Symmetries, Not Disorder, Explain Exponential Growth in Quantum Systems

Thea Budde and colleagues, based at Berkeley and the Institute for Theoretical Physics, are challenging established understandings of ergodicity breaking with research into Hilbert space fragmentation. They demonstrate that generalised symmetries, including higher-form, subsystem, and gauge symmetries, can cause exponential growth in the number of symmetry sectors within a Hilbert space. This finding sharply reframes the interpretation of Krylov sector proliferation, suggesting that an abundance of these sectors does not automatically indicate a lack of ergodicity. Furthermore, the work reveals a natural connection between Krylov-restricted thermalisation and disorder-free localisation, offering insights into scenarios where systems avoid both ergodicity breaking and the need for gauge symmetry.

Mapping fragmented Hilbert spaces using iteratively generated vector sequences

Krylov subspace methods proved central to discerning the role of generalised symmetries in Hilbert space fragmentation. The technique constructs a sequence of vectors by repeatedly applying the Hamiltonian, a mathematical description of the system’s total energy, to an initial state, effectively exploring the system’s evolution step-by-step. The Hamiltonian operator, denoted as H, acts on the initial state |ψ₀⟩, generating a series of states: |ψ₀⟩, H|ψ₀⟩, H²|ψ₀⟩, and so on. Each application of the Hamiltonian expands the accessible state space. The Krylov subspace, spanned by these generated states, represents the portion of the Hilbert space reachable from the initial state within a defined number of Hamiltonian applications. Analysing the resulting Krylov subspace allows mapping out the fragmented sectors, identifying disconnected regions within the system’s vast state space, akin to charting the separate ‘rooms’ of a complex building. This is achieved by examining the overlap between different Krylov subspaces originating from different initial states; minimal overlap indicates fragmentation. This enabled detailed examination of systems too large for direct diagonalization, identifying fragmentation even when traditional methods failed, and is particularly useful for investigating Hilbert space fragmentation, a phenomenon indicating potential ergodicity breaking within quantum systems. Direct diagonalization, while powerful, scales exponentially with system size, quickly becoming intractable for even moderately sized systems. Krylov methods offer a polynomial scaling advantage, allowing access to larger systems and providing a more efficient means of detecting fragmentation.

Exponential Hilbert space fragmentation induced by generalised symmetries in quantum link models

An exponential growth in the number of Krylov sectors of exp(cLf) has been observed, representing a substantial increase from previous observations limited to polynomial growth. This threshold, determined by a constant ‘c, system size ‘L, and a value ‘f greater than or equal to one, unlocks the analysis of sharply larger and more complex quantum systems previously inaccessible. The constant ‘c encapsulates system-specific parameters influencing the rate of sector growth, while ‘L represents the system size, and ‘f dictates the order of the Krylov expansion. This exponential scaling signifies a fundamental shift in the system’s behaviour, moving beyond the limitations of polynomial scaling observed in systems with only conventional symmetries. The team’s findings reveal that generalised symmetries, including higher-form and non-invertible types, directly induce Hilbert space fragmentation, the exponential proliferation of disconnected dynamical sectors, without necessarily causing ergodicity breaking, challenging long-held assumptions. Previously, the emergence of many-body localized phases, characterised by a lack of thermalisation, was often associated with strong disorder or special symmetries preventing the system from exploring its entire Hilbert space. This research demonstrates that fragmentation can occur even in the absence of these factors, suggesting a more nuanced relationship between symmetry, fragmentation, and ergodicity.

Generalised symmetries induce exponential growth in the number of Krylov sectors, reaching at least 7 L 3/2 sectors within a three-dimensional U pure-gauge quantum link model; this fragmentation arises from the model’s local U gauge symmetry and 2-form partial isometry. Quantum link models are lattice gauge theories used to study quantum field theories, and the U pure-gauge model specifically focuses on the dynamics of gauge fields. The 2-form partial isometry introduces a non-local symmetry, contributing to the fragmentation. Further analysis of the same system revealed subsystem 1-form symmetries generating an additional (L 2 +1)L symmetry sectors, demonstrating fragmentation even within the physical sector. Subsystem symmetries act only on a subset of the system’s degrees of freedom, leading to a further increase in the number of symmetry sectors. The PXP model, originally studied for Rydberg blockade dynamics, exhibits exponential sector growth due to locally frozen states and a conserved 2-local projector, analogous to a Z2 gauge theory, confirming fragmentation can be predicted by identifying local conserved operators. Rydberg blockade describes the interaction between atoms where excitation of one atom inhibits the excitation of nearby atoms. The conserved 2-local projector restricts the system’s dynamics, leading to fragmentation. However, current calculations focus on idealized models and do not yet demonstrate fragmentation in realistic, noisy quantum systems, nor do they provide a clear pathway to using this effect for practical quantum computation. The presence of noise and imperfections in real-world quantum systems can disrupt the delicate balance of symmetries and potentially mask or alter the observed fragmentation. Further research is needed to determine the robustness of these findings in more realistic scenarios.

Symmetry’s role in explaining fragmentation within quantum Hilbert space

Researchers are revealing how symmetries within quantum systems dictate their behaviour, potentially reshaping our understanding of complex materials. A proliferation of disconnected sectors within a system’s Hilbert space, previously assumed to signal a loss of predictable behaviour, can instead arise from inherent symmetries. While generalised symmetries explain fragmentation, it remains unclear whether all instances of this phenomenon can be attributed to these symmetries alone. Establishing a link between symmetry and quantum behaviour advances both condensed matter and high-energy physics, allowing for a refined understanding of complex quantum systems, particularly those exhibiting locally frozen states or conserved projectors. In condensed matter physics, understanding fragmentation could lead to the design of novel materials with tailored quantum properties. In high-energy physics, it may provide insights into the behaviour of quantum fields in extreme conditions. Future research will focus on whether all fragmentation instances stem from these symmetries, or if additional mechanisms contribute to this behaviour, fundamentally altering how we view complex materials. Investigating the interplay between generalised symmetries, disorder, and interactions is crucial for a complete understanding of Hilbert space fragmentation and its implications for quantum systems. Determining the precise conditions under which fragmentation occurs and its impact on physical observables will be essential for harnessing this phenomenon in future quantum technologies.

The research demonstrated that generalized symmetries can cause exponential fragmentation of a quantum system’s Hilbert space. This means that what was previously considered evidence of unpredictable behaviour can instead be explained by the system’s underlying symmetries. Researchers found that fragmentation arises from higher-form, subsystem, and gauge symmetries, and can even occur within individual symmetry sectors due to non-invertible symmetries. The study suggests that the presence of many disconnected sectors does not automatically indicate ergodicity breaking, and that disorder-free localisation can occur through restricted thermalisation without requiring such a break in behaviour.