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arXiv:0806.1588 · Physical Review B

Spin- S Kitaev model: Classical ground states, order from disorder, and exact correlation functions

G. Baskaran, Diptiman Sen, Ravi Shankar

Published Sep 22, 2008• 142 citations
arXivHTMLPDF

Abstract

In the first part of this paper, we study the spin-$S$ Kitaev model using spin-wave theory. We discover a remarkable geometry of the minimum-energy surface in the $N$-spin space. The classical ground states, called Cartesian or CN-ground states, whose number grows exponentially with the number of spins $N$, form a set of points in the $N$-spin space. These points are connected by a network of flat valleys in the $N$-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 $N$. In the second part, we present some exact results. For arbitrary spin $S$, we show that localized ${Z}_{2}$ 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 $S$ and find a complete set of commuting link operators similar to the spin-1/2 model, thereby making the ${Z}_{2}$ 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.

DOI
10.1103/physrevb.78.115116

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•Jul 27, 2020•1 citation
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±\else\textpm\fi1\ifmmode\pm\else\textpm\fi{}1\ifmmode±\else\textpm\fi1, 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 ${\ensuremath{\lambda}}_{2}^{c}$.
A family of two-dimensional (2D) spin-1/2 models have been constructed to realize Kitaev’s sixteen-fold way of anyon theories. Defining a one-dimensional (1D) path through all the lattice sites, and performing the Jordan-Wigner transformation with the help of the 1D path, we find that such a spin-1/2 model is equivalent to a model with \nu ν species of Majorana fermions coupled to a static \mathbb{Z}_2 ℤ 2 gauge field. Here each species of Majorana fermions gives rise to an energy band that carries a Chern number \mathcal{C}=1 𝒞 = 1 , yielding a total Chern number \mathcal{C}=\nu 𝒞 = ν . It has been shown that the ground states are three (four)-fold topologically degenerate on a torus, when \nu ν is an odd (even) number. These exactly solvable models can be achieved by quantum simulations.
These lecture notes focus on the Kitaev honeycomb model, a two-dimensional quantumsystem with three distinct Ising-type interactions, depending on the direction of eachbond. This model has proven particularly important for describing quantum spin liquids,as well as for applications in topological quantum computing, due to its exotictopological properties. It constitutes the first exactly solvable theoretical model thatfully captures the behavior of a quantum spin liquid. We begin by presenting the geometryof the model, the Hamiltonian that describes the system, and Alexei Kitaev's effort toobtain an exact analytical solution, based on the decomposition of the Hilbert spaceinto flux sectors. We then present Ettore Majorana's observation that the Dirac equationalso admits real solutions, leading to the theoretical prediction of Majorana fermions-- particles that are identical to their antiparticles. We examine how the spatialseparation of Majorana operators leads to the rise of unbound Majorana fermions, andshowcase the exotic topological properties that manifest in physical systems such assuperconducting wires. Next, we represent the Hamiltonian in the basis of Majoranafermions, by expressing spin operators using four Majorana operators. The transitionfrom the extended space L~\tilde{\mathcal{L}}L~ to the physical subspace L\mathcal{L}Lis achieved through a gauge transformation of the Z2\mathbb{Z}_2Z2​ group. We thenreformulate the Hamiltonian in the basis of complex bond and matter fermions, transitionto momentum space, and assume that the system lies in the zero-flux sector. Through theBogoliubov--de Gennes transformation, we arrive at the final diagonalized Hamiltonianin the quasiparticle basis. Finally, we represent the possible phases -- the gaplessphase (phase BBB) and the gapped phases (phase AαiA_{\alpha_i}Aαi​​) -- compute thecorresponding energy spectrum, and present the complete phase diagram. By employing thecorrelation function, we show that all phases are characterized by strong quantumentanglement and spin fractionalization, classifying the system as a quantum spin liquid.
The spin-1/2 Kitaev model offers the exactly solvable example of quantum spin liquids.Possible material realizations of the spin-1/2 Kitaev systems and the prospect of using the Majorana fermion excitations for quantum computations have revolutionized quantum spin liquids research.Recently it has been suggested that higher-spin, especially spin-1, Kitaev exchange interactions can be realized in a variety of materials.Numerical computations on small clusters indicate that the ground state of the spin-1 Kitaev model may also be a quantum spin liquid.On the other hand, the nature of the ground state remains elusive since the spin-1 model is not exactly solvable in contrast to the spin-1/2 model.In this work, using the quantum-entanglement based tensor network approach, we construct an explicit ground-state wave function for the spin-1 Kitaev model, which is written only in terms of physical spin operators.We establish the existence of distinct topological sectors on a torus by constructing the minimally entangled states in the degenerate ground-state manifold and evaluating topological entanglement entropy.Our results suggest that the ground state of the spin-1 Kitaev model is a gapped quantum spin liquid with Z 2 gauge structure and Abelian quasiparticles.We explain the subtle differences between the spin-1/2 and spin-1 Kitaev quantum spin liquids.
The higher spin Kitaev model prominently features the extensive locally conserved quantities the same as the spin-1/2 Kitaev honeycomb model, although it is not exactly solvable. It remains an open question regarding the physical meaning of these conserved quantities in the higher spin model. In this Letter, by introducing a Majorana parton construction for a general spin-S we uncover that these conserved quantities are exactly the Z_{2} gauge fluxes in the general spin-S model, including the case of spin-1/2. Particularly, we find an even-odd effect that the Z_{2} gauge charges are fermions in the half integer spin model, but are bosons in the integer spin model. We further prove that the fermionic Z_{2} gauge charges are always deconfined; hence, the half integer spin Kitaev model would have nontrivial spin liquid ground states regardless of interaction strengths in the Hamiltonian. The bosonic Z_{2} gauge charges of the integer spin model, on the other hand, could condense, leading to a trivial product state, and this is indeed the case at the anisotropic limit of the model.
In the field of frustrated magnetism, Kitaev models provide prototypical examples of exactly solvable quantum spin liquids, in which the spin degrees of freedom fractionalize into Majorana fermions coupled to an emergent Z2 gauge field. While the ground states of these models are already well understood as exhibiting different kinds of Majorana (semi)metals, it has still been an open task to achieve a detailed understanding of the thermodynamics, and, namely, the low-temperature ordering of the gauge field into different Z2 flux configurations. In this thesis, we investigate the thermodynamics of Kitaev systems in two and three spatial dimensions with sign-problem-free quantum Monte Carlo simulations. In a first study of elementary 3D Kitaev models, we verify that the ground state Z2 flux sectors of these systems are entirely determined by their elementary plaquette length – a result which shows the validity of Lieb’s theorem for lattice geometries which lack the geometric requirements for its proof. We closely investigate the low-temperature phase transition associated with gauge-ordering, which is a particular realization of an inverted Ising phase transition, as it occurs in general lattice Z2 gauge theories. Our results corroborate the understanding of this transition as separating different regimes in terms of vison-loop excitations, by showing a clear correlation between the critical temperature and the vison gap. We also introduce the concept of “gauge frustration”, which, for a particular 3D Kitaev model, leads to the suppression of the phase transition and a more complex physical behavior at low temperatures. A second quantum Monte Carlo study focuses on a generalized 2D Kitaev model on a five-coordinated lattice system. Here, the gauge ordering is accompanied by the spontaneous breaking of time-reversal symmetry, a scenario which not only allows for a variety of topological ground states, but also for the occurrence of a phase transition in two spatial dimensions. We show that this model exhibits such a transition at a particularly high temperature scale. Moreover, the system possesses a number of phases where the Z2 fluxes are only partially ordered.