, Chatterjee, 2016).

1. Quantitative Formulation of the Lattice Maxwell Free Energy

Lattice Maxwell theory is defined on a finite hypercube $$\Lambda_n = \{0,1,\ldots,n\}^d \subset \mathbb{Z}^d$$, with oriented nearest-neighbor edges $E_n$ and a collection of real-valued gauge fields $(u_e)_{e\in E_n^1}$, where $E_n^1$ are the “free” edges not fixed by the maximal tree $T_n$ (imposing axial gauge). The quadratic action is prescribed by the discrete circulation $u_p$ of $u$ around each plaquette $p\in P_n$,

$\Sigma_n^0(u,u) = \sum_{p \in P_n} u_p^2,$

with $u_e=0$ for constrained edges $e\in E_n^0$. The finite-volume Maxwell partition function is

$$Z_n^M = \int_{\mathbb{R}^{E_n^1}} \exp\!\left(-\tfrac{1}{2}\Sigma_n^0(u,u)\right) \prod_{e\in E_n^1} du_e = (2\pi)^{|E_n^1|/2} [\det \Sigma_n^0]^{-1/2}.$$

The number of unconstrained edges satisfies $|E_n^1| = (d-1) n^d - O(n^{d-1})$, and in the thermodynamic limit the normalized log-partition function

$$\lim_{n\to\infty} \frac{1}{n^d} \log Z_n^M = \frac{d-1}{2} \log(2\pi) + K_d$$

is well defined. The universal constant,

$$K_d = - \lim_{n \to \infty} \frac{1}{2 n^d} \mathrm{Tr} \log \Sigma_n^0,$$

is the focus of the explicit characterization.

2. Operator-Theoretic Reduction and Spectral Structure

The quadratic form $\Sigma_n^0$ corresponds, via extension and symmetrization of edge variables, to a discrete one-form $w$ defined on edges at each site. The key result is that $\Sigma_n^0$ can be written as

$$\Sigma_n^0(u,u) = \langle w, Q_d w \rangle - \langle w, R_d w \rangle,$$

where $Q_d$ is a translation-invariant Maxwell operator acting on $\ell^2(\mathbb{Z}^d; \mathbb{R}^d)$: $$(Q_d w)_i(x) = -\Delta w_i(x) - \sum_{j=1}^d \partial_i \partial_j^* w_j(x),$$ with $\Delta$ the usual lattice Laplacian and $\partial_i$ the forward difference operator. The boundary-correction term $R_d$ is supported near the lattice boundary and has rank $O(n^{d-1})$, rendering its contribution to the free energy density vanishing in the infinite-volume limit.

Projection onto the physical (axial gauge) subspace $\Omega_n^{1,a}$ (imposed by $T_n$) leads to the reformulation: $$K_d = -\lim_{n\to\infty} \frac{1}{2 n^d} \mathrm{Tr} \log (\Pi_{\Omega_n^{1,a}} Q_d \Pi_{\Omega_n^{1,a}}).$$ This isolates the contribution from the non-gauge-redundant sector.

3. Boundary Conditions and Zero Modes

Through a unitary embedding into a larger discrete torus $T_{n+5}^d$ with periodicity, the original operator can be replaced (up to vanishing boundary corrections) by the periodic Maxwell operator $Q_d^{\mathrm{per}}$ on the periodic subspace $\Omega_n^{1,a,p}$. Crucially, $Q_d^{\mathrm{per}}$ possesses a finite-dimensional kernel corresponding to pure gradients (arising from gauge invariance), but this kernel impacts only an $O(n^{d-1})$-dimensional subspace.

By orthogonally projecting away the zero modes, the calculation reduces further: $$K_d = -\lim_{n\to\infty} \frac{1}{2 n^d} \mathrm{Tr} \log (\Pi_{\Omega_n^{1,p,+}} Q_d^{\mathrm{per}} \Pi_{\Omega_n^{1,p,+}}),$$ where $\Omega_n^{1,p,+} = (\ker Q_d^{\mathrm{per}})^\perp$.

4. Explicit Evaluation and Closed-Form Formula for $K_d$

On the finite torus $T_n^d = \mathbb{Z}^d / n\mathbb{Z}^d$, the forward/backward differences are diagonalized by plane waves $e_p(x) = n^{-d/2} e^{2\pi i p \cdot x}$ for $p \in \Gamma_n^* = [0,1)^d \cap (1/n)\mathbb{Z}^d$. The Laplacian eigenvalue is $$\varepsilon(p) = 2 \sum_{k=1}^d [1 - \cos(2\pi p_k)]$$. For $p$ such that all $p_k \neq 0$, the spectrum of $Q_d^{\mathrm{per}}$ on the orthogonal complement to gradients consists of one eigenvalue $2[1-\cos(2\pi p_d)]$ and $(d-2)$ copies of $\varepsilon(p)$. Summing over momenta and passing to Riemann integrals, one obtains

$$K_d = -\frac{d-1}{2} \log 2 - \frac{1}{2} \int_0^1 dx \log[1-\cos(2\pi x)] - \frac{d-2}{2} \int_{[0,1]^d} d^d x \, \log\left( \sum_{k=1}^d [1-\cos(2\pi x_k)] \right),$$

where all integrals are over the unit $d$-cube. This expression is universal and independent of lattice artifacts or boundary conditions, capturing purely bulk contributions.

5. Role in Lattice Yang-Mills Theory and Continuum Limits

The constant $K_d$ appears additively in the leading-order free energy density for lattice $\mathrm{U}(N)$ Yang-Mills theories. In $d$ dimensions and for $N\ge1$,

$$f(a) = a^{-d} \biggl[ (d-1) N^2 \log [g^2 a^{4-d}] + (d-1) \log \frac{\prod_{j=1}^{N-1} j!}{(2\pi)^{N/2}} + N^2 K_d \biggr] + o(a^{-d}),$$

for lattice spacing $a$, generalized coupling $g_0^2 = a^{4-d} g^2$, and $K_a\to K_d$ as $a\to 0$ (Chatterjee, 2016). For $N=1$, the formulas specialize directly to Abelian lattice Maxwell theory. In three dimensions,

$$f(a) = a^{-3} \left[ 2\log(e^2) - \log(2\pi) + K_3 \right] + o(a^{-3}),$$

where $e^2$ is the continuum electric charge.

The evaluation of $K_d$ completes the explicit formula for the free energy's leading term, an advance made possible without recourse to phase-cell or block-spin renormalization, but instead by precise spectral analysis in fixed gauge.

6. Methodological Context and Analytical Techniques

The approach is characterized by several interconnected methodologies:

• Gauge fixing to the axial gauge by a maximal tree, drastically reducing the number of degrees of freedom and removing gauge redundancy.
• Quadratic approximation of the action in the weak-coupling regime ($g_0\ll 1$), exploiting the concentration of holonomies near the identity.
• Reduction to a finite-dimensional Gaussian integral, computation of determinants via spectral range separation (removal of zero modes), and boundary correction estimates of rank $O(n^{d-1})$.
• Application of plane wave expansion and Riemann-sum arguments to extract infinite-volume and continuum limits.
• Avoidance of multi-step block-spin renormalization or phase-cell decompositions, streamlining derivation of explicit constants.

The core result rests on the ability to characterize and control the kernel and range of $Q_d^{\mathrm{per}}$, to estimate the negligible effects of boundaries, and to calculate spectral densities in the large-$n$ thermodynamic limit.

7. Broader Implications and Interpretive Remarks

The limiting free energy of lattice Maxwell theory, through the explicit evaluation of $K_d$, serves as a universal additive correction to the thermodynamic free energy for a broad range of lattice gauge theories, including non-Abelian Yang-Mills models in weak-coupling or continuum scaling limits (Brennecke, 10 Nov 2025, Chatterjee, 2016). The explicit nature of $K_d$ is of further significance as it applies across dimensions $d\ge2$, is insensitive to ultraviolet regularization details, and admits computation by elementary spectral means.

A common misconception is that such universal constants might depend heavily on the choice of gauge or on lattice boundary effects; in fact, rigorous estimates show the $O(n^{d-1})$ dependence of gauge and boundary corrections, ensuring the universality of $K_d$ in the infinite-volume limit. The analytical strategy demonstrates that precise spectral and linear-algebraic techniques are sufficient to provide fully explicit, closed-form formulas for quantities previously thought to require non-constructive or renormalization-based arguments.

A plausible implication is that analogous constants for other lattice field theories with Gaussian (quadratic) sectors are similarly approachable by this spectral-projection technology, suggesting routes to explicit formulations in related statistical or quantum field-theoretic models.