Piroli and Colleagues Develops Quasi-Momentum Distribution Function for Generalized Gibbs Ensembles

Dávid Szász-Schagrin of the Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, and colleagues investigate the behaviour of spin chains following a quantum quench, utilising a mapping to free fermions. These models present a unique challenge due to the exponential degeneracy of their energy eigenvalues, prompting a re-evaluation of current understandings of how integrable systems evolve outside of equilibrium. The research details an analytical method for calculating the quasi-momentum distribution function and provides a modified formula to account for entropy arising from the energy degeneracy, influencing entanglement growth. Validated against tensor-network computations, the findings demonstrate strong agreement for local observables and offer a key initial step towards broadening the scope of integrable systems theory to encompass models with free fermions in disguise.

Analytic computation of local Hamiltonian densities in exponentially degenerate spin chains

Szász-Schagrin and colleagues have computed the quasi-momentum distribution function for “free fermions in disguise” models, achieving a level of precision previously unattainable. Traditional methods, reliant on techniques like the Bethe ansatz and coordinate transformations, failed to yield results for any observable beyond those at the edge of a chain, limiting their applicability to bulk properties. However, this new analytic method successfully calculates local Hamiltonian densities, providing insights into the system’s internal energy distribution. This breakthrough extends the established framework of integrable systems to a class of models exhibiting exponential degeneracy, where the complexity of each energy eigenvalue increases at a rate proportional to the system’s size, specifically, the number of degenerate states grows exponentially with the number of lattice sites. This exponential growth presents significant computational hurdles, as standard techniques become intractable due to the sheer number of states that must be considered.

Mapping the behaviour of these spin chains onto a generalised Gibbs ensemble (GGE) now allows a new understanding of nonequilibrium dynamics in systems beyond those solvable by the Jordan-Wigner transformation. The GGE is a statistical ensemble used to describe the stationary states of isolated quantum systems following a quench, effectively representing the system’s long-term equilibrium state. The quasi-momentum distribution function for “free fermions in disguise” models was successfully computed by Szász-Schagrin, Paris Cité, Sorbonne Université Paris, and CNRS, representing an important step in understanding their behaviour following a quantum quench. A quantum quench is a sudden change in a system’s Hamiltonian, initiating a period of nonequilibrium dynamics as the system adjusts to the new conditions. An analytical formula was derived to calculate expectation values for local Hamiltonian densities, demonstrating excellent agreement with numerical tensor-network computations across various initial states and Hamiltonian parameters. Tensor networks, such as the Matrix Product State (MPS) method, provide a powerful numerical approach to simulate the dynamics of many-body quantum systems, offering a benchmark for validating the analytical results.

A modified formula for entanglement growth was also proposed, accounting for the exponential degeneracy. Entanglement, a key feature of quantum mechanics, describes the correlation between different parts of a system. In these degenerate systems, the standard formulas for entanglement growth require modification to accurately reflect the increased number of accessible states. Simulations showed only small deviations from this conjecture, suggesting the modified formula provides a reasonable approximation of the true entanglement dynamics. Szász-Schagrin and colleagues have refined our ability to model the aftermath of disturbances in complex quantum systems, specifically those exhibiting a peculiar characteristic: an exponentially increasing number of possible energy states. This advance addresses a longstanding challenge in understanding how these “free fermions in disguise” models evolve, moving beyond the well-understood area of simpler, solvable systems like the isotropic Heisenberg spin chain.

Despite concerns about the approximate nature of their entanglement predictions, Szász-Schagrin and colleagues offer an important step forward in the field. The limitations in accurately predicting entanglement growth stem from the approximations inherent in the GGE approach and the difficulty in capturing all correlations within the exponentially degenerate space. Their analytic method for calculating energy distributions and expectation values provides a strong framework for understanding complex quantum systems. Extending established principles to a broader class of models, those featuring a rapidly growing number of energy states, this work offers valuable insights despite minor discrepancies in entanglement calculations. The significance lies in providing a tractable analytical approach where previously only numerical methods were viable, opening avenues for further theoretical investigation.

Szász-Schagrin and colleagues have established a new analytical approach to characterise the behaviour of “free fermions in disguise” models, complex quantum systems where each energy level possesses an exponentially increasing number of possibilities. This work allows investigation of systems previously inaccessible to standard solution methods by extending the established theoretical framework for integrable systems. Integrable systems are characterised by an infinite number of conserved quantities, simplifying their analysis and allowing for exact solutions in certain cases. The team computed the quasi-momentum distribution function, a measure of energy and particle motion, to describe these models within a generalised Gibbs ensemble, a statistical representation of all possible system states. The quasi-momentum distribution function provides information about the distribution of momenta within the system, crucial for understanding its transport properties and response to external stimuli. This distribution is calculated analytically, offering a significant advantage over purely numerical approaches.

The implications of this research extend to various areas of condensed matter physics, including the study of one-dimensional quantum materials and the development of novel quantum technologies. Understanding the nonequilibrium dynamics of these systems is crucial for controlling and manipulating quantum states, potentially leading to advancements in quantum computation and information processing. Furthermore, the methodology developed in this study could be applied to other complex quantum systems exhibiting similar characteristics, broadening our understanding of fundamental quantum phenomena. The ability to accurately compute local observables and approximate entanglement growth provides a powerful tool for investigating the behaviour of these systems under various conditions and exploring their potential applications.

Future research will likely focus on refining the entanglement predictions and extending the analytical method to more complex scenarios, such as systems with long-range interactions or disorder. Investigating the limitations of the GGE approach and developing more accurate approximations for the entanglement dynamics will be crucial for advancing our understanding of these fascinating quantum systems. The work by Szász-Schagrin and colleagues represents a significant contribution to the field of quantum many-body physics, paving the way for further exploration of the intricate behaviour of integrable systems and their potential applications.

The researchers successfully developed an analytical method to compute the quasi-momentum distribution function, which describes energy and particle motion in spin chains. This provides a way to understand the behaviour of these complex quantum systems and calculate expectation values for specific observables. The study also proposes a modification to existing formulas for entanglement growth, accounting for the unique properties of these models. Results were verified using numerical computations, showing good agreement for local observables and suggesting the entanglement conjecture is approximately correct.