Exact Correlation Functions for Dual-Unitary Lattice Models in 1+1 Dimensions

Abstract: We consider a class of quantum lattice models in $1+1$ dimensions represented as local quantum circuits that enjoy a particular "dual-unitarity" property. In essence, this property ensures that both the evolution "in time" and that "in space" are given in terms of unitary transfer matrices. We show that for this class of circuits, generically non-integrable, one can compute explicitly all dynamical correlations of local observables. Our result is exact, non-pertubative, and holds for any dimension $d$ of the local Hilbert space. In the minimal case of qubits ($d = 2$) we also present a classification of all dual-unitary circuits which allows us to single out a number of distinct classes for the behaviour of the dynamical correlations. We find "non-interacting" classes, where all correlations are preserved, the ergodic and mixing one, where all correlations decay, and, interestingly, also classes that are are both interacting and non-ergodic.

Exact Correlation Functions for Dual-Unitary Lattice Models in 1+1 Dimensions

This essay provides an analytical summary of the paper by Bertini, Kos, and Prosen, which explores a class of quantum lattice models within the framework of dual-unitary circuits. The research leverages the dual-unitarity property to derive exact expressions for correlation functions in one-dimensional quantum many-body systems, elucidating the dynamics of non-integrable models.

Core Concepts and Methodology

In quantum many-body physics, the computation of correlation functions is vital for understanding the physical behavior of systems, including ergodic properties and transport coefficients. Traditionally, exact solutions for correlation functions are achievable only in integrable models or free theories. This paper breaks new ground by focusing on a broader class of non-integrable models represented through local quantum circuits with dual-unitary gates.

Dual-unitary circuits are characterized by unitary evolution in both temporal and spatial dimensions, offering significant advantages in accessing spatiotemporal correlation functions of local operators. This study presents a novel framework to compute these functions exactly, non-perturbatively, across any local Hilbert space dimension $d$.

Key Results

The paper introduces two main structural properties of dual-unitary models:

1. Lightcone Structure: Correlation functions are restricted to the edges of a propagating lightcone, allowing simplification in calculations. The exact dynamics and correlation functions are shown to be bounded by causality constraints inherent in both space and time dimensions.
2. Unistochastic Maps: The correlation functions are governed by linear unistochastic maps determined solely by unitary matrices. These maps are trace-preserving and completely positive, providing a valuable classification framework for dynamical behavior.

Strong numerical results include a classification of quantum circuits based on their ergodicity levels, revealing potential non-ergodic and mixing scenarios. The classification includes:

• Non-interacting behavior with constant correlations.
• Non-ergodic behavior allowing constant correlations for specific eigenmodes.
• Ergodic but non-mixing, with unit modulus eigenvalues.
• Mixing with strictly decaying correlation functions.

Particularly, the dual-unitary framework facilitates exact finite-time correlation calculations in non-integrable models like the self-dual kicked Ising model (SDKI), offering insight into the distinction between dynamical mixing and mere decay of local correlators.

Implications and Future Work

The theoretical implications are profound, as the methodology provides a reliable means to investigate non-integrable systems holistically, enhancing the foundational understanding of quantum ergodicity. Practically, this could advance the development of quantum computational models and support experimental validation through directly measurable correlation functions.

Future research directions suggested by the paper include:

• Exploring ergodicity in higher-dimensional quantum circuits.
• Developing perturbative expansions surrounding dual-unitary models, potentially uncovered via exponential space-time clustering instances.
• Identification of more complex local conservation laws within exactly solvable circuits with larger support.

The paper strategically challenges traditional limitations and opens up fresh perspectives for understanding quantum dynamics. It is a concerted effort towards establishing a rigorous ergodic theory in quantum many-body systems.

Conclusion

Bertini, Kos, and Prosen have made substantial contributions to quantum dynamics through the formalization of dual-unitary circuits, facilitating exact correlation function computation. The classification provided surfaces underlined behaviors within quantum systems, leading the way for further explorations in non-integrable systems and showcasing the potential impact on the theory of quantum ergodicity. This work represents a significant step in theoretical physics, pointing towards more generalized frameworks that could reshape the landscape of future quantum research.