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Higher Berry curvature and higher gerbes
Jan. 14th, 2020 09:09 am Just put a paper on the arXiv about a "classical" subject which I like very much: the Berry connection. In math, it is common to consider topological invariants of families of things. In physics, this goes back to the work of Michael Berry. He noticed that given a family of Hamiltonians on a fixed Hilbert space, one gets a natural vector bundle on the space of parameters: the bundle of ground states. (Here I assume that there is a gap between the lowest eigenvalue of the Hamiltonian and the next one which does not close as one varies the parameters. Only then is the bundle of ground states well-defined.) Moreover, there is a distinguished unitary connection in this bundle now called the Berry connection. It is obtained by projecting the trivial connection on the trivial Hilbert space bundle to the sub-bundle of ground states. Chern classes of the Berry connection can be expressed through the curvature of the Berry connection via Chern-Weil theory serve as topological invariants of the family of Hamiltonians. Typically one assumes that the ground state is unique (i.e. the bundle of ground states is a line bundle), then the only invariant is the 1st Chern class which can be thought of as the cohomology class of a closed 2-form on the parameter space (the curvature of the Berry connection).
A little known fact is that none of this makes sense for many-body systems, i.e. systems with an infinite-dimensional Hilbert space. For example, if we consider a family of infinite spin-chains, with an energy gap for all values of the parameters, the Berry curvature is ill-defined. This happens because the volume of the system is infinite. In our paper we show that nevertheless for a spin-chain one can define a closed 3-form on the parameter space. We argue that its periods are quantized. More generally, in d spatial dimensions we show how to associate to a family of many-body systems a closed (d+2)-form on the parameter space. These forms are higher-dimensional generalizations of the Berry curvature.
Now, whenever one sees a closed (d+2)-form, d-gerbes come to mind. Of course, one needs to check first that periods of the form are quantized. We argue that this is indeed the case for systems without topological order. This condition is vacuous for d=0,1 but very important for d>1. So, a family of many-body systems in d spatial dimensions without topological order gives rise to a d-gerbe on the parameter space. For d=1 I even know how to construct a model for this gerbe explicitly (as a family of von Neumann algebras). Would be great to do something similar for d>1. Unfortunately, I do not know any nice models for 2-gerbes, etc.
A little known fact is that none of this makes sense for many-body systems, i.e. systems with an infinite-dimensional Hilbert space. For example, if we consider a family of infinite spin-chains, with an energy gap for all values of the parameters, the Berry curvature is ill-defined. This happens because the volume of the system is infinite. In our paper we show that nevertheless for a spin-chain one can define a closed 3-form on the parameter space. We argue that its periods are quantized. More generally, in d spatial dimensions we show how to associate to a family of many-body systems a closed (d+2)-form on the parameter space. These forms are higher-dimensional generalizations of the Berry curvature.
Now, whenever one sees a closed (d+2)-form, d-gerbes come to mind. Of course, one needs to check first that periods of the form are quantized. We argue that this is indeed the case for systems without topological order. This condition is vacuous for d=0,1 but very important for d>1. So, a family of many-body systems in d spatial dimensions without topological order gives rise to a d-gerbe on the parameter space. For d=1 I even know how to construct a model for this gerbe explicitly (as a family of von Neumann algebras). Would be great to do something similar for d>1. Unfortunately, I do not know any nice models for 2-gerbes, etc.
