Entanglement scaling and criticality of infinite-size quantum many-body systems in continuous space addressed by a tensor network approach

TL;DR

This paper develops a tensor network method to simulate infinite continuous-space quantum systems, revealing entanglement and criticality behaviors, including scaling laws and conformal field theory signatures, in ground states of coupled quantum oscillators.

Contribution

It extends tensor network algorithms to continuous-space quantum systems, demonstrating critical entanglement scaling and identifying the breakdown of CFT with three-body couplings.

Findings

Logarithmic entanglement entropy scaling at criticality

Polynomial correlation length scaling with bond dimension

Central charge c=1 indicating free boson CFT

Abstract

Simulating strongly-correlated quantum systems in continuous space belongs to the most challenging and long-concerned issues in quantum physics. This work investigates the quantum entanglement and criticality of the ground-state wave-functions of infinitely-many coupled quantum oscillators (iCQOs). The essential task involves solving a set of partial differential equations (Schr\"odinger equations in the canonical quantization picture) with infinitely-many variables, which currently lacks valid methods. By extending the imaginary-time evolution algorithm with translationally-invariant functional tensor network, we simulate the ground state of iCQOs with the presence of two- and three-body couplings. We determine the range of coupling strengths where there exists a real ground-state energy (dubbed as physical region). With two-body couplings, we reveal the logarithmic scaling law of entanglement entropy (EE) and the polynomial scaling law of correlation length against the virtual bond dimension $\chi$ at the dividing point of physical and non-physical regions. These two scaling behaviors are signatures of criticality, according to the previous results in quantum lattice models, but were not reported in continuous-space quantum systems. The scaling coefficients result in a central charge $c=1$, indicating the presence of free boson conformal field theory (CFT). We further show that the presence of three-body couplings, for which there are no analytical or numerical results, breaks down the CFT description at the dividing point. Our work reveals the scaling behaviors of EE in continuous-space quantum many-body systems. These results provide strong numerical evidence supporting the efficiency of TN in representing continuous-space quantum wave-functions in the thermodynamic limit and offer an efficient approach to studying entanglement properties and criticality in continuous space.