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Physical review. B./Physical review. B

Stevens operator class of exactly solvable one-dimensional spin models

Andrei A. Zvyagin, Victor V. Slavin

Published Jun 9, 2025• 0 citations

Abstract

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 z th 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 ≤ 2 (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 L − 1 , where L 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 L .

DOI: 10.1103/physrevb.111.214417

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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 z th 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 ≤ 2 (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 L − 1 , where L 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 L .

We present a class of exactly solvable tri-junctions of one- and two-dimensional spin systems. Based on the geometric criterion for solvability, we clarify the sufficient condition for the junctions so that the spin Hamiltonian becomes equivalent to Majorana quadratic forms. Then we examine spin tri-junctions using the obtained solvable models. We consider the transverse magnetic field Ising spin chains and reveal how Majorana zero modes appear at the tri-junctions of the chains. Local terms of the tri-junction crucially affect the appearance of Majorna zero modes, and the tri-junction may support Majorana zero mode even if the bulk spin chains do not have Majorana end states. We also examine tri-junctions of two-dimensional SO(5)-spin lattices and discuss Majorana fermions along the junctions.

The Ising model in transverse magnetic field is perhaps the simplest quantum spin model, yet the elements of site disorder, random magnetic fields, and tunable quantum fluctuations permit a rich array of ground states, and the rare ability to emphasize their classical or quantum character. We briefly review in this chapter experimental results from chains to crystals of spins, with magnetic, glassy, and spin liquid ground states. Singlet Ground State Magnets In a crystalline lattice, the crystal fields often take a form such that the ground states of magnetic ions are singlets separated from the higher lying spin multiplets. Sufficiently strong magnetic dipolar or exchange couplings between different sites can mix the singlets with the multiplets enough so as to induce non-zero expectation values for the magnetic dipoles at individual sites. Examples of this phenomenon are generally found among metals and insulators based on rare earths. Such metals were a subject of extensive research in the 1960's and 1970's, and the book by MacKintosh and Jensen [408] gives an excellent account of both the data as well as mean field approaches. With respect to insulating materials, LiTbF 4 emerged as a model system in the 1970's. It is an example of an ideal Ising dipolar coupled ferromagnet; here the long-range nature of the dipolar interactions leads to an upper critical dimension [149] (at which mean field theory with logarithmic corrections calculable using RG methods becomes an exact description of thermal phase transitions) of three rather than four. LiTbF 4 belongs to the LiREF 4 isostructural series of ionic salts. Li and F carry valences +1 and −1 respectively, leaving RE, which can be any rare earth atom or the nonmagnetic element Y, with a valence of +3. Figure 14.1 shows the body-centered tetragonal crystal structure for the family.

We propose an exactly solvable quantum spin model, in which the alternation of the exchange interactions between neighboring spins is accompanied by the multiple spin exchange. The model permits us to obtain exactly thermodynamic characteristics of the considered quantum spin chain. In the ground state, depending on the strength of the three-spin coupling, the model manifests either the spin-gapped behavior of low-lying excitations at low magnetic fields, or the ferrimagnetic ordering with the spontaneous magnetization and with gapless low-lying excitations. The system undergoes second order or first order quantum phase transitions, governed by an external magnetic field and the multiple spin exchange coupling strength. Our exactly solvable quantum spin model demonstrates that multispin interactions in a dimerized quantum spin chain can produce a spontaneous magnetization. On the other hand, for quantum spin chains with a spontaneous magnetization, caused by multispin couplings, the alternation of nearest neighbor exchange interactions can be the reason for destroying of that magnetization and the onset of a spin gap for low-lying excitations.

I study the one-dimensional spin-1 Blume-Emery-Griffiths model with bilinear and biquadratic exchange interactions and single-ion crystal field under an applied magnetic field. This model can be exactly mapped into a tight-binding Hubbard model - extended to include intersite interactions - provided one renormalizes the chemical and the on-site potentials, which become temperature dependent. After this transformation, I provide the exact solution of the Blume-Emery-Griffiths model in one dimension by means of the Green's functions and equations of motion formalism. I investigate the magnetic variations of physical quantities - such as magnetization, quadrupolar moment, susceptibility - for different values of the interaction parameters and of the applied field, focusing on the role played by the biquadratic interaction in the breakdown of the magnetization plateaus.

A new family of exactly solvable models is introduced. These models are generalizations of the XXZ chain where the distance among spins up (

σ^z

-basis) cannot be smaller or equal to t (t=0,1,2,...). The case t=0 recovers the standard XXZ chain. The coordinate Bethe ansatz is applied and the phase diagram is calculated. Exploring the finite-size consequences of conformal invariance the critical exponents are evaluated exactly at the critical regions of the phase diagram