A variational lower bound on the ground state of a many-body system and the squaring parametrization of density matrices

F. Uskov, O. Lychkovskiy

    Research output: Contribution to journalConference articlepeer-review

    4 Citations (Scopus)

    Abstract

    A variational upper bound on the ground state energy E gs of a quantum system, E gs Ψ|H|Ψ, is well-known (here H is the Hamiltonian of the system and Ψ is an arbitrary wave function). Much less known are variational lower bounds on the ground state. We consider one such bound which is valid for a many-body translation-invariant lattice system. Such a lattice can be divided into clusters which are identical up to translations. The Hamiltonian of such a system can be written as , where a term Hi is supported on the i'th cluster. The bound reads , where is some wisely chosen set of reduced density matrices of a single cluster. The implementation of this latter variational principle can be hampered by the difficulty of parameterizing the set , which is a necessary prerequisite for a variational procedure. The root cause of this difficulty is the nonlinear positivity constraint ρ > 0 which is to be satisfied by a density matrix. The squaring parametrization of the density matrix, ρ = τ 2/trτ 2, where τ is an arbitrary (not necessarily positive) Hermitian operator, accounts for positivity automatically. We discuss how the squaring parametrization can be utilized to find variational lower bounds on ground states of translation-invariant many-body systems. As an example, we consider a one-dimensional Heisenberg antiferromagnet.

    Original languageEnglish
    Article number012057
    JournalJournal of Physics: Conference Series
    Volume1163
    Issue number1
    DOIs
    Publication statusPublished - 26 Mar 2019
    Event3rd International Conference on Computer Simulations in Physics and Beyond, CSP 2018 - Moscow, Russian Federation
    Duration: 24 Sep 201827 Sep 2018

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