Many-body localisation implies that eigenvectors are matrix-product states (1409.1252v2)

Abstract: The phenomenon of many-body localisation received a lot of attention recently, both for its implications in condensed-matter physics of allowing systems to be an insulator even at non-zero temperature as well as in the context of the foundations of quantum statistical mechanics, providing examples of systems showing the absence of thermalisation following out-of-equilibrium dynamics. In this work, we establish a novel link between dynamical properties - the absence of a group velocity and transport - with entanglement properties of individual eigenvectors. Using Lieb-Robinson bounds and filter functions, we prove rigorously under simple assumptions on the spectrum that if a system shows strong dynamical localisation, all of its many-body eigenvectors have clustering correlations. In one dimension this implies directly an entanglement area law, hence the eigenvectors can be approximated by matrix-product states. We also show this statement for parts of the spectrum, allowing for the existence of a mobility edge above which transport is possible.

• The paper demonstrates that strong dynamical localisation in quantum systems implies exponentially clustering eigenvector correlations, which in one-dimensional systems allows eigenvectors to be approximated by matrix-product states (MPS).
• The study rigorously establishes a theoretical link between dynamic localisation properties and static eigenvector entanglement structure using advanced mathematical techniques such as Lieb-Robinson bounds.
• This work provides theoretical support for applying tensor network methods like MPS to simulate many-body localised systems and suggests avenues for future research on reversing the implication and characterising local constants of motion.

Many-Body Localisation and Matrix-Product States in Quantum Systems

The paper of many-body localisation (MBL) continues to unveil intricate connections between quantum statistical mechanics and condensed matter physics. This paper explores a novel relationship between dynamical and entanglement properties of quantum systems exhibiting MBL. Specifically, it establishes that strong dynamical localisation implies the description of eigenvectors in the matrix-product state (MPS) formalism, providing a significant link between dynamic and static interpretations of MBL.

Main Contributions

The paper makes two primary contributions:

1. Clustering Correlations of Eigenvectors: The paper demonstrates that in systems exhibiting strong dynamical localisation—characterised by a vanishing group velocity and the absence of transport—eigenvectors display exponentially clustering correlations. This result holds under assumptions of non-degenerate Hamiltonian spectra with locally independent gaps, indicative of a mobility edge. In one-dimensional systems, this clustering directly implies that eigenvectors adhere to an entanglement area law, suggesting they can be efficiently approximated by MPS.
2. Implications for Matrix-Product States: For one-dimensional systems, the clustering result is extended to show that eigenvectors can be approximated by an MPS with a bond dimension that scales polynomially with system size and inverse error. This aligns with the predicted absence of high entanglement in MBL phases and connects with rigorous mathematical descriptions of gapped phases in quantum spin systems.

Methodology

The authors employ advanced mathematical techniques, leveraging Lieb-Robinson bounds and energy filtering. Specifically, Gaussian and high-pass filter functions serve to separate and handle off-diagonal elements in the Hamiltonian energy basis, permitting a step-by-step construction of localised Hamiltonians that reveal exponential clustering. Mobilising these methods, the authors navigate the delicate balance between locality, the decoupling of energy states, and entropy scaling laws to support their claims.

Numerical and Theoretical Implications

Theoretical results find support in numerical analyses, notably demonstrating that the entanglement entropy in models such as the disordered Heisenberg chain distributes in correlation with energy levels. This implies possible scaling efficiencies investigating these systems with tensor network methods. Moreover, the connection to ETH violations and the potential for non-thermalisation in some MBL regimes provide further impetus for exploration in figures of condensed matter and quantum information.

Conclusions

The critical insight of the paper is the rigorous establishment of a relationship between dynamical localisation and the structural entanglement properties of eigenvectors, yielding significant implications for MBL theory. Future research could aim to reverse the current results, establishing that eigenstates approximated by MPS imply dynamical localisation. Furthermore, exploring the utility of local constants of motion derived from energy filters might deepen the understanding of integrability in MBL phases, potentially guiding the design of better simulation algorithms for these systems.

This work represents a stepping stone in a broader quest to precisely characterise the nature of MBL, offering a significant link between dynamical phenomenology and static eigenvector properties in complex quantum many-body systems.