Surveying the quantum group symmetries of integrable open spin chains

Abstract

Using anisotropic R-matrices associated with affine Lie algebras $\hat{g}$ (specifically, $A_{2n}^{(2)},A_{2n-1}^{(2)},B_n^{(1)},C_n^{(1)},D_n^{(1)}$) and suitable corresponding K-matrices, we construct families of integrable open quantum spin chains of finite length, whose transfer matrices are invariant under the quantum group corresponding to removing one node from the Dynkin diagram of $\hat{g}$. We show that these transfer matrices also have a duality symmetry (for the cases $C_n^{(1)}$ and $D_n^{(1)}$) and additional $Z_2$ symmetries that map complex representations to their conjugates (for the cases $A_{2n-1}^{(2)},B_n^{(1)},D_n^{(1)}$). A key simplification is achieved by working in a certain "unitary" gauge, in which only the unbroken symmetry generators appear. The proofs of these symmetries rely on some new properties of the R-matrices. We use these symmetries to explain the degeneracies of the transfer matrices.

48 pages

References in corpus (3)

• Gauge Theory and Integrability, II
• The integrable quantum group invariant A_{2n-1}^(2) and D_{n+1}^(2) open spin chains
• Quantum group symmetries and completeness for A_{2n}^(2) open spin chains

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• Integrability of open boundary driven quantum circuits
• Symmetry protected topological phases beyond groups: The q-deformed bilinear-biquadratic spin chain
• On the classification of rational K-matrices
• Inhomogeneous SU(2) symmetries in homogeneous integrable U(1) circuits and transport
• Reflection $K$ matrices associated with an Onsager coideal of $U_p(A_{n-1}^{(1)})$, $U_p(B_n^{(1)})$, $U_p(D_n^{(1)})$ and $U_p(D_{n+1}^{(2)})$
• Spin chains with boundary inhomogeneities
• Quantum-group-invariant $D_{n+1}^{(2)}$ models: Bethe ansatz and finite-size spectrum