New Simulation Method Tackles Complex Material Challenges

Simulating strongly correlated systems in two dimensions is notoriously challenging due to rapid entanglement growth and frustration. Fabian J. Pauw from Ludwig-Maximilians-Universit at M unchen and the Munich Center for Quantum Science and Technology, in collaboration with Thomas Köhler from Heriot-Watt University, and Ulrich Schollwöck and Sebastian Paeckel also from Ludwig-Maximilians-Universit at M unchen and the Munich Center for Quantum Science and Technology, present a new approach to tackle this problem. They introduce the adaptive projected-purified pseudoboson density-matrix renormalization group (A3P-DMRG) tailored to explore the ground states of dilute lattice models. This method compresses cluster Hilbert spaces by retaining only the most probable low-occupation Fock states, identified via probabilistic bounds and refined through a self-consistent mean-field basis optimization. Demonstrating advantages in low-filling and weak-coupling regimes for large system sizes where conventional DMRG struggles, this work establishes A3P-DMRG as a versatile tool for studying dilute many-body systems relevant to ultra-cold atom simulators, photonic lattices, Moiré materials and beyond.

For decades, accurately modelling complex materials has remained a major computational hurdle. This advance tackles a long-standing limitation in the study of dilute two-dimensional systems. Scientists are continually challenged when simulating strongly correlated systems in two dimensions, a difficulty stemming from the rapid growth of quantum entanglement and the presence of frustrating interactions.

Researchers have developed the adaptive projected-purified pseudoboson density-matrix renormalization group, or A3P-DMRG, a method specifically designed to explore the ground states of dilute lattice models. This technique efficiently compresses the computational space by focusing on the most probable configurations of particles within clusters, identified using probabilistic limits and then refined through a self-consistent optimisation process.

Initial tests reveal A3P-DMRG performs well in scenarios with low particle density and weak interactions, offering advantages over conventional DMRG methods when dealing with larger system sizes. Exact diagonalisation techniques are limited to small systems, while quantum Monte Carlo methods often encounter problems with the ‘fermionic sign problem’ or similar issues in bosonic systems.

As a result, tensor network methods have become essential for studying interacting lattice models in lower dimensions. Matrix product states, forming the basis of the density-matrix renormalization group, are highly effective in one dimension due to limited entanglement. However, applying these to two-dimensional systems is restricted by the entanglement entropy, which increases with system width, leading to an exponential growth in computational demand.

Alternative approaches are needed to overcome these limitations. Tree tensor networks and projected entangled pair states offer extensions to two dimensions, but each has drawbacks, including constraints on entanglement representation or high computational complexity. Instead, the new A3P-DMRG method exploits the physical structure of dilute systems, recognising that the low-energy behaviour can be captured by focusing on a limited number of particle configurations.

This reduction in dimensionality improves convergence and allows for simulations of larger systems. In dilute systems, the method constructs bases on dilute subsystems, drastically reducing the effective dimensionality of the problem. Probabilistic arguments suggest that the most probable low-particle-number configurations dominate the relevant Fock space when fluctuations are suppressed.

By reformulating lattice models in terms of ‘pseudobosons’, the method creates a representation where these relevant configurations form an optimised basis for describing the low-energy sector. While this approach introduces challenges related to symmetry and local Hilbert space dimensions, these are addressed through projected purification and a self-consistent optimisation scheme.

Simulations using A3P-DMRG have been performed on lattices up to 20×20, even at low filling, demonstrating its potential. This method has broad implications for several areas of physics, including ultra-cold atom quantum simulators, photonic lattices, Moiré materials, and quantum chemistry. The ability to accurately model these systems opens doors to understanding emergent phenomena like unconventional superconductivity and topologically ordered states, and could aid in the design of new quantum technologies.

Cluster decomposition and pseudoboson mapping with projected purification refine lattice model calculations

A3P-DMRG, the adaptive projected-purified pseudoboson density-matrix renormalization group, underpins the methodology employed to investigate dilute lattice models. The research constructs a cluster representation of the two-dimensional lattice, focusing on localized regions to manage entanglement growth. Each cluster comprises a defined number of lattice sites and serves as the fundamental building block for the subsequent transformation.

The work reformulates the lattice model using pseudobosons, effectively redefining the system in terms of renormalized many-body degrees of freedom. Mapping local sites to pseudobosons risks breaking particle-number symmetry, a potential complication addressed through projected purification. By applying this technique, the research restores symmetry control and systematically truncates the effective many-body Hilbert space during each DMRG sweep.

For determining the optimal basis, probabilistic bounds, specifically the Bennett bound, are calculated to identify the most probable low-occupation Fock states. These states, representing the dominant configurations within a narrow particle-number corridor, form the truncated basis for describing the system’s low-energy sector. Assessing the influence of cluster fluctuations necessitates careful consideration of particle-number distributions.

The study examines the probability distribution of cluster particle numbers to quantify the extent of fluctuations and refine the basis optimisation process. The discarded weight, a parameter controlling the truncation accuracy, is fixed to ensure a balance between computational cost and precision. By combining pseudobosons with projected purification, the work aims to create a flexible tool for studying dilute quantum many-body systems.

The method’s advantage lies in its ability to compress the cluster Hilbert spaces, retaining only the most probable Fock states. A3P-DMRG excels in low-filling and weak-coupling regimes, particularly for larger system sizes where entanglement poses a significant challenge. The effective dimensionality of the problem is reduced by focusing on a suitably chosen many-body basis, improving convergence and enabling the exploration of systems relevant to ultra-cold atom simulators, photonic lattices, and Moiré materials.

Fock State Retention and Pseudoboson Mapping for Reduced Hilbert Space Complexity

Achieving a maximum discarded cumulative probability mass of 10−5 requires retaining Fock states up to five occupations, irrespective of cluster size. This result stems from applying Bennett’s inequality, a probabilistic bound used to constrain the relevant size of the cluster Hilbert space. This a priori bound, scaling polynomially with cluster size, limits the computational complexity when simulating strongly correlated systems.

The research introduces an adaptive projected-purified pseudoboson density-matrix renormalization group (A3P-DMRG) method. The pseudoboson representation transforms a cluster Hilbert space of dimension 2W into an effective single site with a local dimension of M = 2W. For instance, a cluster of three hard-core bosons, possessing eight possible configurations, is mapped onto a pseudoboson with a dimension of eight.

This transformation, detailed in Table I, allows for a more efficient representation of the system’s quantum state. The method relies on truncating configurations with more than one occupied cluster site to maintain computational tractability. The adaptive basis optimisation further refines this approach. By self-consistently optimising the mean-field basis, the A3P-DMRG method concentrates computational effort on the most probable Fock states.

At a cluster size of W = 10 sites, the study confirms that the Bennett criterion provides a strict upper bound on the effective Hilbert-space dimension for varying values of the particle number fluctuation, δN. A3P-DMRG offers advantages in low-filling and weak-coupling regimes for large system sizes. By retaining only the most probable low-occupation Fock states, the method circumvents the challenges faced by conventional DMRG in these scenarios.

Inside the pseudobosonic MPS ansatz, the purification projection step is important for maintaining accurate representation of the quantum state. Beyond reducing the Hilbert space, this adaptive approach establishes A3P-DMRG as a flexible tool for studying dilute many-body systems.

Advancing simulations of correlated electron systems through adaptive truncation of quantum states

Scientists tackling complex materials have long been hampered by a fundamental limit: the exponential growth of computational demand as systems become more entangled. For decades, simulations of strongly correlated electron behaviour, vital for designing new materials and understanding phenomena like high-temperature superconductivity, have struggled to scale beyond relatively small models.

A new method, adaptive projected-purified pseudoboson density-matrix renormalization group, or A3P-DMRG, offers a potential pathway around this bottleneck. Rather than brute-force computation, this technique intelligently focuses on the most important parts of the quantum state, discarding less relevant information with a carefully controlled approximation.

The significance of A3P-DMRG extends beyond achieving larger system sizes. This method appears particularly well-suited to modelling dilute systems, those with relatively few interacting particles. This is important because many modern material platforms, such as ultra-cold atoms trapped in optical lattices or engineered photonic structures, operate in this regime.

The method’s reliance on probabilistic bounds and mean-field optimisation introduces a degree of approximation that must be carefully considered. Validating the accuracy of these approximations for a wide range of materials remains an open challenge. Once validated, however, the implications are considerable. Beyond the immediate applications in simulating quantum materials, A3P-DMRG could accelerate the design of new devices based on these principles.

For instance, understanding the behaviour of electrons in Moiré materials, twisted layers of graphene, requires accurately modelling strongly correlated systems, a task where this method could prove invaluable. Researchers might soon be able to explore a much wider range of material configurations. The next steps are clear. Further research should focus on refining the mean-field approximations and developing more efficient ways to assess the truncation error.

A broader effort is needed to combine the strengths of different numerical methods, creating a more flexible set of tools for tackling the challenges of quantum materials. By moving beyond the limitations of traditional approaches, scientists are beginning to unlock the full potential of these fascinating systems.