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arXiv:2512.23759

Aliphatic Chains as One-Dimensional XY Spin Chains

Kirill F. Sheberstov

quant-phSubmitted Dec 28, 2025• Announced Dec 30, 2025• 0 citations
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Abstract

Spin waves are propagating disturbances of spin order in lattices with nearest-neighbor interactions. They are traditionally observed in magnetically ordered solids using inelastic neutron, light, or electron scattering, and ferromagnetic resonance. Here, we show that analogous spin dynamics can arise in liquid-state nuclear magnetic resonance (NMR) of molecules containing aliphatic chains. In such molecules, each CH_2 group must have a distinct chemical shift and be magnetically inequivalent via out-of-pair couplings. Under these conditions, singlet state populations of geminal protons propagate along (CH_2)_n segments forming magnetically silent spin waves. For a chain with translational symmetry, the spin Hamiltonian factorizes into subspaces formally equivalent to the one-dimensional XY model. This correspondence yields analytic expressions for eigenstates and eigenenergies in a spectroscopy we term spin-chain zero-quantum NMR. We identify molecular systems in which these conditions are met. Their collective dynamics rapidly exceed classical computational tractability, making them targets for quantum-computer simulations of spin transport and many-body dynamics.

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Spin waves are propagating disturbances of spin order in lattices with nearest-neighbor interactions. They are traditionally observed in magnetically ordered solids using inelastic neutron, light, or electron scattering, and ferromagnetic resonance. Here, we show that analogous spin dynamics can arise in liquid-state nuclear magnetic resonance (NMR) of molecules containing aliphatic chains. In such molecules, each CH_2 group must have a distinct chemical shift and be magnetically inequivalent via out-of-pair couplings. Under these conditions, singlet state populations of geminal protons propagate along (CH_2)_n segments forming magnetically silent spin waves. For a chain with translational symmetry, the spin Hamiltonian factorizes into subspaces formally equivalent to the one-dimensional XY model. This correspondence yields analytic expressions for eigenstates and eigenenergies in a spectroscopy we term spin-chain zero-quantum NMR. We identify molecular systems in which these conditions are met. Their collective dynamics rapidly exceed classical computational tractability, making them targets for quantum-computer simulations of spin transport and many-body dynamics.
Multiple quantum (MQ) nuclear magnetic resonance (NMR) spin dynamics are investigated analytically in infinite one-dimensional (1D) chains of spins 1/2. The representation of spin 1/2 operators with fermion field operators allows to calculate exactly the spin density operator, and hence NMR observables, under a variety of different conditions for 1D spin systems. The exact expressions are valid for all times and for a macroscopic number of coupled spins. The calculations for a 1D spin system initially at thermal equilibrium, and evolving under a 2-quantum/2-spin average dipolar Hamiltonian, in the presence of nearest-neighbor dipolar interactions yield MQ NMR spectra with 0- and 2-quantum coherences only. For a nonequilibrium initial condition with transverse magnetization, the analogous spin dynamics calculations produce MQ NMR spectra with all possible coherences of odd orders. Calculations at the level of perturbation theory, which include next-nearest-neighbor dipolar interactions, generate MQ spectra with higher even order coherences for equilibrium initial condition and evolution under a 2-quantum/2-spin propagator. Consideration of multiple spin correlations, 0-quantum coherences, and rf pulse imperfections are also presented. The relevance and implications of these theoretical results for comparison with the recent MQ NMR experiments of Yesinowski et al. on materials with quasi-one-dimensional distributions of spins, and for MQ NMR of solids in general are discussed.
Multiple-quantum spin dynamics is studied using analytic and numerical methods for one-dimensional finite linear chains and rings of nuclear spins 1/2 coupled by dipole-dipole interactions. An approximation of dipole-dipole interaction between nearest neighbors having the same constants is used to obtain exact expressions for the intensities of the multiple-quantum coherences in the spin systems studied, which are initially in thermal equilibrium and whose evolution is described by a two-spin two-quantum Hamiltonian. An approximation of nearest neighbors with arbitrary dipole-dipole interaction constants is used to establish a simple relationship between the multiple-quantum dynamics and the dynamics of spin systems with an XY Hamiltonian. Numerical methods are developed to calculate the intensities of multiple-quantum coherences in multiple-quantum NMR spectroscopy. The integral of motion is obtained to expand the matrix of the two-spin two-quantum Hamiltonian into two independent blocks. Using the nearest-neighbor approximation the Hamiltonian is factorized according to different values of the projection operator of the total spin momentum on the direction of the external magnetic field. Results of calculations of the multiple-quantum dynamics in linear chains of seven and eight nuclear spins and a six-spin ring are presented. It is shown that the evolution of the intensities of the lowest-order multiple-quantum coherences in linear chains is accurately described allowing for dipole-dipole interaction of nearest and next-nearest neighbors only. Numerical calculations are used to compare the contributions of nearest and remote spins to the intensities of the multiple-quantum coherences.
We extend the picture of a transfer of nuclear spin-1/2 polarization along a homogeneous one-dimensional chain with the XY Hamiltonian to the inhomogeneous chain with alternating nearest neighbor couplings and alternating Larmor frequencies. To this end, we exactly calculate the spectrum of the spin-1/2 XY Hamiltonian of the alternating chain with an odd number of sites. The exact spectrum of the XY Hamiltonian is also applied to study the multiple quantum (MQ) NMR dynamics of the alternating spin-1/2 chain. MQ NMR spectra are shown to have the MQ coherences of zero and ± second orders just as in the case of a homogeneous chain. The intensities of the MQ coherences are calculated.
The three-spin chain with a Heisenberg XYXYXY interaction is simulated in a three-qubit nuclear magnetic resonance quantum computer. The evolution caused by the XYXYXY interaction is decomposed into a series of single-spin rotations and the JJJ-coupling evolutions between the neighboring spins. The perfect state transfer algorithm proposed by Christandl et al.[Phys. Rev. Lett. 92, 187902 (2004)] is realized in the XYXYXY chain.
Transport of quantum information in linear spin chains has been the subject of much theoretical work. Experimental studies by NMR in solid state spin systems (a natural implementation of such models) is complicated since the dipolar Hamiltonian is not solely comprised of nearest-neighbor XY-Heisenberg couplings. We present here a similarity transformation between the XY Hamiltonian and the double-quantum Hamiltonian, an interaction which is achievable with the collective control provided by radio-frequency pulses. Not only can this second Hamiltonian simulate the information transport in a spin chain, but it also creates coherent states, whose intensities give an experimental signature of the transport. This scheme makes it possible to study experimentally the transport of polarization beyond exactly solvable models and explore the appearance of quantum coherence and interference effects.