# Stochastic Hamiltonians for correlated electron models

## TL;DR

This paper revisits Kraichnan's stochastic embedding method to construct explicit Hamiltonians for advanced correlated electron models, enabling better theoretical consistency and analysis of strongly interacting quantum systems.

## Contribution

It extends Kraichnan's approach to derive Hamiltonians for complex correlated models with long-range and short-range interactions, previously lacking a canonical Hamiltonian framework.

## Key findings

- Derived Hamiltonians for models with long-range screening.
- Constructed Hamiltonians incorporating short-range scattering.
- Enhanced theoretical tools for analyzing strongly correlated electrons.

## Abstract

Microscopically conserving reduced models of many-body systems have a long, highly successful history. Established theories of this type are the random-phase approximation for Coulomb fluids and the particle-particle ladder model for nuclear matter. There are also more physically comprehensive approximations such as the induced-interaction and parquet theories. Notwithstanding their explanatory power, some theories have lacked an explicit Hamiltonian from which all significant system properties, static and dynamic, emerge canonically. This absence can complicate evaluation of the conserving sum rules, essential consistency checks on the validity of any model. In a series of papers Kraichnan introduced a stochastic embedding procedure to generate explicit Hamiltonians for common approximations for the full many-body problem. Existence of a Hamiltonian greatly eases the task of securing fundamental identities in such models. I revisit Kraichnan's method to apply it to correlation theories for which such a canonical framework has not been available. I exhibit Hamiltonians for more elaborate correlated models incorporating both long-range screening and short-range scattering phenomena. These are relevant to the study of strongly interacting electrons and condensed quantum systems broadly.