Gaussian continuous tensor network states: short-distance properties and imaginary-time evolution

TL;DR

This work develops and analyzes Gaussian continuous tensor network states (GCTNS) as a continuum, analytically tractable variational class for quadratic quantum field theories. It demonstrates that their short-distance limits flow to free Lifshitz vacua with even dynamical exponent $z$, and introduces two principled ground-state construction schemes: rational dispersion approximants and Trotterized imaginary-time evolution, both benchmarked on Klein–Gordon theory. The study reveals that GCPEPS encode dispersions that are rational in $(kpq)^2$, produce a soft UV cutoff, and exhibit Lifshitz-like entanglement scaling, with a universal finite piece in 1D approaching the expected $g_0=1/3$ for the Klein–Gordon fixed point. While GCPEPS cannot fully capture the linear relativistic dispersion at arbitrarily short distances, they provide controlled, low-energy approximations and concrete insights into entanglement structure, offering a benchmark for extending to interacting QFTs and for connecting to non-Gaussian continuum tensor networks. The results illuminate the strengths and limitations of GCPEPS as relativistic quantum-field variational ansätze and guide future developments toward handling UV physics and potential non-Gaussian extensions.

Abstract

We study Gaussian continuous tensor network states (GCTNS) - a finitely-parameterized subclass of Gaussian states admitting an interpretation as continuum limits of discrete tensor network states. We show that, at short distance, GCTNS correspond to free Lifshitz vacua, establishing a connection between certain entanglement properties of the two. Two schemes to approximate ground states of (free) bosonic field theories using GCTNS are presented: rational approximants to the exact dispersion relation and Trotterized imaginary-time evolution. We apply them to Klein-Gordon theory and characterize the resulting approximations, identifying the energy scales at which deviations from the target theory appear. These results provide a simple and analytically controlled setting to assess the strengths and limitations of GCTNS as variational ansätze for relativistic quantum fields.

Figures (4)

• Figure 1: Imaginary-evolution GCPEPS for a massive deformation of the $z\mathbin=2$ free Lifshitz field theory. Dispersions $\omega_{N,(T/N)}$ from a second-order Trotter decomposition with $mT\mathbin=4$, with and without truncating [\ref{['eq_evo_truncation']} and \ref{['eq_evo_W']} respectively]. The black dotted line depicts the exact dispersion $\omega({ \ProcessList{}{ \StrChar{kpq}{}} }) = m^2 + { \ProcessList{}{ \StrChar{kpq}{}} }^2$.
• Figure 2: (a) rational [\ref{['eq_rational_cf']}] and (b),(c) imaginary-time [\ref{['eq_evo_W']}] GCPEPS approximants $\mathcal{K}$ of the Klein-Gordon dispersion relation $\omega$ (black line) for different number $\chi$ of auxiliary fields. In (b) and (c), different colors denote different decompositions, while darker lines correspond to larger $\chi$. Dotted lines represent the asymptotes in \ref{['tab_asymptotics']}. In (b), the evolution time $mT=2$ is fixed and the Trotter step $m(T/N) =mT/\chi \in\{1/2,\dots,1/32\}$ shrinks with $\chi$. In (c), $m(T/N)=1/64$ is fixed and $mT =m(T/N)\chi \in\{1/16,\dots,2\}$; notice that all decompositions overlap in the IR, visually matching the exact dispersion for $mT\gtrsim2$.
• Figure 3: Lattice-regularized EE $S(\ell)$ of the exact Klein-Gordon vacuum on a ring (black curve), and of its (a) rational and (b) $mT=4$ imaginary-time GCPEPS approximations. Regulators: $a/\xi = e^{-3} \approx 0.05$, $L/\xi \approx 10\pi$ (closest integer multiple of $a/\xi$). The black curve shows the exact (regularized) EE. Dotted grid lines mark $\ell(\xi) \approx \xi$ (vertical) and $S=(1/3)\log(\xi/a)=1$ (horizontal).
• Figure 6: Pointwise convergence of Klein-Gordon's rational approximant $\omega_{\chi}$ in \ref{['fig_dispersion_rational']}. Each color refers to the magnitude of the relative deviation of $\omega_{\chi}({ \ProcessList{}{ \StrChar{kpq}{}} })$ from $\omega({ \ProcessList{}{ \StrChar{kpq}{}} })$ for one value of ${ \ProcessList{}{ \StrChar{kpq}{}} }/m$, as a function of $\chi$. The expansions in \ref{['eq_cf_err_expansions']} (cyan) are accurate already for ${ \ProcessList{}{ \StrChar{kpq}{}} }/m\in\{1/4,4\}$. Dashed lines are guides to the eye.

Theorems & Definitions (5)

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