[Paper Review] The Chern Numbers of Interaction-stretched Monopoles in Spinor Bose Condensates

Tin-Lun Ho, Cheng Li|arXiv (Cornell University)|Apr 12, 2017

Cold Atom Physics and Bose-Einstein Condensates3 citations

TL;DR

This paper demonstrates that interactions in spinor Bose-Einstein condensates stretch point-like monopoles into extended singular manifolds, causing the second Chern number to decrease continuously rather than jump abruptly when the monopole crosses a 4D integration surface. This intrinsic non-integer behavior explains the gradual drop in $C_2$ observed in Spielman's NIST experiment, even under adiabatic conditions, and enables mapping of the singular manifold's shape via Chern number measurements along different parameter-space directions.

ABSTRACT

Using the Dirac and the Yang monopole in spinor condensates as examples, we show that interactions can stretch the point singularity of a monopole into an extended manifold, whose shape is strongly influenced by the sign of interaction. The singular manifold will cause the first and second Chern number to assume non-integer values when it intersects the surface on which the Chern numbers are calculated. This leads to a gradual decrease of the Chern numbers as the monopole moves away from the surface of integration, instead of the sudden jump characteristic of a point monopole. A gradual change in $C_2$ has in fact been observed in the recent experiment by Spielman's group at NIST. By measuring the range of non-integer values of the Chern numbers as the monopole moves away from the surface of integration along different directions, one can map out the shape of the singular manifold in the parameter space.

Motivation & Objective

• To explain the experimentally observed gradual decrease of the second Chern number $C_2$ in the NIST Yang monopole experiment, which contradicts the expected integer jump for a point monopole.
• To investigate how many-body interactions in spinor Bose-Einstein condensates modify the topological structure of monopoles by transforming point singularities into extended manifolds.
• To show that the continuous drop in $C_2$ is an intrinsic feature of interacting systems, not solely due to non-adiabaticity in experiments.
• To demonstrate that measuring the range of non-integer $C_2$ values along different directions in parameter space allows reconstruction of the singular manifold's shape.

Proposed method

• Model the Dirac and Yang monopoles in spinor condensates using a single-mode approximation, treating the system as a two- or four-component spinor with interaction energy dominating near the monopole.
• Define the non-Abelian Berry connection and curvature in the Hilbert space parameterized by momentum $\mathbf{k}$, using projection operators onto the ground state and excited state manifolds.
• Derive the second Chern number $C_2$ via the 4-form integral over a 4D spherical surface $S^4$, expressed in terms of the vector $\hat{\mathbf{m}}$ that defines the spinor orientation in 5D parameter space.
• Analyze the geometry of the integration surface $S^4$ shifted along the $\hat{\mathbf{e}}_5$ direction, leading to a complex relation between angles $\theta$, $\theta'$, and the radial coordinate $k$, which determines the intersection with the singular manifold.
• Use the expression $C_2 = \frac{3}{2} \iint \cos^2\beta \sin\beta \frac{d\beta}{d\theta} \sin\theta' \, d\theta \, d\theta'$ to compute the Chern number, showing its dependence on the singular manifold's extent.
• Show that when the 4D surface $S^4$ intersects the extended singular manifold, $C_2$ becomes non-integer and decreases continuously, rather than jumping from 1 to 0.

Experimental results

Research questions

• RQ1 Why does the second Chern number $C_2$ decrease gradually in the NIST experiment instead of jumping abruptly from 1 to 0 as expected for a point monopole?
• RQ2 How do many-body interactions in spinor Bose-Einstein condensates alter the topological structure of monopoles by modifying the nature of their singularities?
• RQ3 What is the origin of non-integer Chern numbers in interacting systems, and how does it relate to the geometry of the singular manifold?
• RQ4 Can the shape of the singular manifold in parameter space be reconstructed from experimental measurements of $C_2$ as a function of monopole position?
• RQ5 Is the gradual change in $C_2$ an intrinsic property of interacting systems or primarily due to experimental non-adiabaticity?

Key findings

• Interactions in spinor condensates stretch the point singularity of a monopole into an extended manifold, which causes the second Chern number $C_2$ to assume non-integer values when the integration surface intersects this manifold.
• The second Chern number decreases continuously from 1 to 0 as the monopole moves away from the 4D integration surface, rather than jumping abruptly, due to the extended nature of the singular manifold.
• This continuous drop in $C_2$ is an intrinsic feature of interacting systems and persists even under perfectly adiabatic conditions, ruling out non-adiabaticity as the sole explanation for experimental observations.
• The range of displacement $D$ over which $C_2$ decreases reflects the extent of the singular manifold in momentum space, with the size and shape of this interval revealing the geometry of the manifold.
• Measuring $C_2$ along different directions in parameter space allows mapping of the singular manifold's shape, providing a topological probe of interaction-induced structure.
• The analytical expression for $C_2$ in the Yang monopole case, involving integrals over angles $\theta$, $\theta'$, and $\beta$, confirms the continuous descent and shows its dependence on the intersection geometry between $S^4$ and the singular manifold.