bibkey: bytsko2011nonhermitian
authors: A. G. Bytsko
year: 2011
title: "Non-Hermitian spin chains with inhomogeneous coupling"
doi: 10.1090/S1061-0022-2011-01148-3
url: https://arxiv.org/abs/0911.4476v3
claim: "The paper studies open U_q(sl_2)-invariant spin chains H = sum a_n P^{S,s}_{n,n+1} with |q| = 1, builds universal metric operators for them, determines the ranges of gamma with real spectra for short chains, and conjectures (Conjecture 3) that for |gamma| < pi/(2S) the spectrum is real whenever all couplings a_n are positive."
strata_touched:
  - D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation
license: citation-only
triage: anchor

Non-Hermitian spin chains with inhomogeneous coupling

A. G. Bytsko, arXiv:0911.4476v3 (2010-02-10); Algebra i Analiz 22(3), 80–106 (2010); St. Petersburg Math. J. 22(3), 393–410 (2011). Quotations are from the v3 TeX source.

The coproduct:

\Delta(E) = E \otimes K^{-1} + K \otimes E ,, \quad \Delta(F) = F \otimes K^{-1} + K \otimes F ,, \quad \Delta(K) = K \otimes K ,.

The representation:

\pi_S(E),\omega_k &= \sqrt{[S{-}k][S{+}k{+}1]} , \omega_{k+1} , \ \pi_S(F),\omega_k &= \sqrt{[S{+}k][S{-}k{+}1]} , \omega_{k-1} ,

\pi_S(K),\omega_k = q^k , \omega_{k} ,

where $[t]\equiv \frac{\sin \gamma t}{\sin \gamma }$.

The projectors:

Let ${\mathsf{P}}^{S,s}$ denote the projector onto the irreducible submodule $V^s$ in $V^S\otimes V^S$.

The Hamiltonian:

\mathsf H^{S,s}{{a_1,\ldots,a{N-1}}} = \sum_{n=1}^{N-1} a_n , \mathsf P^{S,s}_{n,n+1} ,, \qquad a_n \in \mathbb R ,,

The evidence and Conjecture 3:

in the general $N=3$ case and the two–periodic $N=4$ case the spectra are always real if all coupling constants are positive. This observation is supported by numerical checks in a number of other cases. It is thus tempting to suggest the following.

For $|\gamma |<\frac{\pi }{2S}$, the Hamiltonian (Hal) of a spin chain with inhomogeneous coupling has a real spectrum if all $a_n>0$.

The $N=4$ analysis is the subsection “$N=4$ and $N=5$ for $s=0$”.

The tensor Casimir is introduced as ${\mathcal{C}}^{S,S}=({\pi }_S\otimes {\pi }_S)\Delta (C)$ for the Casimir $C=\frac{1}{2}(EF+FE)-\frac{\cos \gamma }{4{\sin }^2\gamma }(K-K^{-1}{)}^2$ of Eq. (C). The expanded right-hand side printed in Eq. (dC2) does not commute with $\Delta (E)$; the definitional form does.

Verified locator

Source: https://arxiv.org/abs/0911.4476v3

• URL: https://arxiv.org/abs/0911.4476v3 (source main.tex, md5 dc29226c07d058043d2c18f82dbda3a4):
  • the coproduct, l. 666–670;
  • the Casimir (C), l. 1531–1535;
  • the representation, l. 683–691;
  • the projectors, l. 779–780;
  • the Hamiltonian, l. 808–812;
  • the subsection on $N=4$ and $N=5$, l. 994;
  • the evidence and Conjecture 3, l. 1435–1446.
• DOI: 10.1090/S1061-0022-2011-01148-3 (https://doi.org/10.1090/S1061-0022-2011-01148-3).