Notes on exact solvability for rotating and pulsating strings in nonrelativistic Lifshitz background

TL;DR

The paper constructs a one-dimensional Neumann-Rosochatius–type model from a probe string rotating and pulsating in planar Lifshitz spacetime with anisotropy exponent $z$, embedding the worldsheet on a hyperboloid to obtain an NR-like Lagrangian with $r^2$ and $1/r^2$ potentials. It shows that Liouville integrability is conditional due to finite $z$, deriving $z$-dependent deformations of the Uhlenbeck integrals and identifying the precise relations among $z$, string tension, and Lagrange multipliers required for involution. Exact rotating and pulsating string solutions yield explicit energy–momentum dispersions: rotating strings give $E = rac{2\kappa}{\omega} J = \pm \frac{2J}{\omega} \sqrt{\omega^2 z + n}$, while pulsating strings produce an oscillation-number–dependent expression $\mathcal{E}=\mathcal{N}^2 z^2\left[1+\frac{\pi\mathcal{J}}{\mathcal{N}}+\frac{\mathcal{J}^2}{\mathcal{N}^2}\left(\frac{\pi^2}{2}-\frac{2}{z}+\frac{1}{z^2}\right)\right]$. The authors propose spin-chain interpretations, connecting the rotating-string sector to frustrated $J_1$-$J_2$ Heisenberg chains and the pulsating-string sector to Motzkin/Fredkin chains, highlighting a potential Lifshitz–magnet correspondence in the holographic setting. Together, the work provides an exactly solvable, Lifshitz-aware framework linking string dynamics to integrable many-body systems and offers avenues for exploring Lifshitz holography via condensed-matter duals.

Abstract

We construct one dimensional exactly solvable model by choosing a probe fundamental string rotating and pulsating in the planar Lifshitz spacetime that follows nonrelativistic Lifshitz scaling. We present suitable sets of embedding coordinates for rotating and pulsating strings to embed the string worldsheet on a hyperboloid with anisotropy-dependent eccentricity. The resulting worldsheet Lagrangians straightforwardly reduce to the Lagrangian of a Neumann-Rosochatius integrable model. Although the model assumes exact solutions for both the chosen ansatz its classical Liouville integrability is found to be conditional due to the presence of finite anisotropy in the target space geometry. We further use the exact solutions of the model to yield energy-momentum dispersion relations. We interpret those from the perspective of highly degenerate frustrated $J_1-J_2$ spin chain for rotating string and frustration-free Motzkin spin chain for pulsating string.

Figures (1)

• Figure 1: String embeddings in (\ref{['rotating 2']}) for $z=2$, $z=10$ and $z=20$ (Left to right). Here, we fix $n=1$ and $\kappa=1$. The values of the constant $A_1$ varies accordingly with the relation $\kappa^2-A_1z=n$, for each of the chosen $z$ values.