Identifying Criticality in Higher Dimensions by Time Matrix Product State

TL;DR

This paper introduces a novel method using time matrix product states (tMPS) to identify criticality in higher-dimensional quantum many-body systems by analyzing 1D imaginary time states, enabling efficient characterization of critical behavior.

Contribution

The work develops a new approach linking higher-dimensional ground states to 1D tMPS, allowing criticality detection through established 1D scaling schemes, with validation on complex lattice models.

Findings

tMPS accurately distinguishes gapless and gapped phases.

Critical scaling is size-insensitive, effective on small systems.

Identifies gapless ground state of kagome Heisenberg antiferromagnet.

Abstract

Characterizing criticality in quantum many-body systems of dimension $\ge 2$ is one of the most important challenges of the contemporary physics. In principle, there is no generally valid theoretical method that could solve this problem. In this work, we propose an efficient approach to identify the criticality of quantum systems in higher dimensions. Departing from the analysis of the numerical renormalization group flows, we build a general equivalence between the higher-dimensional ground state and a one-dimensional (1D) quantum state defined in the imaginary time direction in terms of the so-called time matrix product state (tMPS). We show that the criticality of the targeted model can be faithfully identified by the tMPS, using the mature scaling schemes of correlation length and entanglement entropy in 1D quantum theories. We benchmark our proposal with the results obtained for the Heisenberg anti-ferromagnet on honeycomb lattice. We demonstrate critical scaling relation of the tMPS for the gapless case, and a trivial scaling for the gapped case with spatial anisotropy. The critical scaling behaviors are insensitive to the system size, suggesting the criticality can be identified in small systems. Our tMPS scheme for critical scaling shows clearly that the spin-1/2 kagom\'e Heisenberg antiferromagnet has a gapless ground state. More generally, the present study indicates that the 1D conformal field theories in imaginary time provide a very useful tool to characterize the criticality of higher dimensional quantum systems.