Nonequilibrium from Equilibrium: Chiral Current-Carrying States in the Spin-1 Babujian-Takhtajan Chain

Abstract

We study the spin-$1$ Babujian-Takhtajan chain deformed by its third conserved charge $Q_3$. We derive $Q_3$ and show that it is a dimensionless energy current and that its local density is a dressed scalar-chirality operator rather than bare chirality alone, as is the case for the spin-$1/2$ Heisenberg chain. The deformation $H_α=H+αQ_3$ therefore provides a local, exactly solvable current bias: it leaves the eigenstates of the original Hamiltonian unchanged, but reorders them so that selected high-energy current-carrying states become ground states of the tilted problem. Using the thermodynamic Bethe ansatz and confirming the analytical calculations with DMRG, we find a quantum phase transition at $α_c={J}/(8π)$. For $α<α_c$, the ground-state remains the undeformed Babujian-Takhtajan phase whose low-energy effective field theory is described by the $SU(2)$ Wess-Zumino-Witten (WZW) model at level $k=2$ representing a critical phase characterized by a central charge $c=3/2$ and $\langle Q_3\rangle=0$. For $α>α_c$, a finite rapidity interval forms, and the system enters a gapless chiral current-carrying sector described by a $c=3/2$ CFT. Near the threshold, the free energy starts quadratically as a function of $α-α_c$, while the energy current turn on linearly. The scalar chirality turns on at the same threshold, showing that the postcritical sector is simultaneously current-carrying and chiral. The most immediate experimental routes are composite spin-1 bosons in optical lattices, and programmable qutrit simulators based on trapped ions or superconducting circuits.

Figures (5)

• Figure 1: Ground-state energy density and chirality across the $Q_3$-driven transition. The blue curve is the exact TBA result for the dimensionless free energy, $f/J$, the orange points are DMRG energies for a periodic chain of length $L=100$, and the green points on the right axis show the magnitude of the DMRG chirality density. The energy is pinned at the undeformed BT value $f/J=-1$ up to $(\alpha_c/J)=1/(8\pi)$ (vertical dashed line) and then decreases smoothly. The chirality turns on at the same threshold, confirming that the postcritical sector is simultaneously current carrying and chiral.
• Figure 2: Dressed two-string energy $\varepsilon_2(\lambda)$ below and above the current-driven transition. Left panel: $\alpha/J=1/(16\pi)<\alpha_c/J$, for which $\varepsilon_2(\lambda)<0$ across the entire real axis and the BT phase has no Fermi boundaries. Right panel: $\alpha/J=1/(4\pi)>\alpha_c/J$, for which $\varepsilon_2(\lambda)$ has two simple zeros at $b_-$ and $b_+$ (vertical dashed lines and red dots), with $\varepsilon_2(\lambda)<0$ only on the interval $b_-<\lambda<b_+$. The steep left crossing and the much shallower right crossing imply that the two Fermi edges of the current-carrying chiral phase are inequivalent. The inset zooms in on the neighborhood of $b_+$ and confirms that the right zero remains simple.
• Figure 3: Conserved-current density and chirality density as functions of $\alpha/J$. Both quantities vanish in the BT phase and become finite at the same critical value, $\alpha_c/J=1/(8\pi)$, shown by the dashed line.
• Figure 4: Difference between the conserved-current density and the chirality density. The difference remains zero below $\alpha_c$ because both quantities vanish in the flat BT phase, but becomes finite immediately above the transition. This directly demonstrates that $Q_3$ is not the bare scalar chirality alone: the BT-specific dressing terms identified in Sec. \ref{['sec:II']} contribute throughout the current-carrying sector.
• Figure 5: Entanglement entropy of the $Q_3$-deformed chain at $\alpha/J=0.1$ for a periodic system of length $L=100$. The DMRG data follow the standard conformal profile with central charge $c=3/2$. The postcritical current-carrying sector is therefore still gapless and consistent with a $c=3/2$ conformal field theory.