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Physical Review A

Semiclassical structure of Hamiltonians

Mario Feingold, David M. Leitner, Oreste Piro

Published Jun 1, 1989• 52 citations
Publisher

Abstract

The structure of few-body Hamiltonian matrices is studied in the semiclassical regime. Given (A${^}_{1}$(q^,p^),A${^}_{2}$ (q^,p^)), a pair of operators, it can be shown that, under quite general conditions, A${^}_{2}$ takes the form of a banded matrix in the ordered eigenrepresentation of A${^}_{1}$. Moreover, the bandwidth depends only on \ensuremath{\Elzxh} and certain generalized microcanonical averages. In particular, if H=${H}_{0}$+\ensuremath{\epsilon}V, this implies that the perturbed Hamiltonian is banded in the appropriately ordered unperturbed basis.

DOI
10.1103/physreva.39.6507

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The structure of few-body Hamiltonian matrices is studied in the semiclassical regime. Given (A{^}_{1}(q^,p^),A{^}_{2} (q^,p^)), a pair of operators, it can be shown that, under quite general conditions, A{^}_{2} takes the form of a banded matrix in the ordered eigenrepresentation of A{^}_{1}. Moreover, the bandwidth depends only on \ensuremath{\Elzxh} and certain generalized microcanonical averages. In particular, if H=H0{H}_{0}H0​+\ensuremath{\epsilon}V, this implies that the perturbed Hamiltonian is banded in the appropriately ordered unperturbed basis.
We study zero-temperature stability of topological phases of matter under weak time-independent perturbations. Our results apply to quantum spin Hamiltonians that can be written as a sum of geometrically local commuting projectors on a D-dimensional lattice with certain topological order conditions. Given such a Hamiltonian H0, we prove that there exists a constant threshold ϵ>0 such that for any perturbation V representable as a sum of short-range bounded-norm interactions, the perturbed Hamiltonian H=H0+ϵV has well-defined spectral bands originating from low-lying eigenvalues of H0. These bands are separated from the rest of the spectra and from each other by a constant gap. The band originating from the smallest eigenvalue of H0 has exponentially small width (as a function of the lattice size). Our proof exploits a discrete version of Hamiltonian flow equations, the theory of relatively bounded operators, and the Lieb–Robinson bound.
We have incorporated the projection operators to the canonical transformation to derive an analytical infinite perturbation-expansion series. This canonical perturbation expansion (CPE) is valid if the unperturbed Hamiltonian H0{H}_{0}H0​ and the perturbation H1{H}_{1}H1​ can be expressed as ${H}_{0}=\ensuremath{\Sigma}{j}^{}{P}_{j}H{P}_{j}$ and ${H}_{1}=\ensuremath{\Sigma}{j\ensuremath{\ne}k}^{}{P}_{j}H{P}_{k}$, where Pj{P}_{j}Pj​ is the projection operator corresponding to a group of closely spaced effective one-electron orbital energies ${E}_{j\ensuremath{\mu}}$ with $\ensuremath{\mu}=1, 2, \dots{}, {d}_{j}$, and if $|{E}_{j\ensuremath{\mu}}\ensuremath{-}{E}_{j\ensuremath{\nu}}|\ensuremath{\ll}|{E}_{j\ensuremath{\omega}}\ensuremath{-}{E}_{k\ensuremath{\delta}}|$ with $j\ensuremath{\ne}k$. We have shown that the CPE is equivalent to the time-dependent perturbation theory. An extremely simple effective Hamiltonian $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ is obtained when the CPE is applied to the sss-band Hubbard model at the atomic limit. An explicit form of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ to the eighth order is given, and the magnetic interaction in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ is of the form of Heisenberg exchange ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{j}$, including far neighbors. We then use this form to compute the antiferromagnetic groundstate energy to the seventh order. Our result is compared with other works.
The properties and structure of the Hamiltonian matrix of a realistic many-body quantum chaotic system (the rare-earth atom of Ce) are analyzed and compared with those assumed in band random matrix theories. The sparsity of the matrix and the behavior of the mean squared matrix elements 〈Hij2{\mathit{H}}_{\mathit{i}\mathit{j}}^{2}Hij2​${\mathrm{〉}}_{\mathrm{\ensuremath{\Vert}}\mathit{i}\mathrm{\ensuremath{-}}\mathit{j}\mathrm{\ensuremath{\Vert}}=\mathrm{\ensuremath{\Delta}}}$ as function of the distance \ensuremath{\Vert}i-j\ensuremath{\Vert} from the diagonal are studied. Fitting 〈Hij2{\mathit{H}}_{\mathit{i}\mathit{j}}^{2}Hij2​${\mathrm{〉}}_{\mathrm{\ensuremath{\Vert}}\mathit{i}\mathrm{\ensuremath{-}}\mathit{j}\mathrm{\ensuremath{\Vert}}=\mathrm{\ensuremath{\Delta}}}$ with the exponent H02{\mathit{H}}_{0}^{2}H02​exp(-\ensuremath{\Delta}/b) yields the bandwidths of b=54 and 79 for the matrices of ${\mathit{J}}^{\mathrm{\ensuremath{\pi}}}$=4+{4}^{+}4+, and ${4}^{\mathrm{\ensuremath{-}}}$ states, respectively.