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arXiv:2302.11267

A spin-energy operator inequality for Heisenberg-coupled qubits

Daniel Ranard, Charles Jess Riedel

Published Feb 22, 2023• 0 citations
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Abstract

We slightly strengthen an operator inequality identified by Correggi et al. that lower bounds the energy of a Heisenberg-coupled graph of $s=1/2$ spins using the total spin. In particular, $ΔH \ge C Δ\vec{S}^2$ for a graph-dependent constant $C$, where $ΔH$ is the energy above the ground state and $Δ\vec{S}^2$ is the amount by which the square of the total spin $\vec{S} = \sum_i \vecσ_i/2$ falls below its maximum possible value. We obtain explicit constants in the special case of a cubic lattice. We briefly discuss the interpretation of this bound in terms of low-energy, approximately non-interacting magnons in spin wave theory and contrast it with another inequality found by Bärwinkel et al.

DOI
10.48550/arxiv.2302.11267

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We give a polynomial-time algorithm for computing upper bounds on some of the smaller energy eigenvalues in a spin-1/2 ferromagnetic Heisenberg model with any graph GGG for the underlying interactions. An important ingredient is the connection between Heisenberg models and the symmetric products of GGG. Our algorithms for computing upper bounds are based on generalized diameters of graphs. Computing the upper bounds amounts to solving the minimum assignment problem on GGG, which has well-known polynomial-time algorithms from the field of combinatorial optimization. We also study the possibility of computing the lower bounds on some of the smaller energy eigenvalues of Heisenberg models. This amounts to estimating the isoperimetric inequalities of the symmetric product of graphs. By using connections with discrete Sobolev inequalities, we show that this can be performed by considering just the vertex-induced subgraphs of GGG. If our conjecture for a polynomial time approximation algorithm to solve the edge-isoperimetric problem holds, then our proposed method of estimating the energy eigenvalues via approximating the edge-isoperimetric properties of vertex-induced subgraphs will yield a polynomial time algorithm for estimating the smaller energy eigenvalues of the Heisenberg ferromagnet.
Abstract. We consider the quantum ferromagnetic Heisenberg model in three di-mensions, for all spins S ≥ 1/2. We rigorously prove the validity of the spin-wave approximation for the excitation spectrum, at the level of the first non-trivial con-tribution to the free energy at low temperatures. Our proof comes with explicit, constructive upper and lower bounds on the error term. It uses in an essential way the bosonic formulation of the model in terms of the Holstein–Primakoff representa-tion. In this language, the model describes interacting bosons with a hard-core on-site repulsion and a nearest-neighbor attraction. This attractive interaction makes the lower bound on the free energy particularly tricky: the key idea there is to prove a differential inequality for the two-particle density, which is thereby shown to be smaller than the probability density of a suitably weighted two-particle random process on the lattice. 1.
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Abstract We consider the ferromagnetic quantum Heisenberg model in one dimension, for any spin S≥1/2S\ge 1/2S≥1/2 S ≥ 1 / 2 . We give upper and lower bounds on the free energy, proving that at low temperature it is asymptotically equal to the one of an ideal Bose gas of magnons, as predicted by the spin-wave approximation. The trial state used in the upper bound yields an analogous estimate also in the case of two spatial dimensions, which is believed to be sharp at low temperature.