Bosonization of nonrelativistic fermions on a circle: Tomonaga’s problem revisited

We use the recently developed tools for an exact bosonization of a finite number $N$ of nonrelativistic fermions to discuss the classic Tomonaga problem. In the case of noninteracting fermions, the bosonized Hamiltonian naturally splits into an $\mathrm{O}(N)$ piece and an O(1) piece. We show that in the large-$N$ and low-energy limit, the $\mathrm{O}(N)$ piece in the Hamiltonian describes a massless relativistic boson, while the O(1) piece gives rise to cubic self-interactions of the boson. At finite $N$ and high energies, the low-energy effective description breaks down and the exact bosonized Hamiltonian must be used. We also comment on the connection between the Tomonaga problem and pure Yang-Mills theory on a cylinder. In the dual context of baby universes and multiple black holes in string theory, we point out that the $\mathrm{O}(N)$ piece in our bosonized Hamiltonian provides a simple understanding of the origin of two different kinds of nonperturbative $\mathrm{O}(e^{-N})$ corrections to the black hole partition function.

Figure 1

The action of ${\sigma }_k^{†}$.

Figure 2

The spectrum with a specified ordering for degenerate levels.

Figure 3

A particle-hole pair state created from vacuum by a single fermion bilinear.

Figure 4

The state with a single particle-hole pair resulting from the action of two fermion bilinears on vacuum.

Figure 5

The states with two particle-hole pairs resulting from the action of two fermion bilinears on vacuum.