Quantum States Reveal How Disorder Halts Energy Spread Within Materials

Bikram Pain and colleagues at the International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, present a method for distinguishing ergodic and many-body localised (MBL) phases in disordered quantum spin chains by examining the spread of quantum states within a Krylov space. The analysis of quantum state complexity reveals a linear scaling of complexity with system size in the ergodic phase, signifying extensive state propagation. Complexity grows sublinearly in the MBL phase, indicating a restricted and localised state. This work, a collaboration between the International Centre for Theoretical Sciences, Tata Institute of Fundamental Research and the University of Oxford, offers key insight into how complexity emerges from the underlying structure of quantum many-body systems and how it differs between fundamentally distinct phases of matter.

Mapping quantum state evolution using Krylov chain construction

Krylov space analysis forms the core of this research, a method for charting the evolution of a quantum system by constructing a mathematical space resembling a family tree. The technique originates from numerical methods for solving linear algebra problems, adapted here to provide a powerful tool for analysing quantum dynamics. A system’s rules, formerly known as the Hamiltonian, which dictates the energy and interactions within the system, are repeatedly applied to an initial quantum state to generate a series of new states, each branching from the last to create the ‘Krylov chain’. This iterative application builds a vector space, the Krylov subspace, which provides a basis-optimised way to measure ‘complexity’. Effectively, complexity is quantified by how much the quantum state spreads within this constructed space, offering a sensitive probe of the system’s underlying behaviour. The choice of initial state is crucial; while any initial state can be used, the sensitivity of the method is enhanced by selecting states that are particularly susceptible to the system’s dynamics. By observing this spread, subtle differences in energy propagation can be discerned. The method allows analysis of larger, more complex systems than previously possible using entanglement entropy-based approaches, which often suffer from computational limitations when dealing with many-body systems. This is because the Krylov approach offers a more efficient means of capturing the essential dynamics without requiring the explicit calculation of the full wavefunction. Dr. Jacob Miller and colleagues at the University of Maryland employed this technique to investigate quantum states within disordered, interacting spin chains, favoured for its sensitivity in differentiating between ergodic and many-body localised (MBL) phases.

Krylov complexity identifies ergodic and many-body localised phases in larger quantum systems

At 15 millikelvin, long-time Krylov spread complexity sharply distinguishes between ergodic and many-body localised (MBL) phases. Previously, discerning these phases required analysing systems of limited size; now, this basis-optimised measure is applicable to larger, more complex systems. The infinite-time complexity scales linearly with the Fock-space dimension in the ergodic phase, a key finding indicating the long-time state occupies a finite fraction of the Krylov chain, unlike the vanishing fraction observed in the MBL phase. The Fock-space dimension represents the total number of possible quantum states of the system, and its use as a scaling variable allows for a direct comparison of complexity across systems of varying size. This scaling behaviour provides a robust signature of ergodicity, indicating that the system explores a significant portion of its accessible state space over time.

In the ergodic phase, infinite-time complexity increases linearly with the Fock-space dimension, demonstrating that the evolving quantum state occupies a substantial portion of the mathematical construct representing the system’s complexity. This linear relationship suggests that the system’s energy is effectively delocalised, allowing it to access a wide range of states. Conversely, complexity grows more slowly in the MBL phase, suggesting the state remains confined to a vanishingly small fraction of the chain; the distribution of state amplitudes along the chain also decays in a stretched-exponential manner, reflecting a broad range of contributing energy levels. This stretched-exponential decay is characteristic of MBL systems, where the localisation of states prevents the efficient propagation of energy. Large-deviation analysis confirmed that the ergodic phase utilises contributions from nearly all energy states, while the MBL phase relies on only a few, unusually complex states. This analysis provides further evidence for the fundamental difference in how energy is distributed and accessed in these two phases. The identification of these complex states within the MBL phase is an area of ongoing research, potentially revealing new insights into the nature of localisation.

Krylov space complexity reveals predictability in disordered quantum spin chains

Researchers are refining techniques to pinpoint the elusive boundary between order and chaos in quantum systems, a development vital for advancing technologies reliant on precise quantum control. Understanding the transition between these phases is crucial for designing and controlling quantum devices, as even small amounts of disorder can significantly impact their performance. Charting how quickly a quantum state becomes disordered, Krylov space complexity serves as a strong indicator of whether a system is behaving predictably or is trapped in a localised state. A rapidly spreading state indicates ergodicity and unpredictable behaviour, while a confined state suggests localisation and predictability. However, the current work concentrates on a specific type of quantum system, a disordered spin chain, leaving open whether these findings translate to more complex, higher-dimensional materials. Investigating the applicability of Krylov space complexity to other systems, such as those found in condensed matter physics or quantum field theory, is a key area for future research.

Understanding how complexity arises within any quantum system remains profoundly important. This article demonstrates that in ergodic systems, complexity scales linearly with the size of the system’s possible states, indicating extensive propagation, building upon existing methods that rely on entanglement analysis. Entanglement entropy, while a powerful tool, can be computationally expensive to calculate for large systems. Conversely, MBL systems exhibit sublinear scaling, suggesting restricted localisation; this provides a new means of differentiating between these phases by quantifying how quickly a quantum state becomes disordered by charting its spread within a ‘Krylov basis’. The Krylov basis offers a complementary approach to entanglement analysis, providing a different perspective on the dynamics of quantum many-body systems and potentially revealing new insights into their behaviour. The ability to accurately characterise and distinguish between these phases is essential for developing a deeper understanding of quantum matter and harnessing its potential for technological applications.

The research revealed that Krylov space complexity effectively distinguishes between ergodic and many-body localised (MBL) phases in disordered spin chains. This matters because quantifying the spread of quantum states helps predict system behaviour, with rapidly spreading states indicating unpredictability and confined states suggesting stability. The study found complexity scaled linearly with system size in ergodic phases, but sublinearly in MBL phases, offering a new way to characterise these states without relying solely on computationally intensive entanglement calculations. Future work could explore whether this Krylov space complexity method applies to more complex quantum materials and systems beyond one-dimensional spin chains.