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Journal of Mathematical Physics

A class of connections governing parallel transport along density matrices

Jochen Dittmann, Gerd Rudolph

Published Dec 1, 1992• 21 citations
Publisher

Abstract

A class of connections governing parallel transport along nondegenerate density matrices is discussed. These connections are given by certain analytic functions. We develop a calculus for operator-valued differential forms, which enables us to calculate the curvature of these connections. The class under consideration contains a non-Abelian generalization of Berry’s phase recently proposed by Uhlmann.

DOI
10.1063/1.529813

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A class of connections governing parallel transport along nondegenerate density matrices is discussed. These connections are given by certain analytic functions. We develop a calculus for operator-valued differential forms, which enables us to calculate the curvature of these connections. The class under consideration contains a non-Abelian generalization of Berry’s phase recently proposed by Uhlmann.
We use the Van Vleck-Primas perturbation theory to study the problem of parallel transport of the eigenvectors of a parameter-dependent Hamiltonian. The perturbative approach allows us to define a non-Abelian connection A\mathcal{A}A that generates parallel translation via unitary transformation of the eigenvectors. It is shown that the connection obtained via the perturbative approach is an average of the Maurer-Cartan 1-form of the one-parameter subgroup generated by the Hamiltonian. We use the Yang-Mills curvature and the non-Abelian Stokes' theorem to show that the holonomy of the connection A\mathcal{A}A is related to the Berry phase.
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A family of finite-dimensional quantum systems with a nondegenerate ground state gives rise to a closed two-form on the parameter space, the curvature of the Berry connection. Its integral over a surface detects the presence of degeneracy points inside the volume enclosed by the surface. We seek generalizations of the Berry curvature to gapped many-body systems in DDD spatial dimensions which can detect gapless or degenerate points in the phase diagram of a system. Field theory predicts that in spatial dimension DDD the analog of the Berry curvature is a closed (D+2)(D+2)(D+2)-form on the parameter space (the Wess-Zumino-Witten form). We construct such closed forms for arbitrary families of gapped interacting lattice systems in all dimensions. We show that whenever the integral of the Wess-Zumino-Witten form over a (D+2)(D+2)(D+2)-dimensional surface in the parameter space is nonzero, there must be gapless edge modes for at least one value of the parameters. These edge modes arise even when the bulk system is in a trivial phase for all values of the parameters and are protected by the nontrivial topology of the phase diagram.