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arXiv:2509.21771 · Physical review. B./Physical review. B

Spin-basis wavefunctions for the one-dimensional Kitaev model

Alwyn Jose Raja, Rajesh Narayanan, R. Ganesh

Published Apr 13, 2026• 1 citation
arXivHTMLPDF

Abstract

Magnetic phases with quantum entanglement are often expressed in terms of parton wavefunctions. Relatively few examples are known where wavefunctions can be directly written down in the spin basis. In this article, we consider the spin-$S$ Kitaev model in one dimension. For $S=1/2$, its eigenstates can be written using a Jordan-Wigner fermionic representation. Here, we present ground state wavefunctions for any $S$ directly in the spin basis. The states we propose are valence bond arrangements, with bonds having singlet or triplet character for $S=1/2$. For $S>1/2$, we use bond-states that serve as analogues of singlets and triplets. We establish the validity of our wavefunctions using a perturbative approach starting from an anisotropic limit, with key features surviving to all orders in perturbation theory. For half-integer $S$ and periodic boundaries, we have exponential ground state degeneracy. The ground states are subject to a nonlocal constraint. They have `triplets' superposed on a background of singlets, but with the total number of triplets constrained to be even. For integer $S$, a unique ground state emerges, composed purely of `triplets'. Our spin-basis wavefunctions, while not exact, capture the dominant weight of the ground state(s). We obtain good agreement against exact diagonalization wavefunctions and Jordan-Wigner spectra.

DOI
10.1103/p2nh-g89v

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Magnetic phases with quantum entanglement are often expressed in terms of parton wavefunctions. Relatively few examples are known where wavefunctions can be directly written down in the spin basis. In this article, we consider the spin-SSS Kitaev model in one dimension. For S=1/2S=1/2S=1/2, its eigenstates can be written using a Jordan-Wigner fermionic representation. Here, we present ground state wavefunctions for any SSS directly in the spin basis. The states we propose are valence bond arrangements, with bonds having singlet or triplet character for S=1/2S=1/2S=1/2. For S>1/2S>1/2S>1/2, we use bond-states that serve as analogues of singlets and triplets. We establish the validity of our wavefunctions using a perturbative approach starting from an anisotropic limit, with key features surviving to all orders in perturbation theory. For half-integer SSS and periodic boundaries, we have exponential ground state degeneracy. The ground states are subject to a nonlocal constraint. They have triplets' superposed on a background of singlets, but with the total number of triplets constrained to be even. For integer $S$, a unique ground state emerges, composed purely of triplets'. Our spin-basis wavefunctions, while not exact, capture the dominant weight of the ground state(s). We obtain good agreement against exact diagonalization wavefunctions and Jordan-Wigner spectra.
In the first part of this paper, we study the spin-SSS Kitaev model using spin-wave theory. We discover a remarkable geometry of the minimum-energy surface in the NNN-spin space. The classical ground states, called Cartesian or CN-ground states, whose number grows exponentially with the number of spins NNN, form a set of points in the NNN-spin space. These points are connected by a network of flat valleys in the NNN-spin space giving rise to a continuous family of classical ground states. Further, the CN-ground states have a correspondence with dimer coverings and with self-avoiding walks on a honeycomb lattice. The zero-point energy of our spin-wave theory picks out a subset from a continuous family of classically degenerate states as the quantum ground states; the number of these states also grows exponentially with NNN. In the second part, we present some exact results. For arbitrary spin SSS, we show that localized Z2{Z}_{2}Z2​ flux excitations are present by constructing plaquette operators with eigenvalues $\ifmmode\pm\else\textpm\fi{}1$, which commute with the Hamiltonian. This set of commuting plaquette operators leads to an exact vanishing of the spin-spin correlation functions beyond nearest-neighbor separation found earlier for the spin-1/2 model [G. Baskaran et al., Phys. Rev. Lett. 98, 247201 (2007)]. We introduce a generalized Jordan-Wigner transformation for the case of general spin SSS and find a complete set of commuting link operators similar to the spin-1/2 model, thereby making the Z2{Z}_{2}Z2​ gauge structure more manifest. The Jordan-Wigner construction also leads, in a natural fashion, to Majorana fermion operators for half-odd-integer spin cases and hard-core boson operators for integer spin cases strongly suggesting the presence of Majorana fermion and boson excitations in the respective low-energy sectors. Finally, we present a modified Kitaev Hamiltonian, which is exactly solvable for all half-odd-integer spins; it is equivalent to an exponentially large number of copies of spin-1/2 Kitaev Hamiltonians.
We study a one-dimensional version of the Kitaev model on a ring of size NNN, in which there is a spin S>1/2S>1/2S>1/2 on each site and the Hamiltonian is $J{\ensuremath{\sum}}_{n}{S}_{n}^{x}{S}_{n+1}^{y}$. The cases where SSS is integer and half-odd integer are qualitatively different. We show that there is a Z2{\mathbb{Z}}_{2}Z2​-valued conserved quantity Wn{W}_{n}Wn​ for each bond (n,n+1)(n,n+1)(n,n+1) of the system. For integer SSS, the Hilbert space can be decomposed into 2N{2}^{N}2N sectors, of unequal sizes. The number of states in most of the sectors grows as dN{d}^{N}dN, where ddd depends on the sector. The largest sector contains the ground state, and for this sector, for S=1S=1S=1, d=(5+1)/2d=(\sqrt{5}+1)/2d=(5​+1)/2. We carry out exact diagonalization for small systems. The extrapolation of our results to large NNN indicates that the energy gap remains finite in this limit. In the ground-state sector, the system can be mapped to a spin-1/2 model. We develop variational wave functions to study the lowest energy states in the ground state and other sectors. The first excited state of the system is the lowest energy state of a different sector and we estimate its excitation energy. We consider a more general Hamiltonian, adding a term $\ensuremath{\lambda}{\ensuremath{\sum}}_{n}{W}_{n}$, and show that this has gapless excitations in the range ${\ensuremath{\lambda}}_{1}^{c}\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}{\ensuremath{\lambda}}_{2}^{c}$. We use the variational wave functions to study how the ground-state energy and the defect density vary near the two critical points ${\ensuremath{\lambda}}_{1}^{c}$ and ${\ensuremath{\lambda}}_{2}^{c}$.
We outline the recent results on the ground state for a class of one- and two-dimensional frustrated quantum spin models with competing ferro(F)- and antiferromagnetic (AF) interactions. Frustrated spin systems are known to have many interesting properties due to large quantum fluctuations. As a result of these fluctuations usual mean-field approach gives quite crude (if not false) description of these systems. Therefore, exactly solvable models are very instructive at investigations of such systems. The exact ground state wave function of proposed models has a structure of the valence-bond state (VBS) type. One of the 1D model describes the transition line between the F and AF phase. The exact singlet ground state on this line has a double-spiral ordering. Using different approximation methods we study the magnetization curve in the AF phase. The second considered set of the 1D and 2D models has an exact non-degenerate ground state with exponentially decaying spin correlations. We also proposed the 1D and 2D electronic models with exact ground state represented in terms of singlet bond functions which are the generalization of the RVB functions including ionic states.
We study a one-dimensional spin-orbital model with Heisenberg symmetry for spins and XY\mathrm{XY}XY symmetry for orbital degrees of freedom. We establish the ground-state phase diagram of this model. The gapless phases of the model consist of fully polarized ferromagnetic spin states together with the filled Jordan-Wigner Fermi sea for the orbitals. Two such phases, with ferromagnetic and antiferromagnetic XY\mathrm{XY}XY correlations for the orbitals, are found. All other phases are found to have a spin gap with degenerate ground states. Two types of correlated spin and orbital dimerizations are observed: (i) spin singlet with orbital singlet in alternating rungs and (ii) spin singlet with orbital triplet in the same rung. The phase boundaries, energy gaps, and order parameters are evaluated using the density matrix renormalization group method.
Inspired by the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, we present exact solutions for a spin-1 chain with Kitaev-like couplings. We consider an expanded Kitaev model with bilinear and biquadratic terms. At an exactly solvable point, the Hamiltonian can be reexpressed as a sum of projection operators. Unlike the AKLT model where projectors act on total spin, we project the component of spin along the bond direction. This leads to exponential ground-state degeneracy, expressed in terms of fractionalized spin-1/2 objects. Each ground state can be expressed concisely as a matrix product state. We construct a phase diagram by varying the relative strength of bilinear and biquadratic terms. The fractionalized states provide a qualitative picture for the spin-1 Kitaev model, yielding approximate forms for the ground state and low-lying excitations.
Recently, there has been a renewed interest in properties of the higher-spin Kitaev models, especially their low-dimensional analogs with additional interactions. These quasi-one-dimensional systems exhibit rich phase diagrams with symmetry-protected topological phases, Luttinger liquids, hidden order, and higher-rank magnetism. However, the nature of the pure spin-SSS Kitaev chains is not yet fully understood. Earlier works found a unique ground state with short-ranged correlations for S=1S=1S=1 and an intriguing double-peak structure in the heat capacity associated with an entropy plateau. To understand the low-energy excitations and thermodynamics for general SSS, we study the anisotropic spin-SSS Kitaev chain. Starting from the dimerized limit, we derive an effective low-energy Hamiltonian at finite anisotropy. For half-integer spins we find a trivial effective model, reflecting a nonlocal symmetry protecting the degeneracy, while for integer SSS we find interactions among the flux degrees of freedom that select a unique ground state. The effective model for integer spins is used to predict the low-energy excitations and thermodynamics, and we make a comparison with the semiclassical limit through linear spin wave theory. Finally, we speculate on the nature of the isotropic limit.