Weak-Coupling Limit of the Lattice Nonlinear Schrödinger Integral Equation

Abstract

We study the ground-state integral equation of the quantum lattice nonlinear Schrödinger model -- equivalently the isotropic Heisenberg XXX spin chain with spin $s = -1$ -- in the weak-coupling limit. Unlike the continuous Lieb--Liniger equation, whose driving term is a constant, the lattice equation is doubly singular: both the driving term and the integral kernel degenerate into $δ$-functions as $κ\to 0$. We develop a matched asymptotic expansion with three regions -- inner, outer, and edge. The Fourier transform of the rescaled inner solution is exactly the Bose--Einstein distribution, and the peak density diverges logarithmically with a constant $C$, which we determine analytically via two independent routes and confirm numerically. An exact duality with the Love integral equation for the circular disc capacitor yields the total density expansion. We prove an identity for the inner energy, allowing us to obtain the ground-state energy per site. From the Wiener--Hopf factorisation of the edge boundary layer, we identify the instanton action and predict a resurgent transseries structure.

Figures (4)

• Figure 1: Numerical solution of the rescaled integral equation \ref{['eq:inner-eq']} for $Q = 20$, $50$, $100$, and $200$. (a) Rescaled density $\tilde{\rho}(\xi;\,Q)$ plotted against $\xi/Q$, showing the full domain $[-Q,Q]$. All curves share the same outer-region (Fermi-sea) plateau at $\tilde{\rho}_{\mathrm{bulk}} = 1/2$ (dashed line), while the central Bose--Einstein peak grows logarithmically with $Q$. The edge boundary layers at $\xi/Q = \pm 1$ are visible as the rapid drop from the plateau to zero. (b) Zoom into the inner region $|\xi| \lesssim \mathcal{O}(1)$, plotted against the unscaled variable $\xi$. The peak height at $\xi = 0$ increases as $\tilde{\rho}(0;\,Q) \sim (\log Q)/\pi + C$ (see \ref{['eq:rho0-asymp']}), with the logarithmic growth clearly visible across the four values of $Q$. The cusp-like shape reflects the $1/|p|$ infrared singularity of the Bose--Einstein distribution \ref{['eq:BE-distribution']} in Fourier space, and the universal profile is controlled by the digamma function $\operatorname{Re}\psi(1+i\xi)$. In panel (b), the dashed curves show the analytical inner approximation $\tilde{\rho}(\xi;Q) \approx [\log(2Q) - \operatorname{Re}\psi(1+i\xi)]/\pi$ from \ref{['eq:inner-profile']}, which matches the numerical solution to graphical accuracy for $|\xi| \lesssim Q/2$.
• Figure 2: Spectral gap of the truncated kernel $\mathcal{K}_Q$. (a) Log--log plot of $\Delta_n(Q) = 2\pi - \lambda_n(Q)$ for $n=0$ (circles) and $n=1$ (squares) versus $Q$. The shaded region indicates the proven bounds $c_1/[Q\,(\log Q)^2] \leq \Delta_0(Q) \leq c_2/Q$. Both gaps close algebraically, with $\Delta_0$ separated below $\Delta_1$ for all $Q$. (b) Compensated gap $Q\cdot\Delta_0(Q)$ versus $Q$ on a semi-logarithmic scale. If $\Delta_0$ scaled as $1/Q$, the product $Q\cdot\Delta_0$ would be constant; instead, it grows slowly, well fitted by $Q\cdot\Delta_0 \approx 6.43 + 0.15\,\log Q$ (dashed line), confirming the logarithmic correction to the $1/Q$ law.
• Figure 3: Edge boundary layer at $\xi = Q$. (a) Raw edge profiles $\tilde{\rho}(Q-s;\,Q)$ versus the distance $s = Q - \xi$ from the Fermi boundary, for $Q = 20$, $50$, $100$, $200$, and $400$. The overall scale decreases with $Q$ (reflecting the $Q$-dependent outer-region density), while the $s^{1/2}$ onset predicted by the Wiener--Hopf factorisation is visible at small $s$ (dashed guide). (b) Shape-collapsed profiles $\Psi(s) = \tilde{\rho}(Q{-}s;\,Q)/\tilde{\rho}(Q{-}s_{\mathrm{ref}};\,Q)$ with $s_{\mathrm{ref}} = 10$. For $Q \geq 100$, the curves collapse onto a single universal profile, confirming the $Q$-independence of the edge boundary layer. The dashed line shows the $s^{1/2}$ behaviour near the Fermi boundary.
• Figure 4: Schematic structure of the rescaled density $\tilde{\rho}(\xi;\,Q)$ showing the three asymptotic regions. Inner region ($|\xi| \lesssim \mathcal{O}(1)$, blue shading): the Bose--Einstein peak with Fourier transform $\hat{\tilde{\rho}}(p) = 1/(e^{|p|}-1)$\ref{['eq:BE-distribution']}, rising $\sim\!\log Q/\pi$ above the Fermi sea. Outer region ($1 \ll |\xi| \ll Q$, green shading): the uniform Fermi sea with $\tilde{\rho}_{\mathrm{bulk}} = 1/2$\ref{['eq:bulk-value']}, contributing the leading $\mathcal{O}(Q)$ term to the total density. Edge boundary layers ($|\xi \mp Q| \lesssim \mathcal{O}(1)$, red shading): the transition from the Fermi sea to zero, governed by the Wiener--Hopf factorisation of $\Sigma(p) = 1 - e^{-|p|}$\ref{['eq:WH-symbol-main']}. Wavy arrows indicate the asymptotic matching between adjacent regions. The rescaled Fermi boundary is $Q = q/\kappa \to \infty$ as $\kappa \to 0$.

Theorems & Definitions (13)

• Proposition A.1
• Theorem A.2: Szegő's first limit theorem
• proof : Proof sketch
• Corollary A.3: Eigenvalue counting function
• proof
• Proposition A.4: Spectral gap vanishing rate
• proof
• Remark A.5
• Proposition A.6: Leading-eigenvalue dominance
• proof