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+---
+title: The Kitaev Honeycomb Model
+excerpt: A short introduction to the weird and wonderful world of exactly solvable quantum models. This is an excerpt from my thesis.
+layout: none
+image:
+
+---
+
+
+
+
+
+
+
+ The Kitaev Honeycomb Model
+
+
+
+
+
+
+
+
+
+{% include header.html %}
+
+
+
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}
+
+
+
+Figure 1:
+(a) The standard Kitaev Model is defined on a honeycomb
+lattice. The special feature of the honeycomb lattice that makes the
+model solveable it is that each vertex is joined by exactly three bonds
+i.e the lattice is trivalent. One of three labels is assigned to each
+(b) We represent the antisymmetric gauge degree of
+freedom \(u_{jk} = \pm 1\) with arrows
+that point in the direction \(u_{jk} =
++1\)(c) The majorana transformation can be
+visualised as breaking each spin into four majoranas which then pair
+along the bonds. The pairs of x,y and z majoranas become part of the
+classical \(\mathbb{Z}_2\) gauge field
+\(u_{ij}\) leaving just a single
+majorana \(c_i\) per site.
+
+
+
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:
I would like to thank my supervisor, Professor Johannes Knolle and
+co-supervisor Professor Derek Lee for guidance and support during this
+long process.
+
Dan Hdidouan for being an example of how to weather the stress of a
+PhD with grace and kindness.
+
Arnaud for help and guidance…
+
Carolyn, Juraci, Ievgeniia and Loli for their patience and
+support.
+
Nina del Ser
+
Brian Tam for his endless energy on our many many calls while we
+served as joint Postgraduate reps for the department.
+
All the students in CMTH, Halvard, Tom, Chris, Krishnan, David,
+Tonny, Emanuele … and particularly to Thank you to the CMT group at TUM
+in Munich, Alex and Rohit.
+
Gino, Peru and Willian for their collaboration on the Amorphous
+Kitaev Model.
+
Mr Jeffries who encouraged me to pursue physics
+
All the gang from Munich, Toni, Mine, Mike, Claudi.
+
Dan Simpson, the poet in residence at Imperial and one of my
+favourite collaborators during my time at Imperial.
+
Lou Khalfaoui for keeping me sane during the lockdown of March 2022.
+Sophie Nadel, Julie Ketcher and Kim ??? for their graphic design
+expertise and patience.
+
All the I-Stemm team, Katerina, Jeremey, John, ….
+
And finally, I’d like the thank the staff of the Camberwell Public
+Library where the majority of this thesis was written.
Johannes had the initial idea to use a long range Ising term to
+stablise order in a one dimension Falikov-Kimball model. Josef developed
+a proof of concept during a summer project at Imperial. The three of us
+brought the project to fruition.
[WARNING] Citeproc: citation abanin_recent_2017 not found abaninRecentProgressManybody2017[WARNING] Citeproc: citation anderson_absence_1958-1not found andersonAbsenceDiffusionCertain1958
diff --git a/_thesis/2.1.2_AMK_Intro.html b/_thesis/2.1.2_AMK_Intro.html
index 340cfae..37ae7e4 100644
--- a/_thesis/2.1.2_AMK_Intro.html
+++ b/_thesis/2.1.2_AMK_Intro.html
@@ -210,9 +210,12 @@ image:
The bond operators \(u_{ij}\) are
useful because they label a bond sector \(\mathcal{\tilde{L}}_u\) in which we can
-easiy solve the Hamiltonian. However the gauge operators move us between
+easy solve the Hamiltonian. However, the gauge operators move us between
bond sectors. Bond sectors are not gauge invariant!
-
Let’s consider instead the properties of the plaquette operators
+
Let us consider instead the properties of the plaquette operators
\(\hat{\phi}_i\) that live on the faces
of the lattice.
-
We already showed that they are conserved. And as one might hope and
-expect, the plaquette operators map cleanly on to the bond operators of
-the Majorana representation:
+
We already showed that they are conserved. As one might hope and
+expect, the plaquette operators also map cleanly onto the bond operators
+of the Majorana representation:
Where the last steps holds because each \(c_i\) appears exactly twice and adjacent to
-its neighbour in each plaquette operator. Note that this is consistent
-with the observation from earlier that each \(W_p\) takes values \(\pm 1\) for even paths and \(\pm i\) for odd paths.
+
Where the last steps hold because each \(c_i\) appears exactly twice and is adjacent
+to its neighbour in each plaquette operator. This is consistent with the
+earlier observation that each \(W_p\)
+takes values \(\pm 1\) for even paths
+and \(\pm i\) for odd paths.
Vortices and their movements
-
Let’s imagine we started from the ground state of the model and
-flipped the sign of a single bond. In doing so we will flip the sign of
-the two plaquettes adjacent to that bond. I’ll call these disturbed
-plaquettes vortices. I’ll refer to a particular choice values
-for the plaquette operators as a vortex sector.
-
If we chain multiple bond flips we can create a pair of vortices at
-arbitrary locations. The chain of bonds that we must flip corresponds to
-a path on the dual of the lattice.
-
Something else we can do is create a pair of vortices, move one
-around a loop and then anhilate it with its partner. This corresponds to
-a closed loop on the dual lattice and applying such a bond flip leaves
-the vortex sector unchanged.
+
+
+
+Figure 1:Dual Loops
+and Vortex Pairs The different kinds of strings and loops that
+we can make by flipping bond variables or transporting vortices around.
+(a) Flipping a single bond makes a pair of vortices on either side. (b)
+Flipping a string of bonds separates the vortex pair spatially. The
+flipped bonds form a path (in red) on the dual lattice. (c) If we create
+a vortex-vortex pair, transport one of them around a loop and then
+annihilate them, we can change the bond sector without changing the
+vortex sector. This is a manifestation of the gauge symmetry of the bond
+sector. (d) If we transport a vortex around the major or minor axes of
+the torus, we create a non-contractable loop of bonds \(\hat{\mathcal{T}}_{x/y}\). Unlike all the
+other dual loops, These operators cannot be constructed from the
+contractable loops created by \(D_j\).
+operators and they flip the value of the topological
+fluxes.
+
+
+
See fig. 1 for a
+diagram of the next three paragraphs.
+
We started from the ground state of the model and flipped the sign of
+a single bond (fig. 1
+(a)). In doing so, we will flip the sign of the two plaquettes adjacent
+to that bond. We will call these disturbed plaquettes vortices.
+We will refer to a particular choice values for the plaquette operators
+as a vortex sector.
+
If we chain multiple bond flips, we can create a pair of vortices at
+arbitrary locations (fig. 1 (b)). The chain of bonds
+that we must flip corresponds to a path on the dual of the lattice.
+
We can also create a pair of vortices, move one around a loop and
+finally annihilate it with its partner (fig. 1 (c)). This corresponds to
+a closed loop on the dual lattice. Applying such a bond flip leaves the
+vortex sector unchanged. We can also do the same thing but move the
+vortex around one the non-contractible loops of the lattice (fig. 1 (d)).
+
Dual Loops and gauge
+symmetries
+
+
+
+Figure 2:Dual Loops
+and Gauge Symmetries A honeycomb lattice with edges in light
+grey, along with its dual, the triangle lattice in light blue. The
+vertices of the dual lattice are the faces of the original lattice and,
+hence, are the locations of the vortices. (Left) The action of the gauge
+operator \(D_j\) at a vertex is to flip
+the value of the three \(u_{jk}\)
+variables (black lines) surrounding site \(j\). The corresponding edges of the dual
+lattice (red lines) form a closed triangle. (Middle) Composing multiple
+adjacent \(D_j\) operators produces a
+large closed dual loop or multiple disconnected dual loops. Dual loops
+are not directed like Wilson loops. (Right) A non-contractable loop
+which cannot be produced by composing \(D_j\) operators. All three operators can be
+thought of as the action of a vortex-vortex pair that is created, one of
+them is transported around the loop, and then the two annihilate again.
+Note that every plaquette has an even number of \(u_{ij}\)s flipped on its edge. Therefore,
+all retain the same value.
+
+
+
See fig. 2 for a diagram of the
+next few paragraphs.
Notice that the \(D_j\) operators
-flip three bonds around a vertex. This is the smallest closed loop
-around which one can move a vortex pair and anhilate it with itself.
-
Such operations compose in the sense that we can build any larger
-loop by applying a series of \(D_j\)
-operations. Indeed the symetrisation procedure
+
Such operations compose, so we can build any larger loop (almost) by
+applying a series of \(D_j\)
+operations. The symmetrisation procedure \(\prod_i \left( \frac{1 + D_i}{2}\right)\)
-that maps from the bond sector to a physical state is applying
-constructing a superposition over every such loop that leaves the vortex
-sector unchanged.
-
The only loops that we cannot build out of \(D_j\)s are non-contractible loops, such as
-those that span the major or minor circumference of the torus.
-
The plaquette operators are the gauge invariant quantity that
-determines the physics of the model
-
Composition of \(u_{jk}\) loops
+that maps from the bond sector to a physical state is really
+constructing a superposition over every such dual loops that leaves the
+vortex sector unchanged.
+
There is one kind of dual loop that we cannot build out of \(D_j\)s, the non-contractible loops.
+
The plaquette operators and topological fluxes are the gauge
+invariant quantities which determine the physics of the
+model
+
Composition of Wilson loops
-Figure 1: In the product of
-individual plaquette operators shared bonds cancel out. The product is
+alt="Figure 3: In the product of individual plaquette operators, shared bonds cancel out. The product is equal to the enclosing path." />
+Figure 3: In the product of
+individual plaquette operators, shared bonds cancel out. The product is
equal to the enclosing path.
-
Second it is now easy to show that the loops and plaquettes satisfy
-nice composition rules, so long as we stick to loops that wind in a
+
Second, one can now easily show that the loops and plaquettes satisfy
+nice composition rules, so long as we keep to loops that wind in a
particular direction.
Consider the product of two non-overlapping loops \(W_a\) and \(i u_{12}\) and the other \(i u_{21}\). Since the \(u_{ij}\) commute with one another, they
square to \(1\) and \(u_{ij} = -u_{ji}\) we see have \(i u_{12} i u_{21} = 1\) and we can repeat
-this for any number of shared edges. Hence, we get a version of Stokes’
+class="math inline">\(u_{ij} = -u_{ji}\), we have \(i u_{12} i u_{21} = 1\). We can repeat this
+for any number of shared edges. Hence, we get a version of Stokes’
theorem: the product of \(i u_{jk}\)
around any closed loop \(\partial A\)
is equal to the product of plaquette operators \[\prod_{u_{jk} \in \partial A} i \; u_{jk} =
-Figure 2: The loop
-composition rule extends to arbitrary numbers of vortices giving a
+alt="Figure 4: The loop composition rule extends to arbitrary numbers of vortices which gives a discrete version of Stoke’s theorem." />
+Figure 4: The loop
+composition rule extends to arbitrary numbers of vortices which gives a
discrete version of Stoke’s theorem.
@@ -350,11 +419,43 @@ discrete version of Stoke’s theorem.
plaquettes operators unless they are non-contractable
Gauge Degeneracy and
the Euler Equation
+
+
+
+Figure 5: (Bond Sector) A
+state in the bond sector is specified by assigning \(\pm 1\) to each edge of the lattice.
+However, this description has a substantial gauge degeneracy. We can
+simplify things by decomposing each state into the product of three
+kinds of objects: (Vortex Sector) Only a small number of bonds need to
+be flipped (compared to some arbitrary reference) to reconstruct the
+vortex sector. Here, the edges are chosen from a spanning tree of the
+dual lattice, so there are no loops. (Gauge Field) The ‘loopiness’ of
+the bond sector can be factored out. This gives a network of loops that
+can always be written as a product of the gauge operators \(D_j\). (Topological Sector) Finally, there
+are two loops that have no effect on the vortex sector, nor can they be
+constructed from gauge symmetries. These can be thought of as two fluxes
+\(\Phi_{x/y}\) that thread through the
+major and minor axes of the torus. Measuring \(\Phi_{x/y}\) amounts to constructing Wilson
+loops around the axes of the torus. We can flip the value of \(\Phi_{x}\) by transporting a vortex pair
+around the torus in the \(y\)
+direction, as shown here. In each of the three figures on the right,
+black bonds correspond to those that must be flipped. Composing the
+three together gives back the original bond sector on the
+left.
+
+
We can check this analysis with a counting argument. For a lattice
with \(B\) bonds, \(P\) plaquettes and \(V\) vertices we can count how many bond
-sectors, vortices sectors and gauge symmetries there are and check them
+class="math inline">\(V\) vertices, we can count the number of
+bond sectors, vortices sectors and gauge symmetries and check them
against Euler’s polyhedra equation.
Euler’s equation states for a closed surface of genus \(g\), i.e that has \[B = P + V + 2 - 2g\]
-Figure 3: In periodic
+alt="Figure 6: In periodic boundary conditions the Kitaev model is defined on the surface of a torus. Topologically, the torus is distinct from the sphere in that it has a hole that cannot be smoothly deformed away. Associated with each such hole are two non-contractible loops on the surface, here labelled x and y, which cannot be smoothly deformed to a point. These two non-contractible loops can be used to construct two special pairs of operators: The two topological fluxes \Phi_x and \Phi_y that are the expectation values of u_{jk} loops around each path. There are also two operators \hat{\mathcal{T}}_x and \hat{\mathcal{T}}_y that transform one half of a vortex pair around the loop before annihilating them together again, see later." />
+Figure 6: In periodic
boundary conditions the Kitaev model is defined on the surface of a
-torus. Topologically the torus is distinct from the sphere in that it
+torus. Topologically, the torus is distinct from the sphere in that it
has a hole that cannot be smoothly deformed away. Associated with each
-such hole are two non-contractible loops on the surface, here labeled A
-and B, that cannot be smoothly deformed to a point. These two
-non-contracible loops can. be used to construct two symmetry operators
-\(\hat{A}\) and \(\hat{A}\) that flip \(u_{jk}\)s along their paths.
+such hole are two non-contractible loops on the surface, here labelled
+\(x\) and \(y\), which cannot be smoothly deformed to a
+point. These two non-contractible loops can be used to construct two
+special pairs of operators: The two topological fluxes \(\Phi_x\) and \(\Phi_y\) that are the expectation values of
+\(u_{jk}\) loops around each path.
+There are also two operators \(\hat{\mathcal{T}}_x\) and \(\hat{\mathcal{T}}_y\) that transform one
+half of a vortex pair around the loop before annihilating them together
+again, see later.
For the case of the torus where \(g =
-1\) we can rearrange this to read: \[B
-= (P-1) + (V-1) + 2\]
-
Each \(u_{ij}\) takes two values and
-there is one associated with each bond so there are exactly , we can rearrange this to read: \[B = (P-1) + (V-1) + 2\]
+
Bond Sectrors: Each \(u_{ij}\) takes two values and there is one
+associated with each bond so there are exactly \(2^B\) distinct configurations of the bond
-sector. Let’s see if we can factor those configurations out into the
-cartesian product of vortex sectors, gauge symmetries and
+sector. Let us see if we can factor those configurations out into the
+Cartesian product of vortex sectors, gauge symmetries and
non-contractible loop operators.
-
Vortex sectors: each plaquette operator Vortex sectors: Each plaquette operator \(\phi_i\) takes two values (\(\pm 1\) or \(\pm
-i\)) and there are \(P\) of them
-so naively one would think there are \(2^P\). However vortices can only be created
-on pairs so there are really \(\tfrac{2^P}{2}
-= 2^{P-1}\) vortex sectors.
-
Gauge symmetries: As discussed earlier these correspond to the all
-possible compositions of the \(D_j\)
-operators. Again there are only \(2^{V-1}\) of these because, as we will see
-in the next section, \(\prod_{j} D_j =
-\mathbb{1}\) in the physical space, and we enforce this by
-chooising the correct product of single particle fermion states. You can
-get an intuitive picture for why \(\prod_{j}
-D_j = \mathbb{1}\) by imagining larger and larger patches of
-\(D_j\) operators on the torus. These
-patches correspond to transporting a vortex pair around the edge of the
-patch. At some point the patch wraps around and starts to cover the
-entire torus, as this happens the bounday of the patch disappears and
-hence it which corresponds to the identity operation. See Fig ??
-(animated in the HTML version).
-
Finally the torus has two non-contractible loop operators asscociated
-with its major and minor diameters.
-
Putting this all together we see that there are ) and there are \(P\) of
+them. Vortices can only be created in pairs so there are \(\tfrac{2^P}{2} = 2^{P-1}\) vortex sectors
+in total. Denoting the number of pairs of vortices as \(N_v\), the vortex parity \(1 - 2*(N_v \mod 2)\) will be relevant in
+the projector later.
+
Gauge symmetries: As discussed earlier, these
+correspond to all possible compositions of the \(D_j\) operators. Again, there are only
+\(2^{V-1}\) of these because, as we
+will see in the next section, \(\prod_{j} D_j
+= \mathbb{1}\) in the physical space. We enforce this by choosing
+the correct product of single particle fermion states. One can get an
+intuitive picture for why \(\prod_{j} D_j =
+\mathbb{1}\) by imagining larger and larger patches of \(D_j\) operators on the torus. These patches
+correspond to transporting a vortex pair around the edge of the patch.
+At some point, the patch wraps around and starts to cover the entire
+torus. As this happens, the boundary of the patch disappears and, hence,
+it corresponds to the identity operation. See fig. 7 and fig. 8.
+
Topological Sectors: Finally, the torus has two
+non-contractible loop operators associated with its major and minor
+diameters. These give us two extra fluxes \(\Phi_x\) and \(\Phi_y\) each with two distinct values.
+
Putting this all together, we see that there are \(2^B\) bond sectors a space which
-can be decomposed into the cartesian product of \(2^{P-1}\) vortex sectors,
\(2^{V-1}\) gauge
symmetries and \(2^2 =
-4\) topological sectors associated with the
-non-contractible loop operators. This last factor forms the basis of
-proposals to construct topologically protected qubits since the 4
-sectors cold only be mixed by a highly non-local perturbation, ref
-?????.
-
+4\) topological sectors.
+
The topological sector forms the basis of proposals to construct
+topologically protected qubits since the four sectors can only be mixed
+by a highly non-local perturbations1.
+
The Extended Hilbert Space decomposes into a direct product
+of Flux Sectors, four Topological Sectors and a set of gauge
+symmetries.
+
+
-
-
+src="/assets/thesis/figure_code/amk_chapter/intro/flood_fill/flood_fill.gif"
+style="width:100.0%"
+alt="Figure 7: A honeycomb lattice (in black) along with its dual (in red). (Left) Taking a larger and larger set of D_j operators (Bold Vertices) leads to an outward expanding boundary dual loop. Eventually every lattice on the torus is included and the boundary contracts to a point and disappears. This is a visual proof that \prod_i D_i \propto \mathbb{1}. We’ll see later that it takes values \pm 1 and is a key part of the projection to the physical subspace. (Right) In black and red the edges and dual edges that must be flipped to add vortices at the sites highlighted in orange. Flipping all the plaquettes in the system is not equivalent to the identity. Not that the edges that must be flipped can always be chosen from a tree since loops can be removed by a gauge transformation." />
+Figure 7: A honeycomb
+lattice (in black) along with its dual (in red). (Left) Taking a larger
+and larger set of \(D_j\) operators
+(Bold Vertices) leads to an outward expanding boundary dual loop.
+Eventually every lattice on the torus is included and the boundary
+contracts to a point and disappears. This is a visual proof that \(\prod_i D_i \propto \mathbb{1}\). We’ll see
+later that it takes values \(\pm 1\)
+and is a key part of the projection to the physical subspace. (Right) In
+black and red the edges and dual edges that must be flipped to add
+vortices at the sites highlighted in orange. Flipping all the
+plaquettes in the system is not equivalent to
+the identity. Not that the edges that must be flipped can always be
+chosen from a tree since loops can be removed by a gauge
+transformation.
+
+
Counting edges,
plaquettes and vertices
-
It will be useful to know how the trivalent structre of the lattice
-constraints the number of bonds \(B\),
+
It is useful to know how the trivalent structure of the lattice
+constrains the number of bonds \(B\),
plaquettes \(P\) and vertices \(V\) it has.
-
We can immediately see that the lattice is built from vertices that
-each share 3 edges with their neighbours. This means each vertex comes
-with \(\tfrac{3}{2}\) bonds i.e The lattice is built from vertices that each share three edges with
+their neighbours. This means that each vertex comes with \(\tfrac{3}{2}\) bonds i.e \(3V = 2B\). This is consistent with the fact
-that in the Majorana representation on the torus each vertex brings
+that, in the Majorana representation on the torus, each vertex brings
three \(b^\alpha\) operators which then
pair along bonds to give \(3/2\) bonds
per vertex.
If we define an integer \(N\) such
that \(V = 2N\) and \(B = 3N\) and substitite this into the
-polyhedra equation for the torus we see that \(P = N\). So if is a trivalent lattice on
-the torus has \(N\) plaquettes, it has
-\(2N\) vertices and \(B = 3N\) and substitute this into the
+polyhedra equation for the torus, we see that \(P = N\). Therefore, if a trivalent lattice
+on the torus has \(N\) plaquettes, it
+has \(2N\) vertices and \(3N\) bonds.
We can also consider the sum of the number of bonds in each plaquette
\(S_p\), since each bond is a member of
exactly two plaquettes \[S_p = 2B =
6N\]
The mean size of a plaquette in a trivalent lattice on the torus is
-exactly 6. Since the sum is even, this also tells us that all odd
+exactly six. As the sum is even, this also tells us that all odd
plaquettes must come in pairs.
+
+
+
+Figure 8: The same as
+fig. 7 but for the amorphous
+lattice.
+
+
+
The Projector
+
The projection from the extended space to the physical space will not
+be particularly important for the results presented here. However, the
+theory remains useful to explain why this is.
-Figure 4: The relationship
-between the different Hilbert spaces used in the solution is slightly
-complex.
+alt="Figure 9: The relationship between the different Hilbert spaces used in the solution. needs updating" />
+Figure 9: The relationship
+between the different Hilbert spaces used in the solution. needs
+updating
-
The Projector
-
It will turn out that the projection from the extended space to the
-physical space is not actually that important for the results that I
-will present. However it it useful to go through the theory of it to
-explain why this is.
-
The physicil states are defined as those for which The physical states are defined as those for which \(D_i |\phi\rangle = |\phi\rangle\) for all
\(D_i\). Since \(D_i\) has eigenvalues \(\tfrac{(1+D_i)}{2}\) has eigenvalue \(1\) for physical states and \(0\) for extended states so is the local
projector onto the physical subspace.
-
The global projector is therefore \[
+
Therefore, the global projector is \[
\mathcal{P} = \prod_{i=1}^{2N} \left( \frac{1 +
D_i}{2}\right)\]
for a toroidal trivalent lattice with \(N\) plaquettes \(2N\) vertices and \(3N\) edges. As I pointed out before the
+class="math inline">\(3N\)
edges. As discussed earlier, the
product over \((1 + D_j)\) can also be
thought of as the sum of all possible subsets \(\{i\}\) of the
Since the gauge operators \(D_j\)
commute and square to one, we can define the complement operator \(C = \prod_{i=1}^{2N} D_i\) and see that it
-take each set of \(\prod_{i \in \{i\}}
-D_j\) operators and gives us the complement of that set. I said
-earlier that \(C\) is the identity in
-the physical subspace and we will shortly see why.
+takes each set of \(\prod_{i \in \{i\}}
+D_j\) operators and gives us the complement of that set. We will
+shortly see why \(C\) is the identity
+in the physical subspace, as noted earlier.
We use the complement operator to rewrite the projector as a sum over
-half the subsets of \(\{i\}\) let’s
-call that \(\Lambda\). The complement
-operator deals with the other half
+half the subsets of \(\{i\}\) -
+referred to as \(\Lambda\). The
+complement operator deals with the other half
To compute \(\mathcal{P}_0\), the
main quantity needed is the product of the local projectors \(D_i\)\[\prod_i^{2N} D_i = \prod_i^{2N} b^x_i b^y_i b^z_i
@@ -531,8 +669,8 @@ c_i \] for a toroidal trivalent lattice with \(N\) plaquettes \(2N\) vertices and \(3N\) edges.
-
First we reorder the operators by bond type, this doesn’t require any
-information about the underlying lattice.
+
First, we reorder the operators by bond type. This does not require
+any information about the underlying lattice.
The product over \(c_i\) operators
@@ -540,14 +678,14 @@ reduces to a determinant of the Q matrix and the fermion parity,
see1. The only difference from the
-honeycomb case is that we cannot explicitely compute the factors 2. The only difference from the
+honeycomb case is that we cannot explicitly compute the factors \(p_x,p_y,p_z = \pm\;1\) that arise from
reordering the b operators such that pairs of vertices linked by the
corresponding bonds are adjacent.
However they are simply the parity of the permutation from one
+
However, they are simply the parity of the permutation from one
ordering to the other and can be computed in linear time with a cycle
decompositionapp:cycle_decomp?.
where \(p_x\;p_y\;p_z = \pm 1\) are
-lattice structure factors. \(det(Q^u)\)
-is the determinant of the matrix mentioned earlier that maps \(\mathrm{det}(Q^u)\) is the determinant of
+the matrix mentioned earlier that maps \(c_i\) operators to normal mode operators
\(b'_i, b''_i\). These
depend only on the lattice structure.
@@ -568,113 +707,156 @@ class="math inline">\(n_i\). As discussed in +1 is 2, \(\hat{\pi}\) is gauge invariant in the sense
that \([\hat{\pi}, D_i] = 0\).
This implies that \(det(Q^u) \prod -i
-u_{ij}\) is also a guage invariant quantity. In translation
+u_{ij}\) is also a gauge invariant quantity. In translation
invariant models this quantity which can be related to the parity of the
number of vortex pairs in the system2. However it is not so simple to
-evaluate in the amorphous case.
-
More general arguments3,4 imply that \(det(Q^u) \prod -i u_{ij}\) has an
-interesting relationship to the topological fluxes. In the non-Abelian
-phase we expect that it will change sign in exactly on of the four
-topological sectors. This forces that sector that contain a fermion and
-hence gives the model a three-fold degerenate ground state. In the
-Abelian phase this doesn’t happen and we get a fourfold degerate ground
-state. Whether this analysis generalises to the amorphous case in
-unclear.
-
An alternate way to view this is to consider the adiabatic insertion
-of the fluxes \(\Phi_{x,y}\) as the
-operations that undo vortex transport around the lattice. In this
-picture the three fold degeneracy occurs because transporting a vortex
-around both the major and minor axes of the torus
-changes its fusion channel such that the two vortices fuse into a
-fermion excition rather than the vacuum.
+role="doc-biblioref">3. I am unsure if this is true for
+the amorphous case.
All these factors take values \(\pm
1\) so \(\mathcal{P}_0\) is 0 or
1 for a particular state. Since \(\mathcal{S}\) corresponds to symmetrising
-over all the gauge configurations and cannot be 0, this tells use that
-once we have determined the single particle eigenstates of a bond
-sector, the true many body ground state has the same energy as either
-the empty state with \(n_i = 0\) or a
-state with a single fermion in the lowest level.
-
Let’s think about where are with the model now. We can map the spin
+over all the gauge configurations and cannot be 0, once we have
+determined the single particle eigenstates of a bond sector, the true
+many body ground state has the same energy as either the empty state
+with \(n_i = 0\) or a state with a
+single fermion in the lowest level.
+
Ground State Degeneracy
+
+
+
+Figure 10: (Left) The two
+topological flux operators of the toroidal lattice. These do not
+correspond to any face of the lattice, but rather measure flux that
+threads through the major and minor axes of the torus. This shows a
+particular choice. Yet, any loop that crosses the boundary is gauge
+equivalent to one of or the sum of these two loop. (Right) The two ways
+to transport vortices around the diameters. These amount to creating a
+vortex pair, transporting one of them around the major or minor
+diameters of the torus and, then, annihilating them again.
+
+
+
More general arguments4,5 imply that \(det(Q^u) \prod -i u_{ij}\) has an
+interesting relationship to the topological fluxes. In the non-Abelian
+phase, we expect that it will change sign in exactly one of the four
+topological sectors.
+
This means that the lowest state in three of the topological sectors
+contain no fermions, while in one of them there must be one fermion to
+preserve product of fermion vortex parity. So overall the non-Abelian
+model has a three-fold degenerate ground state rather than the fourfold
+of the Abelian case (and of my intuition!). In the Abelian phase, this
+does not happen and we get a fourfold degenerate ground state. Whether
+this analysis generalises to the amorphous case is unclear.
+
An alternative way to view this is to imagine we start in one state
+of the ground state manifold. We then attempt to construct other ground
+states by creating vortex pairs, transporting one vortex around one or
+both non-contractible loops and then annihilating them. This works for
+either of the two non-contractible loops but when we try to do it for
+both something strange happens. When we transport a vortex
+around both the major and minor axes of the torus this
+changes its fusion channel. Normally two vortices fuse to the vacuum but
+after this operation they fuse into a fermion excitation. And hence our
+attempt to construct that last ground state doesn’t yield a ground state
+at all, leaving us with just three.
+
NOTE to self: This argument seems to involve adiabatic
+insertion of the fluxes \(\Phi_{x,y}\)
+as the operations that undo vortex transport around the lattice. I don’t
+understand why that part is necessary
+
+
+
+Figure 11: In the
+non-Abelian phase one of the lowest energy state in one of the
+topological sectors contains a fermion and hence is slightly higher in
+energy than the other three. This manifests as a fourfold ground state
+degeneracy in the Abelian phase and a threefold degeneracy in the
+non-Abelian phase.
+
+
+
Quick Breather
+
Let’s consider where are with the model now. We can map the spin
Hamiltonian to a Majorana Hamiltonian in an extended Hilbert space.
Along with that mapping comes a gauge field \(u_{jk}\) defining bond
sectors. The gauge symmetries of \(u_{jk}\) are generated by the set of \(D_j\) operators. The gauge invariant and
-therefore physically relevant variables are the plaquette operators
+class="math inline">\(D_j\) operators. The gauge invariant, and,
+therefore, physically, relevant variables are the plaquette operators
\(\phi_i\) which define as a
-vortex sector. In order to practically solve the
-Majorana Hamiltonian we must remove hats from the gauge field by
-restricting ourselves to a particular bond sector. From there the
-Majorana Hamiltonian becomes non-interacting and we can solve it like
-any quadratic theory. This lets us construct the single particle
-eigenstates from which we can also construct many body states. However
-the many body states constructed this way are not in the physical
-subspace!
-
However for the many body states within a particular bond sector,
-\(\mathcal{P}_0 = 0,1\) tells us which
-of those have some overlap with the physical sector.
+vortex sector. To solve the Majorana Hamiltonian, we
+must remove hats from the gauge field by restricting ourselves to a
+particular bond sector. At this stage, the Majorana Hamiltonian becomes
+non-interacting and we can solve it like any quadratic theory. This lets
+us construct the single particle eigenstates from which we can also
+construct many body states. Yet, the many body states constructed in
+this way are not in the physical subspace!
+
For the many body states within a particular bond sector, \(\mathcal{P}_0 = 0,1\) tells us which of
+those overlap with the physical sector.
We see that finding a state that has overlap with a physical state
only ever requires the addition or removal of one fermion. There are
-cases where this can make a difference but for most observables such as
-ground state energy this correction scales away as the number of
+cases where this can make a difference, but for most observables, such
+as ground state energy, this correction scales away as the number of
fermions in the system grows.
If we wanted to construct a full many body wavefunction in the spin
-basis we would need to include the full symmetrisation over the gauge
-fields. However this was not necessary for any of the results that will
+basis, we would need to include the full symmetrisation over the gauge
+fields. However, this was not necessary for any of the results that will
be presented here.
The Ground State
-
As we have shown that the Hamiltonian is gauge invariant, only the
-flux sector and the two topological fluxes affect the spectrum of the
-Hamiltonian. Thus we can label many body ground state by a combination
-of flux sector and fermionic occupation numbers.
-
By studying the projector we saw that the fermionic occupation
+
We have shown that the Hamiltonian is gauge invariant. As a result,
+only the flux sector and the two topological fluxes affect the spectrum
+of the Hamiltonian. Thus, we can label the many body ground state by a
+combination of fluxes and fermionic occupation numbers.
+
By studying the projector, we saw that the fermionic occupation
numbers of the ground state will always be either \(n_m = 0\) or \(n_0 = 1, n_{m>1} = 0\) because the
-projector really just enforces vortex and fermion parity.
+projector only enforces combined vortex and fermion parity.
I refer to the flux sector that contains the ground state as the
-ground state flux sector. Recall that we call the excitations of the
-fluxes away from the ground ground state configuration
-vortices, so that the ground state flux sector is the
-vortex free sector by definition.
-
On the Honeycomb, Lieb’s theorem implies that the the ground state
+ground state flux sector. Recall that the excitations of the fluxes away
+from the ground ground state configuration are called
+vortices, so that the ground state flux sector is, by
+definition, the vortex free sector.
+
On the Honeycomb, Lieb’s theorem implies that the ground state
corresponds to the state where all \(u_{jk} =
-1\) implying that the flux free sector is the ground state
+1\). This implies that the flux free sector is the ground state
sector5.
Lieb’s theorem does not generalise easily to the amorphous case.
-However we can get some intuition by examining the problem that will
+However, we can get some intuition by examining the problem that will
lead to a guess for the ground state. We will then provide numerical
evidence that this guess is in fact correct.
-
Let’s consider the partition function of the Majorana hamiltonian:
-\[ \mathcal{Z} = \mathrm{Tr}\left( e^{-\beta
+
Consider the partition function of the Majorana hamiltonian: \[ \mathcal{Z} = \mathrm{Tr}\left( e^{-\beta
H}\right) = \sum_i \exp{-\beta \epsilon_i}\] At low temperatures
\(\mathcal{Z} \approx \beta
\epsilon_0\) where \(\epsilon_0\) is the lowest energy fermionic
state.
How does the \(\mathcal{Z}\) depend
-on the Majorana hamiltonian? Expanding the exponential out gives: \[ \mathcal{Z} = \sum_n \frac{(-\beta)^n}{n!}
\mathrm{Tr(H^k)} \]
-
Now there’s an interesting observation to make here. The Hamiltonian
-is essentially a scaled adjacency matrix. An adjacency matrix being a
+
This makes for an interesting observation. The Hamiltonian is
+essentially a scaled adjacency matrix. An adjacency matrix being a
matrix \(g_{ij}\) such that \(g_{ij} = 1\) if vertices \(i\) and \(j\) and joined by an edge and 0
otherwise.
Powers of adjacency matrices have the property that the entry \((g^n)_{ij}\) corresponds to the number of
-paths of length n on the graph that begin at site \(i\) and end at site \(j\). These include somewhat degenerate
-paths that go back on themselves etc.
-
The trace of an adjacency matrix \(n\) on the graph that
+begin at site \(i\) and end at site
+\(j\). These include somewhat
+degenerate paths that go back on themselves.
+
Therefore, the trace of an adjacency matrix \[\mathrm{Tr}(g^n) = \sum_i (g^n)_{ii}\]
-therefore counts the number number of loops of size \(n\) that can be drawn on the graph.
Applying the same treatment to our Majorana Hamiltonian, we can
interpret \(u_ij\) to equal 0 if the
@@ -696,40 +878,36 @@ two sites are not joined by a bond and we put ourselves in the isotropic
phase where \(J^\alpha = 1\)\[ \tilde{H}_{ij} = \tfrac{1}{2} i
u_{ij}\]
-
We then see that the trace of the nth power of H is a sums over
-Wilson loops of size \(n\) with an
-additional factor of \(2^{-n}\). We
-showed earlier that the Wilson loop operators can always be written as
-products of the plaquette operators that they enclose.
-
Lumping all the prefactors together, we can write: \[ \mathcal{Z} = c_A \hat{A} + c_B \hat{B} + \sum_i
-c_i \hat{\phi}_i + \sum_{ij} c_{ij} \hat{\phi}_i \hat{\phi}_j +
-\sum_{ijk} c_{ijk} \hat{\phi}_i \hat{\phi}_j \hat{\phi}_k +
-...\]
+
We then see that the trace of the nth power of H is a sum over Wilson
+loops of size \(n\) with an additional
+factor of \(2^{-n}\). We showed earlier
+that the Wilson loop operators can always be written as products of the
+plaquette operators that they enclose.
+
Lumping all the prefactors together, we will get something
+schematically like: \[ \mathcal{Z} = c_A
+\hat{A} + c_B \hat{B} + \sum_i c_i \hat{\phi}_i + \sum_{ij}
+c_{ij} \hat{\phi}_i \hat{\phi}_j + \sum_{ijk} c_{ijk} \hat{\phi}_i
+\hat{\phi}_j \hat{\phi}_k + ...\]
Where the \(c\) factors would be
something like \[c_{ijk...} = \sum_n
-\tfrac{(-\beta)^n}{n!} \tfrac{1}{2^n} K_{ijk...}\] which is a sum
-over all loop lengths \(n\) and for
-each we have a combinatoral factor This is a sum
+over all loop lengths \(n\) with, for
+each, a combinatorial factor \(K_{ijk...}\) that counts how many ways
-there are to draw a loop of length \(n\) that only encloses plaquettes \(n\)
+that only encloses plaquettes \(ijk...\).
-
We also have the pesky non-contractible loop operators \(\hat{A}\) and \(\hat{B}\). Again the prefactors for these
-are very complicated but we can intuitively see that for larger and
-larger loops lengths there will be a combinatorial explosion of possible
-ways that they appear in these sums. These are suppressed exponentially
-with system size but at practical lattice sizes they cause significant
-finite size effects. The main evidence of this is that the 4 loop
-sectors spanned by the \(\hat{A}\) and
-\(\hat{B}\) operators are degenerate in
-the infinite system size limit, while that degeneracy is lifted in
-finite sized systems.
-
We don’t have much hope of actually evaluating this for an amorphous
-lattice. However it lead us to guess that the ground state vortex sector
-might be a simple function of the side length of each plaquette.
+
We also have the pesky topological fluxes \(Phi_x\) and \(\Phi_y\). Again, the prefactors for these
+are very complicated. However, we can intuitively see that for larger
+and larger loops lengths, there will be a combinatorial explosion of
+possible ways that they appear in these sums. We know that explosion
+will be suppressed exponentially for sufficiently large system sizes but
+for practical lattices they cause significant finite size effects.
+
We do not have much hope of actually evaluating this for an amorphous
+lattice. However, we can guess that the ground state vortex sector might
+be a simple function of the side length of each plaquette.
The ground state of the Amorphous Kitaev Model is found by setting
the flux through each plaquette \(\phi\) to be equal to
This conjecture is consistent with Lieb’s theorem on regular
lattices5 and
+href="#ref-lieb_flux_1994" role="doc-biblioref">6
and
is supported by numerical evidence. As noted before, any flux that
-differs from the ground state is an excitation which I call a
+differs from the ground state is an excitation which we call a
vortex.
Finite size effects
-
This guess only works for larger lattices because of the finite size
-effects. In order to rigorously test it we would like to directly
-enumerate the \(2^N\) vortex sectors
-for a smaller lattice and check that the lowest state found is the
-vortex sector predicted by ???.
-
To do this we tile use an amorphous lattice as the unit cell of a
+
This guess only works for larger lattices. To rigorously test it, we
+would want to directly enumerate the \(2^N\) vortex sectors for a smaller lattice
+and check that the lowest state found is the vortex sector predicted by
+our conjecture.
+
To do this we tile an amorphous lattice as the unit cell of a
periodic \(N\times N\) system. Bonds
that originally crossed the periodic boundaries now connect adjacent
-unit cells. Using Bloch’s theorem the problem then essnetially reduces
-back to the single amorphous unit cell but now the edges that cross the
+unit cells. Using Bloch’s theorem, the problem essentially reduces back
+to the single amorphous unit cell. However, now the edges that cross the
periodic boundaries pick up a phase dependent on the crystal momentum
-\(\vex{q} = (q_x, q_y)\) and the
+\(\vec{q} = (q_x, q_y)\) and the
lattice vector of the bond \(\vec{x} = (+1, 0,
-1, +1, 0, -1)\). Assigning these lattice vectors to each bond is
-also a very conveninent way to store and plot toroidal graphs.
+also a very convenient way to store and plot toroidal graphs.
This can then be solved using Bloch’s theorem. For a given crystal
momentum \(\textbf{q} \in [0,2\pi)^2\),
we are left with a Bloch Hamiltonian, which is identical to the original
@@ -776,233 +954,219 @@ q_{jk}},\end{aligned}\] where \(q_{jk}
class="math inline">\(x\)-periodic boundary in the positive
direction, with the analogous definition for \(y\)-crossing bonds. We also have \(q_{jk} = -q_{kj}\). Finally \(q_{jk} = -q_{kj}\). Finally, \(q_{jk} = 0\) if the edge does not cross any
-boundaries at all – in essence we are imposing twisted boundary
+boundaries at all. In essence, we are imposing twisted boundary
conditions on our system. The total energy of the tiled system can be
calculated by summing the energy of \(M(
\textbf{q})\) for every value of \(\textbf{q}\). The use of a fourier series
-then allows us to compute the diagonalisation with a penalty only linear
-in the number of tiles used compared to diagonalising a single lattice.
-With this technique the finite size effects related to the
+class="math inline">\(\textbf{q}\).
+
With this technique, the finite size effects related to the
non-contractible loop operators are removed with only a linear penalty
in computation time compared to the exponential penalty paid by simply
-simply diagonalising larger lattices.
-
Using this technique we verified that
+
This technique verifies that \(\phi_0\) correctly predicts the ground
-state for hundreds of thousands of lattices with upto 20 plaquettes. For
-larger lattices we verified that random perturbations around the
-predicted ground state never yield a lower energy state.
+state for hundreds of thousands of lattices with up to twenty
+plaquettes. For larger lattices, we verified that random perturbations
+around the predicted ground state never yield a lower energy state.
Chiral Symmetry
-
In the discussion above we see that the ground state has a twofold
-chiral degeneracy that comes about because the global
-sign of the odd plaquettes does not matter.
-
This happens because by adding odd plaquettes we have broken the time
-reversal symmetry of the original modelThe discussion above shows that the ground state has a twofold
+chiral degeneracy which arises because the global sign
+of the odd plaquettes does not matter.
+
This happens because we have broken the time reversal symmetry of the
+original model by adding odd plaquettes6–7–13.
-
Similar to the behaviour of the original Kitaev model in response to
-a magnetic field, we get two degenerate ground states of different
-handedness. Practicaly speaking, one ground state is related to the
+role="doc-biblioref">14.
+
Similarly to the behaviour of the original Kitaev model in response
+to a magnetic field, we get two degenerate ground states of different
+handedness. Practically speaking, one ground state is related to the
other by inverting the imaginary \(\phi\) fluxes7.
+role="doc-biblioref">8.
Phases of the Kitaev Model
-
discuss the abelian A phase / toric code phase / anisotropic
+
discuss the Abelian A phase / toric code phase / anisotropic
phase
the isotropic gapless phase of the standard model
The isotropic gapped phase with the addition of a magnetic field
-
What’s so great about two
-dimensions?
+
What is so great about
+two dimensions?
Topology, chirality and edge
modes
Most thermodynamic and quantum phases studied can be characterised by
a local order parameter. That is, a function or operator that only
requires knowledge about some fixed sized patch of the system that does
not scale with system size.
-
However there are quantum phases that cannot be characterised by such
-a local order parameter. These phases are intead said to posess
+
However, there are quantum phases that cannot be characterised by
+such a local order parameter. These phases are instead said to possess
‘topological order’.
-
One property of topological order that is particularly easy to
-observe that the ground state degeneracy depends on the topology of the
-manifold that we put the system on to. This is referred to as
-topological degeneracy to distinguish it from standard symmetry
-breaking.
-
The Kitaev model will be a good example of this, we have already
-looked at it defined on a graph that is embedded either into the plane
-or onto the torus. The extension to surfaces like the torus but with
-more than one handle is relatively easy.
+
One easily observable property of topological order is that the
+ground state degeneracy depends on the topology of the manifold that we
+put the system on to. This is referred to as topological degeneracy to
+distinguish it from standard symmetry breaking.
+
The Kitaev model is a good example. We have already looked at it
+defined on a graph that is embedded either into the plane or onto the
+torus. The extension to surfaces like the torus but with more than one
+handle is relatively easy.
Anyonic Statistics
+
NB: I’m thinking about moving this section to the overall
+intro, but it’s nice to be able to refer to specifics of the Kitaev
+model also so I’m not sure. It currently repeats a discussion of the
+ground state degeneracy from the projector section.
In dimensions greater than two, the quantum state of a system must
pick up a factor of \(-1\) or \(+1\) if two identical particles are
swapped. We call these Fermions and Bosons.
This argument is predicated on the idea that performing two swaps is
equivalent to doing nothing. Doing nothing should not change the quantum
-state at all, so doing one swap can at most multiply it by \(\pm 1\).
-
However there are many hidden parts to this argument. Firstly, this
-argument just isn’t the whole story, if you want to know why Fermions
-have half integer spin, for instance, you have to go to field
+
However, there are many hidden parts to this argument. First, this
+argument does not present the whole story. For instance, if you want to
+know why Fermions have half integer spin, you have to go to field
theory.
-
There is also a second niggle, why does this argument only work in
-dimensions greater than two? What we’re really saying when we say that
-two swaps do nothing is that the world lines of two particles that have
-been swapped twice can be untangled without crossing. Why can’t they
-cross? Well because if they cross then the particles can interact and
-the quantum state could change in an arbitrary way. We’re implcitly
-using the locality of physics here to argue that if the worldlines stay
-well separated then the overall quantum state cannot too much.
-
In two dimensions we cannot untangle the worldlines of two particles
-that have swapped places, they are braided together. See fig. 5 for a diagram.
+
Second, why does this argument only work in dimensions greater than
+two? When we say that two swaps do nothing, we in fact say that the
+world lines of two particles that have been swapped twice can be
+untangled without crossing. Why can’t they cross? Because if they cross,
+the particles can interact and the quantum state could change in an
+arbitrary way. We are implicitly using the locality of physics to argue
+that, if the worldlines stay well separated, the overall quantum state
+cannot change.
+
In two dimensions, we cannot untangle the worldlines of two particles
+that have swapped places. They are braided together (see fig. 12).
From this fact flows a whole new world of behaviours, now the quantum
-state can aquire a phase factor \(e^{i\phi}\) upon exchange of two identical
-particles, which we now call Anyons.
-
The Kitaev Model is a good demonstration of the connection beween
-Anyons and topological degeneracy. In the Kitaev model we can create a
+
From this fact flows a whole of behaviours. The quantum state can
+acquire a phase factor \(e^{i\phi}\)
+upon exchange of two identical particles, which we now call Anyons.
+
The Kitaev Model is a good demonstration of the connection between
+Anyons and topological degeneracy. In the Kitaev model, we can create a
pair of vortices, move one around a non-contractable loop \(\mathcal{T}_{x/y}\) and then anhilate them
-together. Without topology this should leave the quantum state
-unchanged. Instead it moves us to another ground state in a
-topologically degenerate ground state subspace. Practically speaking it
+class="math inline">\(\mathcal{T}_{x/y}\) and then annihilate
+them together. Without topology, this should leave the quantum state
+unchanged. Instead, we move towards another ground state in a
+topologically degenerate ground state subspace. Practically speaking, it
flips a dual line of bonds \(u_{jk}\)
going around the loop which we cannot undo with any gauge transformation
made from \(D_j\) operators.
If the ground state subspace is multidimensional, quasiparticle
exchange can move us around in the space with an action corresponding to
-a matrix. These matrices do not in general commmute and so these are
-known as non-Abelian anyons.
-
From here things get even more complex, the Kitaev model has a
-non-Abelian phase when exposed to a magnetic field, and the amorphous
+a matrix. In general, these matrices do not commute so these are known
+as non-Abelian anyons.
+
From here, the situation becomes even more complex. The Kitaev model
+has a non-Abelian phase when exposed to a magnetic field. The amorphous
Kitaev Model has a non-Abelian phase because of its broken chiral
symmetry.
-
The way that we have subdivided the Kitaev model into vortex sectors,
-we have a neat separation beween vortices and fermionic excitations.
-However if we looked at the full many body picture we would see that a
-vortex caries with it a cloud of bound majorana states.
+
By subdividing the Kitaev model into vortex sectors, we neatly
+separate between vortices and fermionic excitations. However, if we
+looked at the full many body picture, we would see that a vortex carries
+with it a cloud of bound Majorana states.
-Figure 6: (Left) A large
+alt="Figure 13: (Left) A large amorphous lattice in the ground state save for a single pair of vortices shown in red, separated by the string of bonds that we flipped to create them. (Right) The density of the lowest energy Majorana state in this vortex sector. These Majorana states are bound to the vortices. They ‘dress’ the vortices to create a composite object." />
+Figure 13: (Left) A large
amorphous lattice in the ground state save for a single pair of vortices
shown in red, separated by the string of bonds that we flipped to create
them. (Right) The density of the lowest energy Majorana state in this
-vortex sector. The state is clearly bound to the vortices.
+vortex sector. These Majorana states are bound to the vortices. They
+‘dress’ the vortices to create a composite object.
Consider two processes
We transport one half of a vortex pair around either the x or y
-loops of the torus before anhilating back to the ground state vortex
+loops of the torus before annihilating back to the ground state vortex
sector \(\mathcal{T}_{x,y}\).
-
We flip a line of bond operators coresponding to measuring the
+
We flip a line of bond operators corresponding to measuring the
flux through either the major or minor axes of the torus \(\mathcal{\Phi}_{x,y}\)
-
-
-
-Figure 7: (Left) The two
-topological flux operators of the toroidal lattice, these don’t
-correspond to any face of the lattice, but rather measure flux that
-threads through the major and minor axes of the torus. This shows a
-particular choice but any loop that crosses the boundary is gauge
-equivalent to one of or the sum of these two loop. (Right) The two ways
-to transport vortices around the diameters. These correspond to creating
-a vortex pair, transporting one of them around the major or minor
-diameters of the torus and then anhilating them again.
-
-
The plaquette operators \(\phi_i\)
are associated with fluxes. Wilson loops that wind the torus are
associated with the fluxes through its two diameters \(\mathcal{\Phi}_{x,y}\).
-
In the Abelian phase we can move a vortex along any path we like and
-then when we bring them back together they will anhilate back to the
+
In the Abelian phase, we can move a vortex along any path at will
+before bringing them back together. They will annihilate back to the
vacuum, where we understand ‘the vacuum’ to refer to one of the ground
-states, though not necesarily the same one we started in. We can use
-this to get from the \((\Phi_x, \Phi_y) = (+1,
-+1)\) ground state and construct the set \((+1, +1), (+1, -1), (-1, +1), (-1,
--1)\).
+states. However, this will not necessarily be the same ground state we
+started in. We can use this to get from the \((\Phi_x, \Phi_y) = (+1, +1)\) ground state
+and construct the set \((+1, +1), (+1, -1),
+(-1, +1), (-1, -1)\).
-Figure 8: Wilson loops that
+alt="Figure 14: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus or through the filling. If they made doughnuts that both had a jam filling and a hole, this analogy would be a lot easier to make15." />
+Figure 14: Wilson loops that
wind the major or minor diameters of the torus measure flux winding
-through the hole of the donut/torus or through the filling. If they made
-donuts that had both a jam filling and a hole this analogy would be a
-lot easier to make14.
+role="doc-biblioref">15.
-
However in the non-Abelian phase we have to wrangle with
+
However, in the non-Abelian phase we have to wrangle with
monodromy3,4,4. Monodromy is behaviour of
+role="doc-biblioref">5. Monodromy is the behaviour of
objects as they move around a singularity. This manifests here in that
the identity of a vortex and cloud of Majoranas can change as we wind
-them around the torus in such a way that rather than anhilating to the
-vacuum the anhilate to create an excited state instead of a ground
-state. This means we end up with only three degenerate ground states in
-the non-Abelian phase \((+1, +1), (+1, -1),
-(-1, +1)\)\((+1,
++1), (+1, -1), (-1, +1)\)15. The way that this shows up
-concretly is that the projector enforces both flux and fermion parity.
-When we wind a vortex around both non-contractible loops of the torus,
-it flips the flux parity which forces means we have to introduce a
-fermionic excitation to make the state physical. Hence the process does
-not give a fourth ground state.
-
-
One reason the topology has gained interest recently is there have
+role="doc-biblioref">16. Concretely, this is because
+the projector enforces both flux and fermion parity. When we wind a
+vortex around both non-contractible loops of the torus, it flips the
+flux parity. Therefore, we have to introduce a fermionic excitation to
+make the state physical. Hence, the process does not give a fourth
+ground state.
+
Recently, the topology has notably gained interest because of
proposals to use this ground state degeneracy to implement both
passively fault tolerant and actively stabilised quantum computations
[16;1;17;
hastingsDynamicallyGeneratedLogical2021].
The material in this chapter expands on work presented in
+
Insert citation of amorphous Kitaev paper here
+
which was a joint project of the first three authors with advice and
+guidance from Willian and Johannes. The project grew out of an interest
+Gino, Peru and I had in studying amorphous systems, coupled with
+Johannes’ expertise on the Kitaev model.
Introduction
-
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 \(\alpha\mathrm{-RuCl}_3\) were quickly
-foundThe Kitaev Honeycomb model is remarkable because it combines three
+key properties.
+
First, this model is a plausible tight binding Hamiltonian. The form
+of the Hamiltonian could be realised by a real material. Candidate
+materials are known that are expected to behave according to the Kitaev
+with small corrections such as \(\alpha\mathrm{-RuCl}_3\)1,2 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 computations2.
+
Second, this 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, they can be braided through spacetime to
+achieve noise tolerant quantum computations3.
-
Third and perhaps most importantly, it a rare many body interacting
-quantum system that can be treated analytically. It is exactly
-solveable. We can explicitly write down its many body ground states in
-terms of single particle statesThird, and perhaps most importantly, this model is a rare many body
+interacting quantum system that can be treated analytically. It is
+exactly solvable. We can explicitly write down its many body ground
+states in terms of single particle states4. Its solubility comes about
-because the model has extensively many conserved degrees of freedom that
-mediate the interactions between quantum degrees of freedom.
-
In this chapter I will discuss the physics of the Kitaev Model on
-amorphous lattices.
-
I’ll start by discussing the physics of the Kitaev model in much more
-detail. Here I will look at the gauge symmetries of the model as well as
-its solution via a transformation to a Majorana hamiltonian. From this
-discusssion we will see that for the the model to be sovleable it need
-only be defined on a trivalent, tri-edge-colourable lattice5.
-
In the methods section, I will discuss how to generate such lattices
-and colour them as well as how to map back and forth between
-configurations of the gauge field and configurations of the gauge
-invariant quantities.
-
In results section, I will begin by looking at the zero temperature
-physics. I’ll present numerical evidence that the ground state of the
-model is given by a simple rule. I’ll make an assessment of the gapless,
-abelian and non-abelian phases that are present as well as spontaneous
-chiral symmetry breaking and topological edge states. We will also
-compare the zero temperature phase diagram to that of the Kitaev
-Honeycomb Model. Next I will take the model to finite temperature and
-demonstrate that there is a phase transition to a thermal metal
-state.
-
In the Discussion I will consider possible physical realisations of
-this model as well the motivations for doing so. I will alao discuss how
-a well known quantum error correcting code defined on the Kitaev
-Honeycomb could be generalised to the amorphous case.
-
Various generalisations have been made, one mode replaces pairs of
-hexagons with heptagons and pentagons and another that replaces vertices
-of the hexagons with triangles . When we generalise this to the
-amorphous case, the key property that will remain is that each vertex
-interacts with exactly three others via an x, y and z edge. However the
-lattice will no longer be bipartite, breaking chiral symmetry among
-other things.
-
Kitaev-Heisenberg Model In real materials there will generally be an
-addtional small Heisenberg term \[H_{KH} = -
-\sum_{\langle j,k\rangle_\alpha}
-J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha} +
-\sigma_j\sigma_k\]
+because the model has many conserved degrees of freedom that mediate the
+interactions between quantum degrees of freedom.
Amorphous Systems
Insert discussion of why a generalisation to the amorphous
-case is intersting
+case is interesting
+
This chapter details the physics of the Kitaev model on amorphous
+lattices.
+
It starts by expanding on the physics of the Kitaev model. It will
+look at the gauge symmetries of the model as well as its solution via a
+transformation to a Majorana hamiltonian. This discussion shows that,
+for the the model to be solvable, it needs only be defined on a
+trivalent, tri-edge-colourable lattice5.
+
The methods section discusses how to generate such lattices and
+colour them. It also explain how to map back and forth between
+configurations of the gauge field and configurations of the gauge
+invariant quantities.
+
The results section begins by looking at the zero temperature
+physics. It presents numerical evidence that the ground state of the
+Kitaev model is given by a simple rule depending only on the number of
+sides of each plaquette. It assesses the gapless, Abelian and
+non-Abelian, phases that are present, characterising them by the
+presence of a gap and using local Chern markers. Next it looks at
+spontaneous chiral symmetry breaking and topological edge states. It
+also compares the zero temperature phase diagram to that of the Kitaev
+Honeycomb Model. Next, it takes the model to finite temperature and
+demonstrates that there is a phase transition to a thermal metal
+state.
+
The discussion considers possible physical realisations of this model
+and the motivations for doing so. It also discusses how a well known
+quantum error correcting code defined on the Kitaev Honeycomb model
+could be generalised to the amorphous case.
+
Glossary
+
+
Lattice: The underlying graph on which the models are defined.
+Composed of sites (vertices), bonds (edges) and plaquettes
+(faces).
+
The model : Used when I refer to properties of the the Kitaev
+model that do not depend on the particular lattice.
+
The Honeycomb model : The Kitaev Model defined on the honeycomb
+lattice.
+
The Amorphous model : The Kitaev Model defined on the amorphous
+lattices described here.
+
The Hamiltonian: I will use model to refer to the underlying
+physics and Hamiltonian to refer to particular representations of the
+model.
+
+
The Spin Hamiltonian
+
+
Spin Bond Operators: \(\hat{k}_{ij} =
+\sigma_i^\alpha \sigma_j^\alpha\)
The Extended Hilbert space: The larger Hilbert space spanned by the
+Majorana operators.
+
The physical subspace: The subspace of the extended Hilbert space
+that we identify with the Hilbert space of the original spin model.
+
The Projector \(\hat{P}\): The
+projector onto the physical subspace.
+
+
Flux Sectors
+
+
Odd/Even Plaquettes: Plaquettes with an odd/even number of
+sides.
+
Fluxes \(\phi_i\): The
+expectation values of the plaquette operators \(\pm 1\) for even and \(\pm i\) for odd plaquettes.
+
Flux Sector: A subspace of Hilbert space in which the fluxes take
+particular values.
+
Ground state flux sector: The Flux Sector containing the lowest
+energy many body state.
+
Vortices: Flux excitations away from the ground state flux
+sector.
+
Dual Loops: A set of \(u_{jk}\)
+that correspond to loops on the dual lattice.
+
non-contractible loops or dual loops: The two loops topologically
+distinct loops on the torus that cannot be smoothly deformed to a
+point.
+
Topological Fluxes \(\Phi_{x},
+\Phi_{y}\): The two fluxes associated with the two
+non-contractible loops.
+
Topological Transport Operators: \(\mathcal{T}_{x}, \mathcal{T}_{y}\): The two
+vortex-pair operations associated with the non-contractible
+dual loops.
+
+
Phases
+
+
The A phase: The three anisotropic regions of the phase diagram
+\(A_x, A_y, A_z\) where \(A_\alpha\) means \(J_\alpha >> J_\beta, J_\gamma\).
+
The B phase: The roughly isotropic region of the phase diagram.
+
The Kitaev Model
Commutation relations
-
Before diving into the Hamiltonian of the Kitaev Model, here is a
-quick refresher of the key commutation relations of spins, fermions and
+
Before diving into the Hamiltonian of the Kitaev model, the following
+describes the key commutation relations of spins, fermions and
Majoranas.
Spins
-
Skip this is you’re super familiar with the algebra of the Pauli
-martrices. Scalars like \(\delta_{ij}\)
-should be understood to be multiplied by an implicit identity Skip this is you are familiar with the algebra of the Pauli matrices.
+Scalars like \(\delta_{ij}\) should be
+understood to be multiplied by an implicit identity \(\mathbb{1}\) where necessary.
We can represent a single spin\(-1/2\) particle using the Pauli matrices
@@ -329,22 +405,21 @@ class="math display">\[\sigma^\alpha \sigma^\beta = \delta^{\alpha
\beta} + i \epsilon^{\alpha \beta \gamma} \sigma^\gamma\] \[[\sigma^\alpha, \sigma^\beta] = 2 i
\epsilon^{\alpha \beta \gamma} \sigma^\gamma\]
-
Adding a sites indices \(ijk...\),
-spins at different spatial sites commute always \([\vec{\sigma}_i, \vec{\sigma}_j] = 0\) so
-when \(i \neq j\)Adding site indices, spins at different spatial sites always commute
+\([\vec{\sigma}_i, \vec{\sigma}_j] =
+0\) so when \(i \neq j\)\[\sigma_i^\alpha \sigma_j^\beta = \sigma_j^\alpha
\sigma_i^\beta\]\[[\sigma_i^\alpha,
\sigma_j^\beta] = 0\] while the previous equations hold for \(i = j\).
-
Two extra relations that will be useful for the Kitaev model are the
-value of \(\sigma^\alpha \sigma^\beta
+
Two extra relations useful for the Kitaev model are the value of
+\(\sigma^\alpha \sigma^\beta
\sigma^\gamma\) and \([\sigma^\alpha
\sigma^\beta, \sigma^\gamma]\) when \(\alpha \neq \beta \neq \gamma\) these can
-be computed quite easily by appling the above relations yielding: \[\sigma^\alpha \sigma^\beta \sigma^\gamma = i
-\epsilon^{\alpha\beta\gamma}\] and \[\sigma^\alpha \sigma^\beta \sigma^\gamma =
+i \epsilon^{\alpha\beta\gamma}\] and \[[\sigma^\alpha \sigma^\beta, \sigma^\gamma] =
0\]
Majorana operators are the real and imaginary parts of the fermionic
-operators, physically they correspond to the orthogonal superpositions
-of the presence and absence of the fermion and are thus a kind of
+operators. Physically, they correspond to the orthogonal superpositions
+of the presence and absence of the fermion and are, thus, a kind of
quasiparticle.
-
Once we involve multiple fermions there is quite a bit of freedom in
-how we can perform the transformation from Once we involve multiple fermions, there is some freedom in how we
+can perform the transformation from \(n\) fermions \(f_i\) to \(2n\) Majoranas \(c_i\). The property that must be preserved
-however is that the Majoranas still anticommute:
+class="math inline">\(c_i\). The property that must be preserved,
+however, is that the Majoranas still anticommute:
\[ \{c_i, c_j\} =
2\delta_{ij}\]
+
+
+
+Figure 1: A visual
+introduction to the Kitaev Model.
+
+
The Hamiltonian
-
To get down to brass tacks, the Kitaev Honeycomb model is a model of
-interacting spin\(-1/2\)s on the
+
To start from the fundamentals, 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 \(\alpha\)-bond with exchange coupling \(J^\alpha\)4. For notational brevity is is
+role="doc-biblioref">4. For notational brevity, it is
useful to introduce the bond operators \(K_{ij} =
\sigma_j^{\alpha}\sigma_k^{\alpha}\) where \(\alpha\) is a function of \(i,j\) that picks the correct bond type.
-
-
-
-Figure 1:
-
-
This Kitaev model has a set of conserved quantities that, in the spin
language, take the form of Wilson loop operators \(W_p\) winding around a closed path on the
-lattice. The direction doesn’t matter, but I will stick to clockwise
-here. I’ll use the term plaquette and the symbol \(\phi\) to refer to a Wilson loop operator
that does not enclose any other sites, such as a single hexagon in a
honeycomb lattice.
@@ -423,86 +500,92 @@ bond types
In closed loops, each site appears twice in the product with two of
the three bond types. Applying \(\sigma^\alpha
\sigma^\beta = \epsilon^{\alpha \beta \gamma} \sigma^\gamma, \alpha \neq
-\beta\) then gives us a product containing a single pauli matrix
+\beta\) then gives us a product containing a single Pauli matrix
associated with each site in the loop with the type of the
-outward pointing bond. From this we see that the outward pointing bond. This shows that the \(W_p\) associated with hexagons or shapes
-with an even number of sides all square to 1 and hence have eigenvalues
-\(\pm 1\).
-
A consequence of the fact that the honeycomb lattice is bipartite is
-that there are no closed loops that contain an even number of edges1 and hence all the \(\pm 1\).
+
A bipartite lattice is composed of A and B sublattices with no
+intra-sublattice edges, i.e. no A-A or B-B edges. Any closed loop must
+begin and end at the same site. If we start at an A site, the loop must
+go A-B-A-B… until it returns to the original site. It must, therefore,
+contain an even number of edges to end on the same sublattice that it
+started on.
+
As the honeycomb lattice is bipartite, there are no closed loops that
+contain an even number of edges. Therefore, all the \(W_p\) have eigenvalues \(\pm 1\) on bipartite lattices. Later we
+class="math inline">\(\pm 1\) on bipartite lattices. Later, we
will show that plaquettes with an odd number of sides (odd plaquettes
-for short) will have eigenvalues \(\pm
+for short) have eigenvalues \(\pm
i\).
+alt="Figure 2: The eigenvalues of a loop or plaquette operators depend on the number of bonds in its enclosing path." />
Figure 2: The eigenvalues of
-a loop or plaquette operators depend on how many bonds in its enclosing
-path.
+a loop or plaquette operators depend on the number of bonds in its
+enclosing path.
Remarkably, all of the spin bond operators \(K_{ij}\) commute with all the Wilson loop
operators \(W_p\). \[[W_p, J_{ij}] = 0\] We can prove this by
-considering the three cases: 1. neither \(i\) nor \(j\) is part of the loop 2. one of \(i\) or \(j\) are part of the loop 3. both are part
of the loop
-
The first case is trivial while the other two require a bit of
-algebra, outlined in fig. 3.
+
The first case is trivial. The other two require some algebra,
+outlined in fig. 3.
-Figure 3:
+style="width:100.0%"
+alt="Figure 3: Plaquette operators are conserved." />
+Figure 3: Plaquette
+operators are conserved.
-
Since the Hamiltonian is just a linear combination of bond operators,
-it also commutes with the plaquette operators! This is great because it
-means that the there’s a simultaneous eigenbasis for the Hamiltonian and
-the plaquette operators. We can thus work in a basis in which the
-eigenvalues of the plaquette operators take on a definite value and for
-all intents and purposes act like classical degrees of freedom. These
-are the extensively many conserved quantities that make the model
+
Since the Hamiltonian is a linear combination of bond operators, it
+commutes with the plaquette operators. This is helpful because it leads
+to a simultaneous eigenbasis for the Hamiltonian and the plaquette
+operators. We can, thus, work in or “on”??? a basis in which
+the eigenvalues of the plaquette operators take on a definite value and,
+for all intents and purposes, act like classical degrees of freedom.
+These are the extensively many conserved quantities that make the model
tractable.
Plaquette operators measure flux. We will find that the ground state
of the model corresponds to some particular choice of flux through each
-plaquette. I will refer to excitations which flip the expectation value
-of a plaqutte operator away from the ground state as
+plaquette. We will refer to excitations which flip the expectation value
+of a plaquette operator away from the ground state as
vortices.
-
Fixing a configuration of the vortices thus partitions the many-body
+
Thus, fixing a configuration of the vortices partitions the many-body
Hilbert space into a set of ‘vortex sectors’ labelled by that particular
flux configuration \(\phi_i = \pm 1,\pm
i\).
From Spins to Majorana
operators
For a single spin
-
Let’s start by considering just one site and its Let us start by considering only one site and its \(\sigma^x, \sigma^y\) and \(\sigma^z\) operators which live in a two
dimensional Hilbert space \(\mathcal{L}\).
We will introduce two fermionic modes \(f\) and \(g\) that satisy the canonical
+class="math inline">\(g\) that satisfy the canonical
anticommutation relations along with their number operators \(n_f = f^\dagger f, n_g = g^\dagger g\) and
the total fermionic parity operator \(F_p =
-(2n_f - 1)(2n_g - 1)\) which we can use to divide their Fock
-space up into even and odd parity subspaces which are separated by the
-addition or removal of one fermion.
-
From these two fermionic modes we can build four Majorana operators:
+(2n_f - 1)(2n_g - 1)\) which can be used to divide their Fock
+space up into even and odd parity subspaces. These subspaces are
+separated by the addition or removal of one fermion.
+
From these two fermionic modes, we can build four Majorana operators:
\[\begin{aligned}
b^x &= f + f^\dagger\\
b^y &= -i(f - f^\dagger)\\
@@ -510,50 +593,42 @@ b^z &= g + g^\dagger\\
c &= -i(g - g^\dagger)
\end{aligned}\]
The Majoranas obey the usual commutation relations, squaring to one
-and anticommuting with eachother. The fermions and Majorana live in a 4
-dimenional Fock space \(\mathcal{\tilde{L}}\). We can therefore
identify the two dimensional space \(\mathcal{M}\) with one of the partity
+class="math inline">\(\mathcal{M}\) with one of the parity
subspaces of \(\mathcal{\tilde{L}}\)
-which we will call the physical subspacephysical subspace \(\mathcal{\tilde{L}}_p\). Kitaev defines the
operator \[D = b^xb^yb^zc\] which can
-be expanded out to \[D = -(2n_f - 1)(2n_g -
-1) = -F_p\] and labels the physical subspace as the space sanned
-by states for which \[ D|\phi\rangle =
+be expanded to \[D = -(2n_f - 1)(2n_g - 1) =
+-F_p\] and labels the physical subspace as the space spanned by
+states for which \[ D|\phi\rangle =
|\phi\rangle\]
We can also think of the physical subspace as whatever is left after
applying the projector \[P = \frac{1 -
-D}{2}\] to it. This formulation will be useful for taking states
-that span the extended space This formulation will be useful for taking states that
+span the extended space \(\mathcal{\tilde{M}}\) and projecting them
into the physical subspace.
So now, with the caveat that we are working in the physical subspace,
-we can define new pauli operators:
+we can define new Pauli operators:
\[\tilde{\sigma}^x = i b^x c,\;
\tilde{\sigma}^y = i b^y c,\; \tilde{\sigma}^y = i b^y c\]
-
These extended space pauli operators satisfy all the usual
-commutation relations, the only difference being that if we evaluate
-\(\sigma^x \sigma^y \sigma^z = i\) we
+
These extended space Pauli operators satisfy all the usual
+commutation relations. The only difference is that if we evaluate \(\sigma^x \sigma^y \sigma^z = i\), we
instead get \[
\tilde{\sigma}^x\tilde{\sigma}^y\tilde{\sigma}^z = iD \]
-
Which indeed makes sense, as long as we promise to confine ourselves
-to the physical subspace \(D = 1\) and
-this all makes sense.
-
-
-
-Figure 4:
-
-
+
This makes sense if we promise to confine ourselves to the physical
+subspace \(D = 1\).
For multiple spins
-
This construction generalises easily to the case of multiple spins:
-we get a set of 4 Majoranas \(b^x_j,\;
+
This construction easily generalises to the case of multiple spins.
+We get a set of 4 Majoranas \(b^x_j,\;
b^y_j,\;b^z_j,\; c_j\) and a \(D_j =
b^x_jb^y_jb^z_jc_j\) operator for every spin. For a state to be
-physical we require that \(D_j |\psi\rangle =
+physical, we require that \(D_j |\psi\rangle =
|\psi\rangle\) for all \(j\).
From these each Pauli operator can be constructed: \(\hat{u}_{ij} = i b^\alpha_i b^\alpha_j\).
Note that these bond operators are not equal to the spin bond operators
\(K_{ij} = \sigma^\alpha_i \sigma^\alpha_j = -
-\hat{u}_{ij} c_i c_j\). In what follows we will work much more
-frequently with the Majorana bond operators so when I refer to bond
-operators without qualification, I am refering to the Majorana
+\hat{u}_{ij} c_i c_j\)
. In what follows, we will work much more
+frequently with the Majorana bond operators. Therefore, when we refer to
+bond operators without qualification, we are referring to the Majorana
variety.
-
Similar to the argument with the spin bond operators \(K_{ij}\) we can quickly verify by
+
Similarly to the argument with the spin bond operators \(K_{ij}\), we can quickly verify by
considering three cases that the Majorana bond operators \(u_{ij}\) all commute with one another. They
-square to one so have eigenvalues \(\pm
-1\) and they also commute with the \(\pm
+1\). They also commute with the \(c_i\) operators.
-
Another important point here is that the operators \(D_i = b^x_i b^y_i b^z_i c_i\) commute with
-\(K_{ij}\) and therefore with Importantly, the operators \(D_i = b^x_i
+b^y_i b^z_i c_i\) commute with \(K_{ij}\) and, therefore, with \(\tilde{H}\). We will show later that the
action of \(D_i\) on a state is to flip
the values of the three \(u_{ij}\)
bonds that connect to site \(i\).
-Physcially this is telling us that \(u_{ij}\) is a gauge field with a high
degree of degeneracy.
-
In summary Majorana bond operators In summary, Majorana bond operators \(u_{ij}\) are an emergent, classical, \(\mathbb{Z_2}\) gauge field!
Partitioning
the Hilbert Space into Bond sectors
-
Similar to the story with the plaquette operators from the spin
-language, we can break the Hilbert space \(\mathcal{L}\) up into sectors labelled by
-the a set of choices \(\{\pm 1\}\) for
-the value of each \(u_{ij}\) operator
-which I denote by \(\mathcal{L}_u\).
-Since \(u_{ij} = -u_{ji}\) we can
-represent the \(u_{ij}\) graphically
-with an arrow that points along each bond in the direction in which
-\(u_{ij} = 1\).
+
Similarly to the story with the plaquette operators from the spin
+language, we can divide the Hilbert space \(\mathcal{L}\) into sectors labelled by a
+set of choices \(\{\pm 1\}\) for the
+value of each \(u_{ij}\) operator which
+we denote by \(\mathcal{L}_u\). Since
+\(u_{ij} = -u_{ji}\), we can represent
+the \(u_{ij}\) graphically with an
+arrow that points along each bond in the direction in which \(u_{ij} = 1\).
Once confined to a particular \(\mathcal{L}_u\), we can ‘remove the hats’
-from the \(\hat{u}_{ij}\) and the
+from the \(\hat{u}_{ij}\). The
hamiltonian becomes a quadratic, free fermion problem \[\tilde{H_u} = \frac{i}{4} \sum_{\langle
-i,j\rangle_\alpha} 2J^{\alpha} u_{ij} c_i c_j\] the ground state
-of which, \(|\psi_u\rangle\) can be
-found easily via matrix diagonalisation. If you have been paying very
-close attention, you may at this point ask whether the \(\mathcal{L}_u\) are confined entirely
+i,j\rangle_\alpha} 2J^{\alpha} u_{ij} c_i c_j\] The ground state,
+\(|\psi_u\rangle\) can be found easily
+via matrix diagonalisation. At this point, we may wonder whether the
+\(\mathcal{L}_u\) are confined entirely
within the physical subspace \(\mathcal{L}_p\) and indeed we will see that
-they are not. However it will be helpful to first develop the theory of
-the Majorana Hamiltonian a little more.
-
+class="math inline">\(\mathcal{L}_p\) and, indeed, we will see
+that they are not. However, it will be helpful to first develop the
+theory of the Majorana Hamiltonian further.
+
-Figure 5:
-(a) The standard Kitaev Model is defined on a honeycomb
+alt="Figure 4: (a) The standard Kitaev model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each (b). We represent the antisymmetric gauge degree of freedom u_{jk} = \pm 1 with arrows that point in the direction u_{jk} = +1 (c). The Majorana transformation can be visualised as breaking each spin into four Majoranas which then pair along the bonds. The pairs of x,y and z Majoranas become part of the classical \mathbb{Z}_2 gauge field u_{ij}. This leavies a single Majorana c_i per site." />
+Figure 4:
+(a) The standard Kitaev model is defined on a honeycomb
lattice. The special feature of the honeycomb lattice that makes the
-model solveable it is that each vertex is joined by exactly three bonds
-i.e the lattice is trivalent. One of three labels is assigned to each
-(b) We represent the antisymmetric gauge degree of
+model solvable is that each vertex is joined by exactly three bonds,
+i.e. the lattice is trivalent. One of three labels is assigned to each
+(b). We represent the antisymmetric gauge degree of
freedom \(u_{jk} = \pm 1\) with arrows
that point in the direction \(u_{jk} =
-+1\)(c) The Majorana transformation can be
++1\) (c). The Majorana transformation can be
visualised as breaking each spin into four Majoranas which then pair
along the bonds. The pairs of x,y and z Majoranas become part of the
classical \(\mathbb{Z}_2\) gauge field
-\(u_{ij}\) leaving just a single
+\(u_{ij}\). This leavies a single
Majorana \(c_i\) per site.
The Majorana Hamiltonian
-
We now have a quadtratic hamiltonian \[
+
We now have a quadratic Hamiltonian \[
\tilde{H} = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha}
u_{ij} c_i c_j\] in which most of the Majorana degrees of freedom
have paired along bonds to become a classical gauge field 6.
-
As a consequence of the the antisymmetry of the matrix with entries
-\(J^{\alpha} u_{ij}\), the eigenvalues
-of the Hamiltonian \(\tilde{H}_u\) come
-in pairs \(\pm \epsilon_m\). This
+
Because of the antisymmetry of the matrix with entries \(J^{\alpha} u_{ij}\), the eigenvalues of the
+Hamiltonian \(\tilde{H}_u\) come in
+pairs \(\pm \epsilon_m\). This
redundant information is a consequence of the doubling of the Hilbert
-space which occured when we transformed to the Majorana
+space which occurred when we transformed to the Majorana
representation.
-
If we pair organise the eigenmodes of \(H\) into pairs such that If we organise the eigenmodes of \(H\) into pairs, such that \(b_m\) and \(b_m'\) have energies \(\epsilon_m\) and \(-\epsilon_m\) we can construct the
+class="math inline">\(-\epsilon_m\), we can construct the
transformation \(Q\)\[(c_1, c_2... c_{2N}) Q = (b_1, b_1', b_2,
b_2' ... b_{N}, b_{N}')\] and put the Hamiltonian into
@@ -679,9 +753,9 @@ the form \[\tilde{H}_u = \frac{i}{2} \sum_m
The determinant of \(Q\) will be
useful later when we consider the projector from \(\mathcal{\tilde{L}}\) to \(\mathcal{L}\) but otherwise the \(b_m\) are just an intermediate step. From
-them we form fermionic operators \[ f_i =
+class="math inline">\(\mathcal{L}\). Otherwise, the \(b_m\) are merely an intermediate step. From
+them, we form fermionic operators \[ f_i =
\tfrac{1}{2} (b_m + ib_m')\] with their associated number
operators \(n_i = f^\dagger_i f_i\).
These let us write the Hamiltonian neatly as
@@ -690,36 +764,36 @@ These let us write the Hamiltonian neatly as
The ground state \(|n_m = 0\rangle\)
of the many body system at fixed \(u\)
is then \[E_{u,0} = -\frac{1}{2}\sum_m
-\epsilon_m \] and we can construct any state from a particular
-choice of \(n_m = 0,1\).
-
In cases where all we care about it the value of \(E_{u,0}\) it is possible to skip forming
+\epsilon_m \]
We can construct any state from a particular choice
+of \(n_m = 0,1\).
+
If we only care about the value of \(E_{u,0}\), it is possible to skip forming
the fermionic operators. The eigenvalues obtained directly from
diagonalising \(J^{\alpha} u_{ij}\)
come in \(\pm \epsilon_m\) pairs. We
can take half the absolute value of the whole set to recover \(\sum_m \epsilon_m\) easily.
-
The Majorana Hamiltonian is quadratic within a Bond
-Sector.
+
Takeaway: the Majorana Hamiltonian is quadratic within a Bond
+Sector.
Mapping
back from Bond Sectors to the Physical Subspace
At this point, given a particular bond configuration \(u_{ij} = \pm 1\) we are able to construct a
+class="math inline">\(u_{ij} = \pm 1\), we can construct a
quadratic Hamiltonian \(\tilde{H}_u\)
in the extended space and diagonalise it to find its ground state \(|\vec{u}, \vec{n} = 0\rangle\). This is not
-necessarily the ground state of the system as a whole, it just the
+necessarily the ground state of the system as a whole, it is just the
lowest energy state within the subspace \(\mathcal{L}_u\)
However, \(|u, n_m =
0\rangle\) does not lie in the physical subspace. As an
-example let’s take the lowest energy state associated with \(u_{ij} = +1\), this state satisfies \(u_{ij} = +1\). This state satisfies \[u_{ij} |\vec{u}=1, \vec{n} = 0\rangle =
|\vec{u}=1, \vec{n} = 0\rangle\] for all bonds \(i,j\).
-
If we act on it this state with one of the gauge operators \(D_j = b_j^x b_j^y b_j^z c_j\) we see that
+
If we act on it, this state with one of the gauge operators \(D_j = b_j^x b_j^y b_j^z c_j\), we see that
\(D_j\) flips the value of the three
bonds \(u_{ij}\) that surround site
\(k\):
Since \(D_j\) commutes with the
-hamiltonian in the extended space \(\tilde{H}\), the fact that \(D_j\) flips the value of bond operators is
-telling us that there is a gauge degeneracy between the ground state of
+class="math inline">\(D_j\) flips the value of bond operators
+indicates that there is a gauge degeneracy between the ground state of
\(\tilde{H}_u\) and the set of \(\tilde{H}_{u'}\) related to it by gauge
-transformations \(D_j\). I.e we can
+transformations \(D_j\). Thus, we can
flip any three bonds around a vertex and the physics will stay the
same.
We can turn this into a symmetrisation procedure by taking a
@@ -745,27 +819,27 @@ superposition of every possible gauge transformation. Every possible
gauge transformation is just every possible subset of \({D_0, D_1 ... D_n}\) which can be neatly
expressed as \[|\phi_w\rangle = \prod_i
-\left( \frac{1 + D_i}{2}\right) |\tilde{\phi}_u\rangle\] this is
-nice because the quantity \(\frac{1 +
+\left( \frac{1 + D_i}{2}\right) |\tilde{\phi}_u\rangle\] This is
+convenient because the quantity \(\frac{1 +
D_i}{2}\) is also the local projector onto the physical subspace.
Here \(|\phi_w\rangle\) is a gauge
invariant state that lives in \(\mathcal{L}_p\) which has been constructed
from a set of states in different \(\mathcal{L}_u\).
-
This gauge degeneracy leads nicely onto the next topic which is how
-to construct a set of gauge invariant quantities out of the \(u_{ij}\), these will turn out to just be
-the plaquette operators.
-
The Bond Sectors overlap with the physical subspace but are
-not contained within it.
+
This gauge degeneracy leads us to the next topic of discussion,
+namely how to construct a set of gauge invariant quantities out of the
+\(u_{ij}\), these will turn out to just
+be the plaquette operators.
+
Takeaway: The Bond Sectors overlap with the physical subspace but are
+not contained within it.
Open boundary conditions
-
Care must be taken in the definition of open boundary conditions.
-Simply removing bonds from the lattice leaves behind unpaired \(b^\alpha\) operators that need to be paired
-in some way to arrive at fermionic modes. In order to fix a pairing we
-always start from a lattice defined on the torus and generate a lattice
-with open boundary conditions by defining the bond coupling Care must be taken when defining open boundary conditions. Simply
+removing bonds from the lattice leaves behind unpaired \(b^\alpha\) operators that must be paired in
+some way to arrive at fermionic modes. To fix a pairing, we always start
+from a lattice defined on the torus and generate a lattice with open
+boundary conditions by defining the bond coupling \(J^{\alpha}_{ij} = 0\) for sites joined by
bonds \((i,j)\) that we want to remove.
This creates fermionic zero modes
Alternatively, since all the fermionic zero modes are degenerate
anyway, an arbitrary pairing of the unpaired \(b^\alpha\) operators could be performed.
-Is is possible that a lattice constructed and coloured like this
-would have unequal numbers of \(b^x\)
-\(b^y\) and \(b^z\) operators?
Blaizot, J.-P. & Ripka, G. Quantum
theory of finite systems. (The MIT Press, 1986).
-
-
-
-
A bipartite lattice is composed of A
-and B sublattices with no intra-sublattice edges i.e no A-A or B-B
-edges. Any closed loop must begin and at the same site, let’s say it’s
-an A site. The loop must go A-B-A-B… until it returns to the original
-site and must therefore must contain an even number of edges in order to
-end on the same sublattice that it started on.↩︎
-
-
-
-
-
-
-
+
+alt="Figure 2: The eigenvalues of a loop or plaquette operators depend on the number of bonds in its enclosing path." />
-
-