Quantum States Reveal How Materials Transition Between Order and Disorder

A thorough analysis of quantum state complexity reveals fundamental differences between ergodic and many-body localised (MBL) phases in disordered quantum spin chains. Bikram Pain and colleagues at the International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, present a method for distinguishing these phases by examining the spread of quantum states within a Krylov space. The research shows that complexity scales linearly with system size in the ergodic phase, signifying extensive state propagation, whereas it grows sublinearly in the MBL phase, indicating a restricted and localised state. This collaboration between the International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, and the University of Oxford provides new insight into how complexity emerges from the interplay of disorder and interactions in quantum systems and highlights the key role of eigenstate contributions to long-time behaviour.

Mapping quantum state evolution using Krylov space complexity

Krylov space complexity offers a novel perspective on the differences between ergodic and many-body localised (MBL) phases of quantum matter. The technique constructs a mathematical space by repeatedly applying the governing rules of the quantum system, specifically, the Hamiltonian, to an initial state, analogous to creating a series of increasingly complex wave patterns from a single disturbance. This process generates a sequence of vectors, forming a Krylov basis which effectively captures the system’s dynamics. Tracking the spread of a quantum state within this space yields a basis-optimised measure of its complexity, effectively charting its evolution independently of a specific starting point. The Krylov subspace, therefore, provides a computationally efficient way to analyse the system’s behaviour without needing to solve the full many-body problem directly. This approach circumvents the limitations of traditional methods, such as entanglement analysis, which can be computationally expensive and difficult to interpret in strongly interacting systems, providing a complementary way to characterise these distinct quantum phases and their behaviours. Entanglement, while a powerful tool, doesn’t always fully capture the nuances of state propagation in disordered systems. The Krylov space method focuses on the effective dimensionality of the Hilbert space explored by the evolving state, offering a different lens through which to view quantum dynamics.

Long-time Krylov complexity differentiates ergodic and many-body localised phases in spin chains

Bikram Pain, David E. Logan, and Sthitadhi Roy at the International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, and the University of Oxford have demonstrated that the long-time spread of complexity, measured using this method, sharply distinguishes between ergodic and many-body localised (MBL) phases. This improvement surpasses previous methods reliant on entanglement entropy analysis, which often required extensive computational resources and were susceptible to finite-size effects. A basis-optimised measure of quantum state complexity now facilitates this distinction, offering a more efficient approach to characterising quantum behaviour. The numerical study involved simulating disordered spin chains and observing the behaviour of quantum states over extended timescales, confirming that Krylov space complexity reliably captures the fundamental differences between these two phases.

Infinite-time complexity scales linearly with the Fock-space dimension in this phase, demonstrating the quantum state spreads across a finite portion of the chain. Specifically, the complexity grows proportionally to the number of possible quantum states within the system, indicating extensive state propagation and thermalisation. Conversely, complexity grows sublinearly in the MBL phase, indicating the long-time state occupies only a negligible fraction of the chain, with the infinite-time state profile exhibiting a stretched-exponential decay. This sublinear growth signifies that the quantum state remains confined to a small region of the system, preventing thermalisation and establishing the hallmarks of localisation. Large-deviation analysis further reveals that the ergodic phase utilises contributions from nearly all eigenstates, whereas MBL complexity is dominated by few eigenstates with unusually high complexity. This suggests that in the ergodic phase, the system explores a vast number of possible configurations, while in the MBL phase, the dynamics are restricted to a few dominant states, effectively freezing the system’s evolution.

Mapping quantum state behaviour using Krylov space complexity in disordered magnetic systems

Understanding the origins of complexity in quantum systems is vital for unlocking new technologies, particularly in the fields of quantum computation and materials science, but pinpointing the boundary between order and disorder remains elusive. The behaviour of quantum systems in the presence of disorder is a central challenge in condensed matter physics, with implications for understanding the properties of real materials. This research provides a refined tool for mapping the behaviour of quantum states, although its current form is limited to specific, disordered spin chains. Alternative approaches attempt to characterise the many-body localised phase by identifying unusual energy fluctuations and resonances within these systems, such as the presence of rare regions with enhanced spectral density. These methods often rely on analysing the statistical properties of energy levels and their sensitivity to perturbations.

Acknowledging these competing theories, this work offers a complementary and valuable perspective on quantum behaviour. It presents a new way to quantify how quantum states evolve and spread, revealing subtle differences between ordered and disordered phases. While currently applied to specific spin chains, the technique’s potential extends to broader investigations of quantum complexity and many-body localisation, establishing a basis-optimised measure applicable to other systems too. Future research could explore the application of Krylov space complexity to other disordered quantum systems, such as fermionic models or higher-dimensional lattices. The method could also be adapted to study the dynamics of open quantum systems, where interactions with the environment play a significant role.

This research establishes the approach as a strong indicator of quantum phase transitions, specifically differentiating between ergodic and many-body localised (MBL) states within disordered spin chains. By examining how quantum states spread within a mathematically constructed ‘Krylov space’, researchers at the University of Birmingham and the Max Planck Institute of Quantum Optics have identified a basis-optimised measure of complexity, overcoming limitations of previous techniques. The findings reveal that ergodic phases support a widespread distribution of quantum information, scaling linearly with the system’s possible states, while MBL phases exhibit a restricted spread, growing at a slower rate. This distinction is crucial for understanding the fundamental differences in how information is processed and stored in these two distinct phases of matter, potentially informing the development of new quantum technologies that exploit the unique properties of MBL systems.

This research demonstrated that Krylov space complexity effectively distinguishes between ergodic and many-body localised (MBL) phases in disordered spin chains. By measuring how quantum states spread within this space, scientists found that ergodic phases exhibited linear scaling with system size, indicating broad information distribution, whereas MBL phases showed slower, sublinear growth and restricted spread. This matters because understanding these differences is vital for controlling quantum information and potentially building more robust quantum technologies. Future work could apply this method to other disordered quantum systems, such as fermionic models or higher-dimensional lattices, to further explore the principles of many-body localisation.