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Topological Theory of Helium 4

Leonid Lantsman

Abstract

In this paper we attempt to construct the topological theory of superfluid helium $4$ in the framework of the (rigid) $U(1)$ model in which the initial $U(1)$ group is destroyed with appearance of (topologically nontrivial) domains separated by domain walls treated as step voltages between domains (e.g. with neighboring topological numbers). This can explain the superfluid properties in a helium $4$ specimen as well as the appearance of topologically nontrivial vortices therein.

Topological Theory of Helium 4

Abstract

In this paper we attempt to construct the topological theory of superfluid helium in the framework of the (rigid) model in which the initial group is destroyed with appearance of (topologically nontrivial) domains separated by domain walls treated as step voltages between domains (e.g. with neighboring topological numbers). This can explain the superfluid properties in a helium specimen as well as the appearance of topologically nontrivial vortices therein.

Paper Structure

This paper contains 9 sections, 297 equations.

Table of Contents

  1. Introduction. About Topological Defects.
  2. The two-component phenomenological liquid helium theory. Superfluidity and vortices.
  3. Introductory remarks.
  4. Superconductor as a system with attraction forces. Bogolubov transformations.
  5. The gauge $U(1)$ symmetry of the He $^4$ Hamiltonian.
  6. The phenomenology of of the superfluid helium specimen contained in a turning vessel. Global and local vortices.
  7. The topological theory of ${\bf He}^4$.
  8. Discussion.
  9. Appendix. The topological grounding of the discussed theory.