Symbolic Recursion Method Advances Understanding of Strongly Correlated Fermions in Three Dimensions

Understanding the behaviour of interacting electrons, known as strongly correlated fermions, presents a fundamental challenge in condensed matter physics, with implications for materials science and our understanding of quantum systems. Igor Ermakov from the Wilczek Quantum Center, Oleg Lychkovskiy from the Skolkovo Institute of Science and Technology, and their colleagues have developed a new computational method that tackles this problem by employing symbolic recursion, allowing them to model these complex interactions in one, two, and three dimensions. This approach confirms a key theoretical prediction, the universal operator growth hypothesis, for interacting fermions and importantly, enables the precise calculation of material properties like charge transport across a wide range of conditions. By performing the most demanding calculations only once and generating reusable symbolic results, the team demonstrates a powerful new paradigm for studying strongly correlated systems and unlocking insights into their behaviour at a fundamental level.

Skolkovo Institute of Science and Technology, Bolshoy Boulevard 30, bld 0.1, Moscow 121205, Russia. We present a symbolic implementation of a recursion method for the dynamics of strongly correlated fermions on one-, two- and three-dimensional lattices.

Hubbard Model, Transport, and Krylov Complexity

Research in theoretical and computational physics increasingly focuses on many-body systems, the Hubbard model, and transport properties, with a growing interest in Krylov complexity. The Hubbard model, a cornerstone of condensed matter physics, is used to describe interacting electrons in a lattice and understand phenomena like high-temperature superconductivity and metal-insulator transitions. Scientists calculate transport coefficients, such as conductivity and diffusion constants, using methods like the Mori formalism and recursion methods, often comparing results to experiments involving cold atoms. Many-body techniques, including moment methods, Pauli propagation, sparse Pauli dynamics, and diagrammatic techniques, are employed to study these systems.

Pauli propagation and related methods are becoming prominent for simulating quantum circuits and fermionic systems, addressing the challenges of simulating these complex interactions. Symbolic computation is also utilized to derive expressions for moments, furthering theoretical understanding. A significant area of investigation is high-temperature physics, where many-body effects are particularly strong. Krylov complexity, originating from quantum chaos theory, characterizes the scrambling of quantum information and its connection to black hole interiors. Researchers are applying this concept to systems like the Hubbard model and conformal field theories, alongside studying operator growth, which describes the spreading of quantum information over time.

Saddle-point scrambling, a specific type of scrambling, is also under investigation. There is a growing trend towards developing efficient algorithms for simulating many-body quantum systems, both classically and potentially on quantum computers. This includes optimizing Pauli propagation methods and searching for polynomial-time classical algorithms for quantum circuit simulation. Research extends to specific models like the Hubbard model, Heisenberg chain, transverse-field Ising model, Sachdev-Ye-Kitaev model, and scalar field theories, with connections to interpreting experimental results from cold atom systems. The field is characterized by interdisciplinary connections between condensed matter physics, quantum chaos, and quantum information theory, with a particular focus on the challenges of simulating fermionic systems.

Fermion Dynamics and Universal Operator Growth Confirmed

Scientists have achieved a breakthrough in simulating the dynamics of strongly correlated fermions, employing a symbolic implementation of the recursion method across one-, two-, and three-dimensional lattices. This work focuses on the spinless fermion t-V model and the Hubbard model, and directly confirms the universal operator growth hypothesis for interacting fermionic systems, evidenced by the linear growth observed in Lanczos coefficients. The team computed these coefficients and used their asymptotic behavior to calculate infinite-temperature autocorrelation functions, allowing for the observation of thermalization. Experiments revealed a high-precision determination of the charge diffusion constant across a broad range of interaction strengths, V.

The data demonstrates that these results are accurately described by a simple functional dependence of approximately 1/V², universally applicable for both small and large values of V. All calculations were performed directly in the thermodynamic limit, eliminating finite-size effects and providing truly scalable results, representing a significant step forward in understanding complex materials. The research delivers a powerful new technique for exploring strongly correlated electron systems, establishing the recursion method as a viable approach in higher dimensions where real-time dynamics pose a substantial challenge for existing numerical techniques. Measurements confirm the method’s ability to capture dynamics across the entire time domain, from short-time behavior to the onset of thermalization. The symbolic computational paradigm employed in this study offers a substantial advantage, performing the most computationally intensive step only once and generating symbolic outputs that can be readily used to compute physical quantities for various parameter values, accelerating future research.

Fermionic Systems Confirm Universal Operator Growth

This research team successfully implemented a symbolic recursion method to investigate the behaviour of strongly interacting fermions, examining both one-, two-, and three-dimensional systems, specifically the t-V and Hubbard models. The method calculates crucial quantities, including moments and Lanczos coefficients of dynamic autocorrelation functions, as functions of model parameters, offering a detailed picture of how these systems evolve over time. A key achievement is the confirmation that the universal operator growth hypothesis holds true for these locally interacting fermionic systems, demonstrating predictable behaviour even at very long timescales approaching thermal equilibrium. The team then leveraged this understanding to compute transport properties, notably the diffusion constant at infinite temperature and the direct current conductivity at high temperatures, achieving high precision across a broad range of interaction strengths. Results indicate a surprisingly simple functional dependence governing these properties, applicable to both weak and strong interactions, and importantly, all calculations were performed directly in the thermodynamic limit. While acknowledging that the current work primarily utilizes translational symmetry, the authors plan to incorporate additional symmetries within the operator space in future research, extending the method to study quench dynamics and performing calculations at finite temperatures, ultimately aiming to create comprehensive symbolic representations applicable to a wider range of physical scenarios and lattice geometries.