Stevens operator class of exactly solvable one-dimensional spin models

The new class of exactly solvable models of one-dimensional spin chains is found. Interactions of spins with crystalline electric fields (CEF) and spin-spin coupling between neighboring sites are taken into account for those models. Spin-spin coupling includes higher-order spin operators (the Stevens operators) and describes the interaction between <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>z</a:mi><a:mi>th</a:mi></a:mrow></a:math> components of the spins. We show that essential parts of the Hamiltonians of these models can be written in terms of exactly solvable transverse field Ising model (TFIM), in which certain combinations of the parameters of spin-spin interactions play the role of the TFIM exchange constant, and components of the CEFs play the role of external transverse magnetic field of the TFIM. The obtained results are valid for any values of spin (or total moments for rare-earth ions) and for the wide class of CEFs. The requirement for the exact solvability is that the space of single spin states should be split by the CEF into the subspaces with dimensions <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mo>≤</b:mo><b:mn>2</b:mn></b:mrow></b:math> (i.e., the only singlets and doublets are allowed). Cases of integer and half-integer spins are considered. For finite-size chains with integer spins and open ends we show numerically that energy spectra always contain the states whose energies do not depend on the CEF (the magnetic field for the TFIM). The energies (per spin) of these states decay as <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msup><c:mi>L</c:mi><c:mrow><c:mo>−</c:mo><c:mn>1</c:mn></c:mrow></c:msup></c:math>, where <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>L</d:mi></d:math> is the chain length. This behavior qualitatively distinguishes these states from known Majorana edge states and strong zero modes, where the energy decreases exponentially in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>L</e:mi></e:math>.