Invariant subspaces and elliptic spin-helix states in the integrable open spin-$\frac{1}{2} \mathrm{XYZ}$ chain

We derive a criterion under which splitting of all eigenstates of an open $\mathrm{XYZ}$ Hamiltonian with boundary fields into two invariant subspaces, spanned by chiral shock states, occurs. The splitting is governed by an integer number, which has the geometrical meaning of the maximal number of kinks in the basis states. We describe the generic structure of the respective Bethe vectors. We obtain explicit expressions for Bethe vectors, in the absence of Bethe roots, and those generated by one Bethe root, and we investigate the single-particle subspace. We also describe in detail an elliptic analog of the spin-helix state, appearing in both the periodic and the open $\mathrm{XYZ}$ model, and we derive the eigenstate condition. The elliptic analog of the spin-helix state is characterized by a quasiperiodic modulation of the magnetization profile, governed by Jacobi elliptic functions.

Figure 1

Local magnetization profiles for the factorized eigenstate $|{\mathrm{\Psi }}_{+}\rangle$ from (15) discussed in detail in Sec. 4. The $x,y,z$ components of the local magnetization are given by blue, yellow, and green points, respectively. The used parameters are $\eta =0.43,\tau =0.35i, u_1=0.9+0.23i, N=9$. The curves are exact interpolations by elliptic functions (38, 39, 40) and are guides for the eye.

Figure 2

Upper panel: The symmetric set (not all states are linearly independent) of all states in (29) is represented by all possible trajectories (directed paths), lying within the filled green region, including the boundaries. Each individual path starts in one of $M_{+}+1$ points (filled black circles) on site $n=1$ and ends at one of $M_{+}+1$ points at $n=N$. Note that $n_j$ on the left side of the upper panel are all different nonzero integers $0<n_1<n_2<...\le N$. Lower panel: The set of linearly independent states (32), which is a subset of (29) in the upper panel. The allowed trajectories end in one of two points at the right boundary site $n=N$, their total number being $\left(\genfrac{}{}{0pt}{}{N}{0}\right)+\left(\genfrac{}{}{0pt}{}{N}{1}\right)+\cdots \left(\genfrac{}{}{0pt}{}{N}{M_{+}}\right)$. Black and red trajectories indicate two basis states from (32): $|1,3,4,5,8\rangle$ and $|0,0,3,8,N\rangle$, respectively. The blue trajectory in the upper panel represents a state $|4,N,N,N,N\rangle$, which can be expressed via basis states (32). Note that $M_{+}$ gives the maximal number of kinks a trajectory can have.

Figure 3

Upper panel: Location of the Bethe roots in the multiplet $G_1^{+}$ with dimension $dim\left(G_1^{+}\right)=N+1=101$. The respective energies decrease in the counterclockwise direction, starting with the bottom of the “continuum” on the right. Lower panel: The energies for the multiplet, respecting the color code. Bulk parameters are $N=100, \eta =0.43,\tau =0.35i$, and they correspond to $J_x\approx 4.51,J_y\approx 0.244,J_z\approx 0.136$. Boundary parameters are $\{{\alpha }_1^{-},{\alpha }_2^{-},{\alpha }_3^{-}\}=\{0.9,0.86+\frac{\tau }{2},\frac{1}{2}+0.53i\}, \{{\alpha }_1^{+},{\alpha }_2^{+},{\alpha }_3^{+}\}=\{0.96,38.99+1.225i,\frac{1}{2}+0.53i\}$, and satisfy (23) with $L_0=2,K_0=0$, and $M_{+}=1$.

Figure 4

Coefficients $|F_n|$ for the $H$ eigenfunctions corresponding to the two red and the two green points in Fig. 3 combined in one plot, on a logarithmic scale. Red, orange, dark green, and green symbols correspond to the energies $E=81.844,75.07,70,56.43$, respectively.