Studying Criticality and Excitations of One Dimensional Systems by Time Matrix Product States

TL;DR

This paper introduces a novel approach using time matrix product states (tMPS) to analyze the criticality and excitations of one-dimensional quantum systems, enabling accurate extraction of physical properties and insights into critical behavior.

Contribution

It proposes the use of tMPS as a boundary state in tensor networks to study criticality and excitations, unifying existing algorithms and providing new tools for quantum many-body analysis.

Findings

tMPS accurately captures excitation spectra and correlation lengths.

At critical points, tMPS exhibits logarithmic scaling revealing the central charge.

The spectra of transfer matrices relate to low-lying masses in the continuous limit.

Abstract

(Please refer to arXiv:1810.08050, which has completely different aims but contains all the main contents of this paper) In this work, we propose to access the information of criticality and excitations of one-dimensional quantum systems by a matrix product state (MPS) defined in the (imaginary) time direction. This state, dubbed as time MPS (tMPS), is a boundary state of tensor network (TN) that represents the ground-state simulation after Trotter-Suzuki decomposition. We show that the tMPS exhibits the structure of the continuous MPS originally proposed for the field theories. The information of excitations, e.g., dynamic correlation length and energy gap, can be accurately calculated from the tMPS. The non-universal renormalization of velocity of the excitations is given by a ratio between the correlations of the ground state and the tMPS. When the system is at the quantum critical point, the tMPS is found to show the logarithmic scaling law, where the scaling coefficient gives the central charge that characterizes the criticality of the theory. Our work also implies that the spectra of the transfer matrices defined by the ground state and tMPS provide information of the low-lying masses of the theory in the continuous limit. This could help to understand the role of the finiteness of the bond dimension as an infra red regulator. Furthermore, we show from the perspective of methodology that, the tMPS emerges from a generalized TN ab-initio optimization principle scheme, which unifies the infinite density matrix renormalization group and the infinite time-evolving block decimation algorithms.