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arXiv:1009.2583 · Physical Review B

Spin-1 Kitaev model in one dimension

Diptiman Sen, Ravi Shankar, Deepak Dhar, Kabir Ramola

Published Nov 17, 2010• 24 citations
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Abstract

We study a one-dimensional version of the Kitaev model on a ring of size $N$, in which there is a spin $S>1/2$ on each site and the Hamiltonian is $J{\ensuremath{\sum}}_{n}{S}_{n}^{x}{S}_{n+1}^{y}$. The cases where $S$ is integer and half-odd integer are qualitatively different. We show that there is a ${\mathbb{Z}}_{2}$-valued conserved quantity ${W}_{n}$ for each bond $(n,n+1)$ of the system. For integer $S$, the Hilbert space can be decomposed into ${2}^{N}$ sectors, of unequal sizes. The number of states in most of the sectors grows as ${d}^{N}$, where $d$ depends on the sector. The largest sector contains the ground state, and for this sector, for $S=1$, $d=(\sqrt{5}+1)/2$. We carry out exact diagonalization for small systems. The extrapolation of our results to large $N$ 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 ${\ensuremath{\lambda}}_{2}^{c}$.

DOI
10.1103/physrevb.82.195435

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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 ground-state and thermodynamic properties of an S=1S=1S=1 Kitaev model. We first clarify the existence of global parity symmetry in addition to the local symmetry on each plaquette, which enables us to perform the large scale calculations up to 24 sites. It is found that the ground state should be singlet and its energy is estimated as E/N∼−0.65JE/N\sim -0.65JE/N∼−0.65J, where JJJ is the Kitaev exchange coupling. We find that a lowest excited state belongs to the same subspace as the ground state and the gap monotonically decreases with increasing system size, which suggests that the ground state of the S=1S=1S=1 Kitaev model is gapless. By means of the thermal pure quantum states, we clarify finite temperature properties characteristic of the Kitaev models with S≤2S\le 2S≤2.
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.
We study a Heisenberg antiferromagnet (AF) on a one-dimensional strip composed of two rows of corner sharing triangular plaquettes. The geometry of this ladder-like lattice naturally allows for two different exchange couplings, one between pairs of spins on the outer legs (J) and a different one between the spins on the axis and their nearest neighbors on the legs ${(J}^{\ensuremath{'}}).$ For ${J/J}^{\ensuremath{'}}\ensuremath{\lesssim}0.5$ the model is a ferrimagnet. Our main interest is in the region ${J/J}^{\ensuremath{'}}\ensuremath{\gtrsim}0.5,$ where the classical ground state of this system shows the same macroscopic degeneracy as the classical ground state of the antiferromagnet on the kagom\'e lattice. To explore to which extent this similarity between the classical ground states of these two models carries over to their quantum states we have applied exact diagonalization techniques and density-matrix renormalization group (DMRG) methods to our model. Exact diagonalization is restricted to system sizes of up to N=30N=30N=30 sites. For ${J/J}^{\ensuremath{'}}=1,$ the results obtained by this technique, low-energy spectra, correlation functions and the specific heat, agree qualitatively with the results obtained for finite samples of the kagom\'e AF. As in the case of the kagom\'e AF, these finite size results suggest that our model is a spin liquid with a gap between the ground state and the lowest spin excitation. However, extrapolations from DMRG data for strips of up to N=120N=120N=120 sites point towards a vanishing spin gap in a wide range of couplings, $0.5\ensuremath{\lesssim}{J/J}^{\ensuremath{'}}\ensuremath{\lesssim}1.25,$ so that contrary to the above conjecture, our one-dimensional model may in fact be critical in this parameter range. Moreover, our DMRG data indicate that the model undergoes a transition to a gapped state as the ratio ${J/J}^{\ensuremath{'}}$ increases through the value ${J/J}^{\ensuremath{'}}\ensuremath{\simeq}1.25.$ In the vicinity of the transition point, a high density of low-lying singlets develops in the spin gap.
In the present work the numerical density matrix renormalization group (DMRG) algorithm is used to analyze the ground state properties of the Hubbard model with transverse spin-exchange. The DMRG algorithm, which was developed by White in 1992, is based on the following simple but effective concept: the ground-state wavefunction as well as the low energy excitations of a large interacting chain are obtained by increasing the lattice size iteratively, starting with a small one that can be diagonalized exactly. The exponentially growing Hilbert space is controlled by a renormalization procedure in which 'less important' degrees of freedom are integrated out. Motivated by recent experimental findings showing evidence for the competition or even coexistence of triplet superconductivity and ferromagnetism we focused our investigations on a rather simple extension of the Hubbard model including transverse spin exchange between electrons on nearest-neighbour sites. In the half-filled case, we showed that the phase diagram obtained in the weak-coupling limit has to be modified. A new phase, described by spin and charge excitation gap, shows long-range order in the longitudinal spin correlation, whereas superconducting correlations are surpressed and decay exponentially as expected for the case of a finite charge gap. In general, the presence of a repulsive on-site Coulomb interaction U leads to an enlargement of the sectors with nonvanishing charge gap at the expense of the sectors with spin gap. We extend our analysis to the case of a quarter-filled band. Depending on the value J/t the model belongs either to a gapless excitation phase or to a spin gapped phase with gapless charge degrees of freedom.