Here is a footnote reference,1 and another.2
-This paragraph won’t be part of the note, because it isn’t -indented.
-The Kitaev Honeycomb Model
-The Kitaev-Honeycomb model is remarkable because it was the first -such model that combined three key properties.
-First, it is a plausible tight binding Hamiltonian. The form of the -Hamiltonian could be realised by a real material. Indeed candidate -materials such as were quickly found that are expected to behave -according to the Kitaev with small corrections.
-Second, the Kitaev Honeycomb model is deeply interesting to modern -condensed matter theory. Its ground state is almost the canonical -example of the long sought after quantum spin liquid state. Its -excitations are anyons, particles that can only exist in two dimensions -that break the normal fermion/boson dichotomy. Anyons have been the -subject of much attention because, among other reasons, there are -proposals to braid them through space and time to achieve noise tolerant -quantum computations .
-Third and perhaps most importantly, it a rare many body interacting -quantum system that can be treated analytically. It is exactly solveable -meaning that we can explicitly write down its many body ground states in -terms of single particle states~. Its solubility comes about because the -model has extensively many conserved degrees of freedom that mediate the -interactions between quantum degrees of freedom.
-To get down to brass tacks, the Kitaev Honeycomb model is a model of -interacting spin\(-1/2\)s on the -vertices of a honeycomb lattice. Each bond in the lattice is assigned a -label \(\alpha \in \{ x, y, z\}\) and -that bond couples its two spin neighbours along the \(\alpha\) axis.
-This gives us the Hamiltonian \[\mathcal{H} = - \sum_{\langle j,k\rangle_\alpha} -J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha},\] where \(\sigma^\alpha_j\) is a Pauli matrix acting -on site \(j\), (j,k_) is a pair of -nearest-neighbour indices connected by an \(\alpha\)-bond with exchange coupling \(J^\alpha\)~.
-% plaquette operators and wilson loops This model has a set of -conserved quantities that, in the spin language, take the form of Wilson -loops \[W_p = \prod -\sigma_j^{\alpha}\sigma_k^{\alpha}\] following any closed path of -the lattice. In this product each pair of spins appears twice with two -of the three bonds types, using the spin commutation relations we can -replace each pair with the third. For a single hexagonal plaquette this -looks like: \[W_p = \sigma_1^{z}\sigma_2^{z} -\sigma_2^{x}\sigma_3^{x} \sigma_3^{y}\sigma_4^{y} -\sigma_4^{z}\sigma_5^{z} \sigma_5^{x}\sigma_6^{x} -\sigma_6^{y}\sigma_1^{y}\] $\(W_p = -\sigma_1^{x}\sigma_2^{y} \sigma_3^{z} \sigma_4^{x} -\sigma_5^{y}\sigma_6^{z}\) In this latter form can be seen to -commute with all the terms in the Hamiltonian because { why again?}
-The Hamiltonian commutes with the plaquette operators \(W_p\), products of the \(K\)s around a plaquette. The Ks also -commute with one another. \[W_p = -\prod_{<ij> \in P} K_{ij} = K_{12}K_{23}K_{34}K_{56} ... -K_{N1}\]
-Expanding the bond operators \(K_{ij} = -\sigma_i^{\alpha} \sigma_j^{\alpha}\), Pauli operators on each -site appear in adjacent pairs so can be replaced via \(\sigma_i \sigma_j = \delta_{ij} + \epsilon_{ijk} -\sigma_k\) giving a product of Pauli matrices associated with the -outward pointing bonds from the plaquette. In the general case: \[W_p = \prod_{i \in P} i (-1)^{c_i} -\sigma_i\] where \(c_i = 0,1\) -measures the handedness of the edges around vertex i, see Fig \(\ref{fig:handedness}\). Plaquette operators -for plaquettes with even numbers of edges square to 1 and hence have -eigenvalues \(\pm 1\), while those -around odd plaquettes have eigenvalues (i) breaking chiral symmetry. The -values of the plaquette operators partition the Hilbert space of the -Hamiltonian into a set of flux sectors.
-% relationship between wilson loops and topology Such paths can -enclose a collection of faces or `plaquettes’ of the lattice. In the -case of periodic boundary conditions, the system is torioidal and we -also get Wilson loops that wind the whole system without enclosing a -definite area. The loop operator associated with each such path has -eigenvalues \(/pm 1\) and can be -interpreted as measuring the magnetic flux through that region. Without -going into the details of counting them, the number of these conserved -loop operators clearly scales with system size and it is this extensive -number of classical degrees of freedom that ultimately allows us to -decouple this interacting many body hamiltonian into a set of non -interaction quadratic hamiltonians. { add a figure showing the different -kinds of Wilson loops and of an example plaquette}
-In order to actually solve the model we need to figure out how to -leverage these conserved quantities. The trick is not so much a trick as -an almost perfect consequence of the structure of the model and perhaps -this was in fact how Kitaev first came up with it. We know that a single -spin\(-1/2\) can be represented by -fermionic creation and annihilation operators \(\sigma^{\pm} = 1/2(\sigma^x \pm \sigma^y)\) -through a Jordan-Wigner transformation~, this gives one fermion for each -spin. In turn a fermion can be broken into two Majorana fermions \(c_1 = 1/\sqrt{1}(f + f^\dagger)\) and \(c_2 = i/\sqrt{1}(f - f^\dagger)\). If we -double up the Hilbert space we get four Majoranas per spin:
--
-
-
Here is the footnote.↩︎
-Here’s one with multiple blocks.
-Subsequent paragraphs are indented to show that they belong to the -previous footnote.
-
-{ some.code }The whole paragraph can be indented, or just the first line. In this -way, multi-paragraph footnotes work like multi-paragraph list items.↩︎
-