Abelian Tensor Models on the Lattice

TL;DR

This work analyzes lattice realizations of Abelian KT tensor models by assembling KT sites into one-dimensional chains and examining the gauge-singlet spectrum for small lengths. It finds that, unlike their large-N cousins, these Abelian chains show Poisson-like level statistics and numerous mid-spectrum degeneracies, pointing toward quasi-many-body localization rather than maximal chaos. The study systematically characterizes the spectrum across 2–5-site chains, explores large-g and symmetric-hopping limits, and uses spectral form factors and thermodynamics to reinforce the qMBL picture. The results illuminate the finite-N landscape of tensor models, offering a stepping stone toward understanding localization in tensor-based quantum systems and suggesting avenues to extend to Abelian Gurau-Witten and more general tensor networks.

Abstract

We consider a chain of Abelian Klebanov-Tarnopolsky fermionic tensor models coupled through quartic nearest-neighbor interactions. We characterize the gauge-singlet spectrum for small chains ($L=2,3,4,5$) and observe that the spectral statistics exhibits strong evidences in favor of quasi-many body localization.

Figures (19)

• Figure 1: The tetrahedron contraction of the gauge indices in the Hamiltonian of the KT model \ref{['def:KT model Hamiltonian']}. Each vertex of the tetrahedron corresponds to the fermion $\psi_{ijk}$, and the each edge represents the gauge contraction of the corresponding color between two fermions.
• Figure 2: (a) $U(1)^3$ singlet spectrum : Rescaled eigenvalues $E/\sqrt{\frac{1}{2}(\lambda_r^2 + \lambda_g^2 + \lambda_b^2) + g^2}$ of the three-site Abelian KT chain model Hamiltonian with asymmetric hopping couplings against the coupling ratio $\sigma$ (degeneracies not shown). Here $\sigma \equiv \frac{g^2}{\frac{1}{2}(\lambda_r^2 + \lambda_g^2 + \lambda_b^2) + g^2}$. (b) $U(1)^3$ singlet spectrum : Rescaled eigenvalues $E/\sqrt{\frac{3}{2}\lambda^2 + g^2}$ of the three-site Abelian KT chain model Hamiltonian with symmetric hopping couplings against the coupling ratio $\sigma$ (degeneracies not shown). Here $\sigma \equiv \frac{g^2}{\frac{3}{2}\lambda^2 + g^2}$.
• Figure 3: The spectral staircase function $N(E)$ of the three-site KT chain model for asymmetric case of the hopping couplings. The plot is for $\sigma = 0.5$, which corresponds to $\lambda_r/g = 0.8255, \lambda_g/g = 0.3005$ and $\lambda_b/g = 1.1083$. Note that here $E$ is measured in units of the on-site coupling $g$.
• Figure 4: (a) The histogram of nearest neighbor spacing distribution $P(s)$ against $s$ for the three-site Abelian KT chain model with asymmetric hopping couplings. Level clustering is evident in the model, indicating the the system is integrable. (b) The histogram of $r$-parameter distribution $P(r)$ against $r$ for the three-site Abelian KT chain model with asymmetric hopping couplings. In both cases the plots are for the coupling ratio $\sigma = 0.5$, which corresponds to $\lambda_r/g = 0.8255, \lambda_g/g = 0.3005$ and $\lambda_b/g = 1.1083$; and the fits are to Poisson distribution.
• Figure 5: (a) $U(1)^3$ singlet spectrum : Rescaled eigenvalues $E/\sqrt{\frac{1}{2}(\lambda_r^2 + \lambda_g^2 + \lambda_b^2) + g^2}$ of the four-site Abelian KT chain model Hamiltonian with asymmetric hopping couplings against the coupling ratio $\sigma$ (degeneracies not shown). Here $\sigma \equiv \frac{g^2}{\frac{1}{2}(\lambda_r^2 + \lambda_g^2 + \lambda_b^2) + g^2}$. (b) $U(1)^3$ singlet spectrum : Rescaled eigenvalues $E/\sqrt{\frac{3}{2}\lambda^2 + g^2}$ of the four-site Abelian KT chain model Hamiltonian with symmetric hopping couplings against the coupling ratio $\sigma$(degeneracies not shown). Here $\sigma \equiv \frac{g^2}{\frac{3}{2}\lambda^2 + g^2}$.