Variational Equations-of-States for Interacting Quantum Hamiltonians

TL;DR

This paper introduces variational equations of state (VES) for interacting quantum systems, enabling computation of correlation functions and ground state properties with applications to models like the transverse field Ising and Heisenberg models.

Contribution

It develops a novel set of variational equations of state expressed via density operators and correlation functions, applicable to strongly interacting quantum systems.

Findings

Derived algebraic relations between density matrices and their variations.

Applied VES to perturbation calculations in the transverse field Ising model.

Numerically comparable results for the 2D Heisenberg model.

Abstract

Variational methods are of fundamental importance and widely used in theoretical physics, especially for strongly interacting systems. In this work, we present a set of variational equations of state (VES) for pure states of an interacting quantum Hamiltonian. The VES can be expressed in terms of the variation of the density operators or static correlation functions. We derive the algebraic relationship between a known pure state density matrix and its variation, and obtain the VES by applying this relation to the averaged Heisenberg-equations-of-motion for the exact density matrix. Additionally, we provide a direct expression of the VES in terms of correlation functions to make it computable. We present three nontrivial applications of the VES: a perturbation calculation of correlation functions of the transverse field Ising model in arbitrary spatial dimensions, a study of a longitudinal field perturbation to the one-dimensional transverse field Ising model at the critical point and variational calculation of magnetization and ground state energy of the two-dimensional spin-1/2 Heisenberg model on a square lattice. For the second one, our results not only recover the scaling limit, but also indicate the possibility of continuous tuning of the critical exponents by adjusting the longitudinal fields differently from the scaling limit. For the Heisenberg model, we obtained results numerically comparable to established results with simple calculations. The VES approach provides a powerful and versatile tool for studying interacting quantum systems.