## 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 H_{i} 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 language | English |
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Article number | 012057 |

Journal | Journal of Physics: Conference Series |

Volume | 1163 |

Issue number | 1 |

DOIs | |

Publication status | Published - 26 Mar 2019 |

Event | 3rd International Conference on Computer Simulations in Physics and Beyond, CSP 2018 - Moscow, Russian Federation Duration: 24 Sep 2018 → 27 Sep 2018 |