# Quantum spin liquids

Tom Lancastera CONTACT Tom Lancaster. Email<EMAIL_ADDRESS>aDepartment of Physics, Durham University, South Road, Durham, DH1 3LE, UK

###### Abstract

A glance at recent research on magnetism turns up a curious set of articles
discussing, or claiming evidence for, a state of matter called a quantum spin
liquid (QSL). These articles are notable in their invocation of exotic notions
of topological physics, quantum entanglement, fractional quantum numbers,
anyon statistics and gauge field theories. So what is a QSL and why do we need
this complicated technical vocabulary to describe it? Our aim in this article
is to introduce some of these concepts and provide a discussion of what a QSL
is, where it might occur in Nature and why it is of interest. As we’ll see,
this is a rich subject which is still in development, and unambiguous evidence
for the realisation of the QSL state in a magnetic material remains hotly
debated. However, the payoff in terms of the special nature of quantum
entanglement in the QSL, and its diverse spectrum of unusual excitations and
topological status will (at least to some extent) justify the need to engage
with some powerful, occasionally abstract, technical material.

###### keywords:

quantum spin liquid, gauge theory, topology, spinon

††articletype: Review

##  1 Introduction

We start with an attempt at a working definition of a quantum spin liquid
(QSL) taken from Ref. [1]: a quantum spin liquid ground state is an electronic
insulator with spin-rotation symmetry and an odd number of electrons per unit
cell. Although there’s quite a lot going on here, one key feature is that we
have an interacting electronic system comprising localised electrons whose
spin moments do not align into a magnetically-ordered configuration, even down
to $T=0$. Although this notion of a lack of spin order will provide a jumping-
off point, we’ll see that the picture of a disordered magnetic state fails to
capture much of the rich physics that underlies the QSL. However, the general
principle of order will underly our discussion. Specifically, we shall start
from Lev Landau’s description of the order that follows from a symmetry-
breaking phase transition [2], and the subsequently development of this
subject by Philip Anderson [3], since these concepts have provided a means of
classifying and understanding ordered phases of condensed matter.

The discovery and elucidation of the fractional quantum Hall (FQH) fluid
fundamentally challenged this pervasive notion of order [1, 4]. The FQH fluid
is a state of matter realised in a two-dimensional electron system in high
magnetic fields [5]. The fluid has a gap in energy between the ground state
and the lowest-lying excited states of the system.111The existence or absence
of such energy gaps will turn out to be a useful notion in classifying states
of matter. If we put enough energy into the system to excite excitations over
the energy gap, the resulting particle-like states do not carry integer
numbers of elementary electronic charges, but rather fractions of an electron
charge. We say that they are fractionalised. Moreover, they are neither bosons
nor fermions, but are known as anyons, and have their own, unique, quantum
statistics. There is not one FQH fluid, but many, each realised in practice by
applying a magnetic field of a different magnitude. Moreover, we cannot
distinguish or classify the different FQH states by their symmetries, as we
can with other ordered states of matter. Instead, topology, the study of the
shapes of objects and spaces, is invoked, and we treat the FQH fluid as being
topologically ordered (TO). Relative to conventionally-ordered states, TO
states are typified by (i) a ground-state degeneracy that depends on the
topology of the system, (ii) a gap against excitations and (iii) a high degree
of quantum entanglement.

Here the concept of a QSL fits into condensed matter. Like the FQH fluids,
there is not one QSL state, but many that can’t be distinguished by symmetry.
All QSLs have fractionalised excitations. Some QSLs show topological order and
have gapped excitations; some do not. QSLs feature an unusually-high degree of
(long-range) quantum entanglement in their ground states. We will take the
view that it is the nature of the quantum entanglement that characterises the
QSL and explains its properties, rather than its magnetic disorder. Following
X.-G. Wen [1], we call the special entanglement quantum order as a
shorthand,222The entanglement partially explains the Q in QSL. The other
feature is that we are dealing with $T=0$, where fluctuations are purely
quantum mechanical rather than thermal. In contrast, a classical spin liquid
has $T=0$ order destroyed by $T>0$ thermal fluctuations owing to high ground-
state degeneracy. because the unique pattern of entanglements characterises
each QSL. Topological order is then a special case of quantum order in which
all excitations have finite energy gaps between the ground state and the
excited states.333To close the loop on our technical definition above, with
its specification of odd numbers of electrons, we note that the Lieb-Schultz-
Mattis theorem says that a system with a half-odd integer spin per unit cell,
and without symmetry breaking order, has either (i) a unique ground state with
gapless excitations (e.g. a gapless QSL) or (ii) a degenerate ground state
with a gap (e.g. a topologically ordered QSL).

The paper is structured as follows: we start with an introduction to the
treatment of order and its absence in magnets, followed in Section 3 by a
survey of the conceptual ingredients required to understand this field. Two of
these: anyons and gauge theory are introduced in more detail in Sections 4 and
5. We then discuss the toric code, which is the simplest model of a QSL.
Fractionalised spinons are described in Section 7 as a route to finding more
model QSLs, before the exactly solvable Kitaev model is introduced in Section
8. Finally we briefly assess some examples in the search for a realisation of
the QSL state.

##  2 Background: magnetic order, disorder and the RVB

Figure 1: (a) A disordered (paramagnetic) magnetic state. This configuration
has magnetisation $M=0$ and so does (b), which is produced by turning all of
the spins through 180∘. (c) An ordered (ferro)magnetic phase with $M=M_{0}$.
Turning the spins through 180∘ results in (d) which has $M=-M_{0}.$ (e) The
Néel antiferromagnet, with ordered, alternating spins.

A simple picture of a magnetically disordered system is shown in Fig. 1(a).
The arrows represent spins arranged on an ordered lattice. (This structural
order is always assumed, the disorder refers only to the arrangement of
magnetic moment directions.) The system can be understood [6, 3, 2] via its
symmetry: the magnetisation $M$ (that is, the average magnetic moment) is
zero, since as many arrows point up as down. If we turn each of the moments
through 180∘ then we obtain the situation shown in Fig. 1(b), which also has
$M=0$. We therefore cannot tell from the magnetisation that we have
transformed the system. This inability to tell that a change has been made is
a symmetry. An ordered magnet is shown in Fig. 1(c). This has magnetisation
$M_{0}\neq 0$ and has lost its previous symmetry: turning the arrows through
180∘ [Fig. 1(d)] makes a measurable difference in that it reverses the
magnetisation $M_{0}\rightarrow-M_{0}$. The symmetry has been broken on
magnetic ordering. In Nature, this sort of ordering is observed to take place
via a magnetic phase transition at a temperature $T_{\mathrm{c}}$.

Philip Anderson notes some key points here [3] that apply in general to
broken-symmetry states. (i) Once the rotational symmetry of the spin system is
broken, the ordered magnetic state becomes rigid, such that it costs energy to
deform the spin structure. (ii) A new sort of excitation emerges on symmetry
breaking: the magnon. This can be thought of as a single flipped spin, smeared
out to give spin $s=1$ particle-like excitation. Owing to the possibility of
exciting magnons with arbitrarily long wavelength, for a rotationally-
invariant Hamiltonian it costs a vanishing amount of energy to create one, and
so, formally, we say that the excitation is gapless. One way to think about
the gaplessness is that it is “protected” by symmetry and the ordering.

Allowing spins now to point in any direction, a magnet has a microscopic
description using the Heisenberg model with Hamiltonian [6]
$\hat{H}=-\sum_{\langle
ij\rangle}J_{ij}\hat{\boldsymbol{S}}_{i}\cdot\hat{\boldsymbol{S}}.$ We will
consider only $S=1/2$ spin operators $\hat{\boldsymbol{S}}$ and the sum will
be taken only over nearest-neighbour spins, denoted $\langle ij\rangle$. The
ferromagnet (FM) [Fig. 1(c)] is found for constant exchange $J>0$, but more
common than the ferromagnet is the antiferromagnet (AFM) ($J<0$), where
magnetic sublattices break symmetry, with the spins arranged into an
alternating Néel state [Fig. 1(e)].

Antiferromagnets have several differences compared to ferromagnets, especially
when quantum mechanics is taken into account [6, 3]. The crux is that the Néel
state is not an eigenstate of the Heisenberg Hamiltonian. It’s useful at this
stage to take a brief step back and consider the quantum-mechanical fate of a
model two-level system such as the benzene molecule [7]. Benzene has two
energetically-degenerate configurations of alternating double and single
covalent bonds, as shown in Fig. 2(a) (states we shall call $\phi_{1}$ and
$\phi_{2}$). Neither is an eigenstate of the system, and there is a matrix
element for a transition between them. It is straightforward to show [7] that
the ground state is the symmetric superposition of the states
[($\phi_{1}+\phi_{2}$)/$\sqrt{2}$], and the antisymmetric combination
[($\phi_{1}-\phi_{2}$)/$\sqrt{2}$] is an excited state. The ground state
superposition gives rise to the notion of a delocalised $\pi$ orbital [Fig
2(a, right)] that represents the system fluctuating, or resonating, between
the two alternating-bond configurations. Specifically, if we start the system
in state $\phi_{1}$, then the probability of finding the system in this same
state some time $t$ later oscillates as a function of $t$, at a frequency
$\omega=2V/\hbar$, determined by the energy gap $2V$ between the eigenstates.
These fluctuations have nothing to do with temperature, but are purely
quantum-mechanical in nature. The point here is that since the initial state
$\phi_{1}$ is not an eigenstate of the system, probability sloshes around at a
rate determined by the energy-level separation. Clearly, if $V$ is small, the
resulting small energy-level separation means that the oscillations have
correspondingly low frequency. If it approaches zero, the state $\phi_{1}$
becomes stable, even though it is not an eigenstate.

Figure 2: (a) Benzene has bonds that we picture in the $\phi_{1}$ (left) or
$\phi_{2}$ (middle) states. The superposition of these leads to the $\pi$
orbital picture (right). (b) A triangular arrangement of three spins with
exchange interactions between them. (c,d) Two possible RVB configurations of
singlet bonds linking spins on the triangular lattice.

Since, like $\phi_{1,2}$, the Néel state is not an eigenstate of the AFM
Hamiltonian, it was suggested that quantum fluctuations (generalising the
sloshing of the time-dependence of the probability density in the previous
example) would break up an antiferromagnetically-ordered configuration of
spins. However, the Néel state is stable. This is because, for the large
systems that comprise condensed matter, there are lots of eigenstates
arbitrarily close in energy out of which a stable wavepacket can be
constructed. The closeness of eigenstates in energy means that the oscillation
frequency is therefore very small (or zero in an infinite system),
corresponding to a small (or zero) $V$ in the benzene example. One way to
think about this feature in an extended system is by analogy with picking up a
rock, where we might suspect that position-momentum uncertainty would prevent
a solid object from being localised [3, 8]. The reason why the rock is stable
is that we seek to localise it within, say, a lattice spacing $a$, so the
uncertainty in momentum is of order $\hbar/a$ and the uncertainty in energy
for an atom of mass $m$ is then $\Delta E=\hbar^{2}/2ma^{2}$. However, the key
is that the solid is made of $N$ atoms, rigidly stuck together, so $\Delta
E=\hbar^{2}/2Nma^{2}$ and, since $N$ is a macroscopic number, the energy
uncertainty is very small. This implies that the frequency of oscillation
between closely-spaced position states is vanishingly small for a macroscopic
piece of matter. The analogous argument can be made in the case of an
antiferromagnet (given in the slightly more complicated terms of a rigid
rotor), and it turns out that the closely-spaced energy levels in a
macroscopic system similarly allow a very stable wavepacket to be constructed
giving a stable Néel AFM [8, 9]. The smoking-gun experimental verification of
its existence came from magnetic neutron diffraction where the periodicity of
the magnetic sublattices can be directly seen from the measured Bragg peaks
[6].

Although the AFM survives the strictures of quantum mechanics, it’s possible
to postulate a state that shouldn’t. 1973 saw Philip Anderson’s suggestion of
a resonating valence bond (RVB) state [10]. This is based on the notion of
frustration, where it is impossible to satisfy all of the interactions for a
particular geometry.
Example: consider a triangle decorated with Ising spins as shown in Fig. 2(b)
with Hamiltonian $H=-\sum J_{ij}\sigma^{z}_{i}\sigma^{z}_{j}$, where all bonds
favour antiferromagnetism ($J_{ij}=-J<0$). This state is frustrated as there
is no single state that satisfies all of the interactions. In this case the
frustration gives rise to six lowest-energy spin configurations (with energy
$E=-J$) and two excited states (with $E=3J$). By contrast, an unfrustrated
model where all $J_{ij}=J$ has a ferromagnetic, two-fold degenerate ground
state with energy $E=-3J$. (The degeneracy here simply reflects the fact that
all spins can align either up or down.) We see how frustration increases not
only the relative energy of the ground state but, more crucially, the entropy
through the large number of degenerate ground states that result. The
degeneracy becomes macroscopic in the thermodynamic limit and has a
destabilising effect on ordering.

To form Anderson’s RVB state [10, 11, 12, 13] we extend the triangle to a
spin-1/2 Heisenberg model on an infinite two-dimensional triangular lattice
with purely antiferromagnetic interactions on all bonds. We also upgrade from
an Ising spin model to a model of Heisenberg spins.444Reminder: two Heisenberg
spins $|s_{1}s_{2}\rangle$ with isotropic antiferromagnetic exchange coupling
between them will have a singlet ground state:
$\left(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle\right)/\sqrt{2}$,
and a triplet of excited states, comprising $|\uparrow\uparrow\rangle$,
$|\downarrow\downarrow\rangle$ and
$\left(|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle\right)/\sqrt{2}$.
Note first that, quantum mechanically, the ground state of a single
antiferromagnetic bond is represented by an entangled spin singlet
$\left(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle\right)/\sqrt{2}$.
We therefore form a typical state by decorating the lattice with singlets
coupling nearest neighbours [Fig 2(c)]. We then repeat with a different choice
of singlet bonds to form another state with the same energy [Fig 2(d)]. We
continue this process until we have all possible coverings, and then form our
final trial wavefunction by adding together all of these states to form the
RVB state. Notice how the RVB bears a resemblance to the benzene example,
albeit in an infinite system: in both we can think of the choice of bonds
resonating between different degenerate configurations. Ultimately, the RVB
state did not turn out to describe a $s=1/2$ Heisenberg antiferromagnet on a
triangular lattice.555It has, however, since been invoked in several contexts,
not least to describe high $T_{\mathrm{c}}$ cuprate superconductors. However,
the RVB is our first example of a QSL. It does not break rotational symmetry
since each bond represents a $s=0$ spin state. Moreover, the state itself can
be thought of in terms of a complicated dynamic dance of singlet bonds that
represents the underlying pattern of quantum entanglement.

The notion of a dynamic pattern of quantum correlations that underlies a
magnetically-disordered state of macroscopic matter will now be expanded via a
consideration of some of the background concepts needed to understand the QSL
picture.

##  3 Ingredients of a QSL

There are a number of ingredients of QSLs that we describe here and will
feature in the rest of the article. The first is reduced dimensionality. When
interactions are confined to less than three spatial dimensions (3D),
fluctuations become more effective in destabilising order. In fact, the
Mermin-Wagner theorem [6, 4] says that for spins with a continuous degree of
freedom, magnetic order will always be destabilised above $T=0$. We shall
mainly be concerned in this review with the case of two spatial dimensions
(2D) and time [known as (2+1) dimensions]. Although the two-dimensional
Heisenberg magnet will not order for $T>0$, the ferromagnetic certainly does
at $T=0$, and there is good evidence that the antiferromagnet should order at
$T=0$ as well [14]. We therefore need some further source of fluctuations if
we want to prevent long-range magnetic order and instead promote the quantum
order that gives robust (i.e. $T=0$) spin-liquid behaviour. Frustration, as
considered above for the triangle, is one possibility, but there are others,
including higher-order interactions or disorder. (There are also QSL states
that can be realised in 3D, which we return to at the end of the article.)

The second ingredient is topology [4, 1]. Topology enters many-body physics in
a number of ways. One is in theories: some theories make no contact with the
distances and time intervals in a problem (i.e. the geometric features) and
instead rely only on the underlying shape of their space. These are
topological theories: an example is the Chern-Simons theory used to describe
the FQH fluid, along with some QSL states. A consequence of this lack of
geometry turns out to be that the Hamiltonian of a purely topological theory
(such as a the Chern-Simons Hamiltonian) is $\hat{H}=0$, meaning all states
are degenerate with zero energy. The exact size of the large resulting
degeneracy depends on the topology of the underlying space (or manifold) on
which we’re working. The topology of the manifold is an interesting case in
itself. If we impose periodic boundaries on our 2D lattice in both spatial
directions then the shape of the underlying topological space is a torus. This
topology leads to a robust degeneracy: it is very difficult to split the
energies of the degenerate states with perturbations. Topological order (where
there is a nonzero energy gap between the ground state and all of the excited
states) follows from the existence of these robust topological degenerate
ground states, and is sometimes described in terms of topology “protecting”
the gaps that appear in topologically-ordered states. A final appearance of
topology is made by excitations [4, 15]. Some excitations are extended in
space and cost a large amount of energy to remove. Topological excitations,
often called topological defects, include the domain wall in the one-
dimensional (1D) magnet, the vortex in 2D and the monopole in 3D. In each
case, the large energetic penalty involved in removing a topological
excitation can be traced back to their shape (e.g. for the domain wall, we
would need to flip a semi-infinite number of spins to remove the wall).

Entanglement is our next ingredient. A state is entangled if it is in a
quantum superposition that cannot be written as a product, even under an
arbitrary (local) change of basis states. The key example to have in mind here
are two spins in an entangled singlet state
$\left(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle\right)/\sqrt{2}$.
As we’ve said, quantum order describes the pattern of entanglements in a many-
body system, although it’s rather hard to rigorously define a many-particle
entangled state. We saw one example in the RVB of resonating singlets, we’ll
see another in our discussion of the toric code model.

The number of different elementary excitations666The low-energy excited states
that exist close to the ground state of an interacting, many-body system can
be thought of as hosting an assembly of elementary excitations. Examples of
elementary excitations are phonons in a crystal, magnons in a ferromagnet,
quasielectrons in a metal etc. of the QSL are an important means of
classifying the states, especially whether they are gapped or gapless (i.e.
whether or not there is a nonzero energy gap between the ground state and
first excited state). QSLs can support local particle-like excitations that
can be fermionic, bosonic or anyonic, and can have fractional quantum numbers.
For example, many QSLs support gapless, chargeless $s=1/2$ fermionic
excitations known as spinons [16]. In constrast, excitations can also be
topological objects. An example of a (bosonic) excitation found in many QSLs
is the topological vortex.

In order to discuss spin liquids we shall also need some ideas from quantum
field theory (QFT) [17, 4]. In a QFT we describe a system in terms of a field
(an object in which we input a position and output an amplitude or quantum
operator). Particle-like excitations in QFT are quantised excitations in the
field. Operators in QFT act on many-particle states that describe the system.
The most important state is the ground state or vacuum $|0\rangle$. Excited
states, such as particle excitations, are then added to this vacuum state,
using operators like $\hat{c}^{\dagger}$. For example, to add a $c$-particle
to the vacuum we write $\hat{c}_{i}^{\dagger}|0\rangle=|1_{i}\rangle$, where
$|1_{i}\rangle$ is a state containing one $c$-particle at position $i$. (We
can also annihilate particles: $\hat{c}_{i}|1_{i}\rangle=|0\rangle$.) One
important model to have in mind in our discussion is the tight-binding model,
describing the energies of electrons on a lattice with a Hamiltonian

$\hat{H}=\sum_{ij}(-t_{ij})\hat{c}^{\dagger}_{i}\hat{c}_{j},$ (1)

where the subscripts label lattice sites. The idea here is that the operators
annihilate the electron at site $j$ and create it at site $i$, causing an
electron to hop between sites, making a contribution of kinetic energy of
$(-t_{ij})$. This model can often be solved in momentum space to give
$\hat{H}=\sum_{\boldsymbol{k}}E(\boldsymbol{k})\hat{c}^{\dagger}_{\boldsymbol{k}}\hat{c}_{\boldsymbol{k}}$,
where $E(\boldsymbol{k})$ is the electron dispersion. We’ll see that some
important models of QSLs can be boiled down to variations of this basic
picture.

The final, crucial, ingredient is gauge field theory [4]. A gauge field is one
where there is some redundancy of description. This is to say that the same
physical state can be described by several different configurations of a gauge
field. (You can think of this as a little like a language in which a
physically-equivalent scene can be described in English, French, Arabic,
Hindi, Japanese etc.) The most familiar gauge field is the electromagnetic
gauge field, whose components are
$A^{\mu}=(V,A^{x},A^{y},A^{z})=(V,\boldsymbol{A})$. This is often also called
the electromagnetic potential. We define the electric field $\boldsymbol{E}$
in terms of the components of the gauge field as
$\boldsymbol{E}=-\boldsymbol{\nabla}V-\frac{\partial\boldsymbol{A}}{\partial
t}$, and the magnetic field $\boldsymbol{B}$ as
$\boldsymbol{B}=\boldsymbol{\nabla}\times\boldsymbol{A}.$ An important
property of these equations is that there is some freedom in how the
components $A_{\mu}$ are specified. In fact, if we make the gauge
transformations

$\begin{array}[]{cc}V\rightarrow
V-\frac{\partial\chi(t,\boldsymbol{x})}{\partial
t},&\boldsymbol{A}\rightarrow\boldsymbol{A}+\boldsymbol{\nabla}\chi\end{array}$
(2)

where $\chi(t,\boldsymbol{x})$ is some arbitrary function, then the values for
the $\boldsymbol{E}$ and $\boldsymbol{B}$ fields are unchanged.

The notions of anyons and of a gauge theory are so important that we’ll devote
some more discussion to them in the following sections.

##  4 Anyons and topology

One of the most striking properties of QSLs is that they can host excitations
that are neither conventional fermions nor bosons [4]. In quantum mechanics in
3D, changing the labels on two identical particles results in a change of the
wavefunction according to the rule $\psi(x_{1},x_{2})=\pm\psi(x_{2},x_{1})$,
where the $+$ sign applies to bosons and the $-$ sign to fermions. For the
case of 2D space, the exchange of particles needs more careful attention. We
start with two identical particles at positions $x_{1}$ and $x_{2}$ and
identify two distinct ways of exchanging them. The result of processes of type
A is to move $x_{1}\rightarrow x_{1}$ and $x_{2}\rightarrow x_{2}$. Some
examples are shown in Fig. 3(a) and (b). The particles end up where they were
originally, although they may move around each other. The result of type-B
processes is to move $x_{1}\rightarrow x_{2}$ and $x_{2}\rightarrow x_{1}$
[Fig. 3(c) and (d)]. Again particles may move around each other several times
before settling at their final positions [as in Fig. 3(d)].

We ask what the relative quantum mechanical phase difference is between
processes of type A and type B. The key parameter is the angle that one
particle is moved around the other. Topology comes in here: given a set of
particle paths, we can smoothly distort the paths of the particles, but we may
not change the number of times particles wrap around each other without
introducing singularities in the paths. Processes of type A involve rotating
particle 2 around particle 1 by angle $\phi=2\pi p$, where the winding number
$p$ takes an integer value (including zero). Processes of type B involve
rotations of $\phi=\pi(2p+1)$. Each value of $p$ describes a topologically
distinct process. We suppose that these topologically distinct processes make
a multiplicative contribution to the wavefunction of $\Phi(\phi)$ which is
pure phase. If we carry out a sequential string of these processes then we
require that the angles add, whilst the wavefunctions should multiply. That
is, $\Phi(\phi_{1}+\phi_{2})=\Phi(\phi_{1})\Phi(\phi_{2})$, which implies that
$\Phi(\phi)={\rm e}^{\mathrm{i}\eta\phi}$, where $\eta$ is a parameter which,
crucially, doesn’t need to be an integer.

Figure 3:  Examples of ways of exchanging particles. (a) and (b) are type-A
processes, where the particles end up at the same positions. In case (b) one
particle loops around the other once in the exchanging process. (c) and (d)
are type-B processes where the particles exchange positions. In case (d) one
particle loops once around the other during the exchange. [Figure reproduced
from Ref. [4], reprinted with permission of Oxford University Press.]

Compare our upgraded 2D exchange to the old-fashioned definition. If we carry
out the exchange $(x_{1},x_{2})\rightarrow(x_{2},x_{1})$ [shown in Fig. 3(c)]
then the formal definition of exchange tells us that the wavefunction should
be identical (for bosons) or pick up a minus sign (fermions). However, the new
version merely tells us that $\phi=\pi$, resulting in a phase factor of
$\Phi(\phi)={\rm e}^{\mathrm{i}\eta\pi}$. The two versions of exchange are
only identical for the special cases that (a) we have $\eta=\mbox{even
integer}$, when we recover the expected exchange behaviour for bosons or (b)
we have $\eta=\mbox{odd integer}$, when we recover fermion exchange. However,
this analysis shows that there are many more possible values of $\eta$ in 2D,
since it doesn’t have to be an integer. We are therefore not tied simply to
bosons and fermions, the freedom to choose $\eta$ means we can have any
exchange statistics, and particles with such statistics are called (Abelian)
anyons.777Having only two spatial dimensions is vital to the argument. In 3D
all type-A processes are topologically identical since they are all deformable
into paths where the particles don’t move. Similarly all type-B processes are
topologically identical and may be reduced to a simple exchange of particles.
This reduction occurs because the extra dimension allows us to move the paths
past each other in the third dimension, shrinking all loops to zero.

##  5 Gauge theory

QSLs are often named with a mathematical group (e.g. a $Z_{2}$ spin liquid, or
$U(1)$ spin liquid). This is related to the properties of the QSL’s gauge
structure, considered in this section through a number of examples.

Example 1: Consider complex scalar field theory, which is a model built from
continuous complex-number valued fields $\psi(x)$ that are described by a
Hamiltonian

$\hat{H}=(\partial_{x}\psi)^{\dagger}(\partial_{x}\psi)+m^{2}\psi^{\dagger}\psi,$
(3)

where $m$ is a constant. This theory has an internal symmetry known as a
global phase symmetry, or global $U(1)$ symmetry. This is a shorthand for the
observation that with the replacement $\psi(x)\rightarrow\psi(x){\rm
e}^{\mathrm{i}\alpha}$, the Hamiltonian $H$ does not change.888This
transformation generates the elements of the Lie group $U(1)$, hence the name.
This is a global transformation in that the phase changes by the same amount
($\alpha$) at every point.

What if we attempt to change the phase locally, i.e. differently at each
point? That is, change the phase by an amount $\alpha(x)$ that depends in some
arbitrary manner on position. Under the local transformation, the derivatives
change as follows:

$\displaystyle\partial_{x}\psi(x)\rightarrow\partial_{x}\left[\psi(x){\rm
e}^{\mathrm{i}\alpha(x)}\right]=$ $\displaystyle{\rm
e}^{\mathrm{i}\alpha(x)}\partial_{x}\psi(x)+\psi(x){\rm
e}^{\mathrm{i}\alpha(x)}\mathrm{i}\partial_{x}\alpha(x)$ $\displaystyle=$
$\displaystyle{\rm
e}^{\mathrm{i}\alpha(x)}\left\\{\partial_{x}+\mathrm{i}\left[\partial_{x}\alpha(x)\right]\right\\}\psi(x).$
(4)

Similarly $\partial_{x}\psi^{\dagger}(x)\rightarrow{\rm
e}^{-\mathrm{i}\alpha(x)}\left\\{\partial_{x}-\mathrm{i}\left[\partial_{x}\alpha(x)\right]\right\\}\psi^{\dagger}(x)$,
and the first term in the Hamiltonian becomes

$(\partial_{x}\psi)^{\dagger}(\partial_{x}\psi)-\mathrm{i}\psi^{\dagger}(\partial_{x}\alpha)(\partial_{x}\psi)+\mathrm{i}\psi(\partial_{x}\psi^{\dagger})(\partial_{x}\alpha)+\psi^{\dagger}\psi(\partial_{x}\alpha)(\partial_{x}\alpha),$
(5)

which is not what we started with. Perhaps unsurprisingly, the theory is not
therefore invariant with respect to local phase transformations. However, to
fix things we introduce a new gauge field whose job is to cancel out the
effect of the change in internal variable $\alpha(x)$ with position. This
enters into the Hamiltonian as a covariant field derivative $D_{x}$, defined
by $D_{x}\psi(x)=\partial_{x}\psi(x)-\mathrm{i}qA^{x}(x)\psi(x)$, where $q$ is
a coupling constant. We define the gauge field such that if we change the
phase by angle $\alpha(x)$, the components of the gauge field transform
according to $A^{x}(x)\rightarrow A^{x}(x)+\frac{1}{q}\partial_{x}\alpha(x)$.
Now if $\psi(x)\to\psi(x){\rm e}^{{\rm i}\alpha(x)}$, then

$\displaystyle D_{x}\psi=(\partial_{x}-\mathrm{i}qA^{x})\psi\to{\rm
e}^{\mathrm{i}\alpha}(\partial_{x}\psi)+\mathrm{i}\psi{\rm e}^{{\rm
i}\alpha}(\partial_{x}\alpha)-\mathrm{i}qA^{x}\psi{\rm
e}^{\mathrm{i}\alpha}-\mathrm{i}\psi{\rm e}^{{\rm
i}\alpha}(\partial_{x}\alpha)={\rm e}^{\mathrm{i}\alpha}D_{x}\psi.$ (6)

This property makes the whole Hamiltonian invariant under local phase changes
if we replace ordinary derivatives by covariant ones:
$\hat{H}=(D^{x}\psi)^{\dagger}(D_{x}\psi)+m^{2}\psi^{\dagger}\psi,$ since now
with $D_{x}\psi\rightarrow{\rm e}^{\mathrm{i}\alpha}D_{x}\psi$, the first term
is invariant.

The message of these manipulations is that if the phase $\alpha(x)$ is a
function of $x$ then, in order to guarantee local phase invariance, a new
gauge field $A^{x}(x)$ is required to form the covariant field derivative.
Furthermore, we recognise this field as akin to the electromagnetic gauge
field, where we saw that one of its features was that, without changing
$\boldsymbol{E}$ and $\boldsymbol{B}$, the gauge field could be changed by an
arbitrary amount $A^{x}(x)\rightarrow A^{x}(x)+\partial_{x}\chi(x)$. If we
identify $\chi(x)$ with $\alpha(x)/q$, then the transformation demanded by our
argument is simply electromagnetic gauge invariance, confirming that $A^{x}$
has the usual properties of a gauge field as defined in electromagnetism. We
say that the gauge field has its own dynamics, which in electromagnetism are
governed by the Maxwell equations, written in terms of the (3+1)-dimensional
field $A^{\mu}$.

Example 2: In Section 7 we will introduce a model of spinon particles on a
lattice represented in terms of the operator $\hat{f}^{\dagger}_{i}$, which
creates a spinon at site $i$. This is similar to the continuous model in
Example 1, except that position is now a discrete variable. We again demand
invariance under a local $U(1)$ phase transformation
$\hat{f}_{i}\rightarrow\hat{f}_{i}{\rm e}^{\mathrm{i}\phi_{i}}$, which changes
the phase at each lattice point $i$. To ensure this we again introduce a gauge
field, this time $\bar{\chi}_{ij}{\rm e}^{\mathrm{i}\theta_{ij}}$, where
$\theta_{ij}$ is a phase that depends on the bond between sites $i$ and $j$.
The spinons and the gauge field interact according to a Hamiltonian
$\hat{H}=\hat{f}^{\dagger}_{i}\hat{f}_{j}\left(\bar{\chi}_{ji}{\rm
e}^{\mathrm{i}\theta_{ij}}\right)$, so if the gauge field transforms according
to $\theta_{ij}\rightarrow\theta_{ij}+\phi_{i}-\phi_{j}$, then the Hamiltonian
is symmetric under local $U(1)$ transformations of the spinon operators. We
will see later that this gives us a $U(1)$ gauge theory of spinons.

Example 3: $Z_{2}$ (more often written $\mathbb{Z}_{2}$ in the mathematics
literature) is a group with two elements: $1$ and $-1$. For the Kitaev model
on a lattice (Section 8), we identify an operator $\hat{c}_{i}$ and demand
invariance under a local transformation $\hat{c}_{i}\rightarrow
W_{i}\hat{c}_{i}$, where $W_{i}$ is an arbitrary function that outputs 1 or
$-1$ at the $i$th site. We identify a gauge field $\hat{u}_{ij}$ that depends
on two positions $i$ and $j$, and a Hamiltonian
$\hat{H}=(\hat{u}_{jk})\hat{c}_{j}\hat{c}_{k}$. If $\hat{u}_{ij}$ transforms
according to $\hat{u}_{ij}\rightarrow W_{i}\hat{u}_{ij}W^{-1}_{j}$, then this
allows the theory to be locally $Z_{2}$ invariant.

Example 4: Our discussion in the next section relies on another $Z_{2}$ gauge
field $S_{ij}$, where positions are specified by two lattice sites $i$ and
$j$. Here we will again transform via the local transformation
$S_{ij}\rightarrow W_{i}S_{ij}W^{-1}_{j}$ in the same way as Example 3. This
transformation leaves the Hamiltonian invariant, as we shall now discuss.

##  6 Our second spin liquid: the toric code

There is a class of models that allows us to get a handle on some of the
ingredients we’ve met so far: we start with the toric code and then (Section
6.2) consider a simplified version of this model. We shall see how these
models give rise to spin-liquid ground states with topological ground-state
degeneracy, anyonic excitations and a gauge-field structure based on the group
$Z_{2}$.

Figure 4: (a) A torus with three different sorts of paths on its surface.
[Figure reproduced from Ref. [4], reprinted with permission of Oxford
University Press.] (b) Square lattice in which spins lie in the spaces between
lattice points. Example spin positions are shown in green in the upper left.
An example plaquette $p$ and star $s$ are also shown. (c) Flipping a connected
string of spins aligned along $\pm x$ (red dots) excites two $e$ anyon (or
star) excitations at the ends of the string. (d) Flipping spins aligned along
$\pm z$ along a string perpendicular to the bonds (blue dots) creates two $m$
anyon (or plaquette) excitations. (e) Top: the action of the star operator on
a path through the lattice flips spins (purple), giving a loop of flipped
spins on the dual lattice (green). Bottom: a line of flipped spins (blue)
makes an $A$-type path on the torus. (f) Configuration of bonds and plaquettes
that results from applying periodic boundary conditions to a lattice of 4
points. Arabic numerals label bonds, roman numerals label plaquettes.

The toric code [20, 11] describes spins on a 2D lattice with periodic boundary
conditions. The boundary conditions are a key feature: by identifying each of
the two perpendicular directions on the lattice, we obtain a space with the
topology of a torus (hence the name of the model). An example torus is shown
in Fig. 4(a). This surface has two holes: one through the empty middle of the
doughnut, and one through the tubular part (filled with dough in the edible
version). The important thing to note is that there are three main sorts of
paths on the torus [see Fig. 4(a)]: $A$-type and $B$-type paths wrap the two
different holes that characterise the torus, while $C$-type paths wrap neither
of the holes. It is impossible to smoothly deform one type of path into a
different one (that is, they are topologically distinct: we would need to cut
the paths and glue them back together to do this).

The lattice in the toric code model is decorated by $s=1/2$ spins which are,
rather unusually, situated on the centres of the bonds, rather than on the
vertices of the lattice, as one would usually expect, [the spins are
represented by green dots in Fig. 4(b)]. We shall mostly work in a basis of
spins directed up and down along $z$, and operate on a spin between sites $i$
and $j$ with Pauli sigma operators using the usual rules
$\hat{\sigma}_{ij}^{z}|\uparrow_{ij}\rangle=|\uparrow_{ij}\rangle$,
$\hat{\sigma}_{ij}^{z}|\downarrow_{ij}\rangle=-|\downarrow_{ij}\rangle$,
$\hat{\sigma}_{ij}^{x}|\uparrow_{ij}\rangle=|\downarrow_{ij}\rangle$ and
$\hat{\sigma}_{ij}^{x}|\downarrow_{ij}\rangle=|\uparrow_{ij}\rangle$.

There are two contributions to the Hamiltonian [see Fig. 4(b)]: (i) from each
square plaquette (labelled with an index $p$) on the lattice, and (ii) from
each star, which is the cross-shaped structure formed from the nearest-
neighbour bonds of lattice site $s$. The contribution from a plaquette is
given by $\hat{P}_{p}=\prod_{ij\in p}\hat{\sigma}_{ij}^{z}$, which is to say a
product of the $z$-components of spins on the bonds around a plaquette. This
takes values $\pm 1$. The contribution from a star is given by
$\hat{R}_{s}=\prod_{ij\in s}\hat{\sigma}_{ij}^{x}$, (i.e. operate with the
$x$-sigma matrix, flipping spins in our $\sigma^{z}$ basis). This contribution
also takes values $\pm 1$ (most easily seen by transforming to a basis of
spins directed along $\pm x$). Each of these two operators commutes amongst
themselves and, as can be checked, they commute with each other. We put the
two contributions together to form the Hamiltonian for the toric code

$\hat{H}=-g\sum_{p}\hat{P}_{p}-t\sum_{s}\hat{R}_{s},$ (7)

where $g$ and $t$ are positive constants.

###  6.1 Ground state and excitations

On one level, the toric code is quite a simple model whose properties can be
guessed. We can immediately spot a candidate ground state $|0\rangle$: if we
can find a wavefunction such that all eigenvalues $R_{s}=P_{p}=1$ we obtain
the lowest-possible energy. Consider plaquette $q$, to get $P_{q}=1$ we must
have an even number of down spins. (There are, of course, several ways to
arrange this.) We can also see the role of entanglement. Consider the ground
state expressed in a basis of eigenstates of the $\hat{\sigma}^{x}$ operator.
Each spin is shared between two stars. To make sure a star has $R_{s}=1$ there
must be an even number of down (along-$x$) spins in the star. Moreover, in
order that all stars have $R_{s}=1$, these down spins must be must be arranged
in closed loops. (Try it and see!) The ground state $|0\rangle$ is an equal
amplitude superposition of such loop states. Such an entangled ground state
does not have a preferred spin direction and so is magnetically disordered and
is an example of a quantum spin liquid.

Excitations from the ground state have some plaquettes or stars contributing
$-1$ to the Hamiltonian. Excitations with $P_{p}=-1$ are known as vortices.
These are bosons, they cost energy $2g$ to produce, and are also known as
magnetic ($m$) particles (or sometimes as visons). The other sort of
excitation is a single negative star $R_{s}=-1$, which is also a boson, this
time costing energy $2t$, and is known as an electric ($e$) charge. Neither
$m$ nor $e$ can be created locally as individual excitations using a single
operator, as we do with particles (where $c^{\dagger}|0\rangle=|1\rangle$).
For the $e$ particles, for example, the best we can do is flip a string of
spins, which creates two localised $e$ particles: one at each lattice site at
the end of the string of flipped spins [Fig. 4(c)]. This is because each spin
is shared between two lattice sites, and hence shared by two stars. The $m$
particles are also created in pairs, occurring at plaquette centres at the
ends of strings of flipped bonds, this time as shown in Fig. 4(d). Again this
is due to each spin being shared between two plaquettes. This non-locality of
excitations leads to anyon statistics, although in slightly more subtle form
to what we’ve seen so far. Although $e$ and $m$ are both bosonic, moving an
$e$ around an $m$ gives a $\pi$ phase shift, resulting in a negative sign. We
say that the particles have mutual statistics, specifically we call them
mutual semions in this context. Additionally, the composite particle
$\varepsilon$ made up of a boson $e$ and a boson $m$ is actually a fermion.

Using these ideas, we can deduce some basic properties of the toric-code
ground state more formally. Working in a basis of eigenstates of $\sigma^{z}$
operators, we specify spin configurations by writing a wavefunction
$|\\{S_{ij}\\}\rangle$, where $\\{S_{ij}\\}$ are a list of the eigenstates of
each operator $\sigma^{z}_{ij}$ for the bond between sites $i$ and $j$. The
ground state $|0\rangle$ will be built from a superposition of these spin
states $|\\{S_{ij}\\}\rangle$. To find $|0\rangle$, require for the plaquette
term that $\hat{P}_{p}|0\rangle=|0\rangle$. This is only possible for a state
of the form

$|0\rangle=\sum c_{s}\left|\\{S_{ij}\\}\right\rangle,$ (8)

where the $c_{s}$s are constants and the sum is constrained to be over those
spin configurations that feature no vortices (i.e. where $P_{p}=1$).

In this $\sigma^{z}$ basis, the star operators act on an
$|\\{S_{ij}\\}\rangle$ to flip spins. If we start with a state with no
vortices and act with the star operator on sites that lie on paths around the
lattice then we create a pattern of down spins on an up-spin background. (The
star operator never creates vortices owing to the commutation of the two
operations.) If we visualise this by linking the down spins via the dual
lattice (i.e. the lattice formed from points in the middle of each lattice
point on the original lattice), the result is always closed loops of down
spins, of the sort shown in green in the example in Fig. 4(e). A ground state
is formed from the superposition of all such vortex-free states comprising
closed loops of down spins on the dual lattice. These all contribute with
equal coefficients $c_{s}=c$ in eqn 8. The resulting state gives
$\hat{R}_{s}|0\rangle=|0\rangle$, confirming that we have a ground state.

Although this accounts for one ground state, there are actually three more
degenerate states, making four in total. To access the others, let’s consider
the Wilson loop operator $\hat{U}(\mathcal{C})$, which multiplies the spins
around a closed contour $\mathcal{C}$. The operator is given explicitly by

$\hat{U}(\mathcal{C})=\hat{S}_{ij}\hat{S}_{jk}...\hat{S}_{li},$ (9)

where the indices are selected to take us around the contour. The eigenvalue
$U(\mathcal{C})$, which can take values $\pm 1$, is called the $Z_{2}$ flux
through $\mathcal{C}$. (For example, the flux from a vortex excitation (i.e.
$P_{p}=-1$ on a plaquette $p$) is $U=-1$.) Since the star operator flips pairs
of spins along any contour through the lattice, it must commute with
$\hat{U}(\mathcal{C})$. Now, if we take the path around the two distinct paths
that wrap a torus [$\ell_{1}=A$ or $\ell_{2}=B$ in Fig. 4(a)], then we must
have $U(\ell_{i})=\pm 1$. [The ground-state configuration we formed in the
last paragraph has $U(A)=U(B)=1$.]

The four degenerate ground states come from the possibility of adding loops of
down spins that wrap all the way around the torus in either of those two
distinct paths that link the torus [i.e. paths $A$ or $B$ in Fig. 4(c)]. An
example of an $A$-type loop of down spins is shown in Fig. 4(e). A $B$-type
contour that wraps the torus must cross this line of down spins and yield
$U(B)=-1$ [we also have $U(A)=+1$]. Note that there is no way to create the
down-spin loop from the application of the star operators, and also that the
state containing just this loop has no vortices. Compared to the state with no
down-spin loops that wrap the torus, this state will give distinct patterns of
down spins on the dual lattice once we start working on it with the star
operator. As a result, the state formed by all down-spin loops in an
$U(A)=1,U(B)=-1$, system yields a second degenerate ground state. The other
degenerate states have $U(A)=-1,U(B)=+1$ [i.e. a vertical line of flipped
spins on Fig. 4(e)] and $U(A)=U(B)=-1$ [i.e. both a vertical and horizontal
line of flipped spins].

The resulting 4-fold degeneracy is a topological degeneracy, meaning that it
occurs for all lattices on the torus. In the language of flux, we might call
$U(A)$ the flux through one hole of the torus and $U(B)$ the flux through the
other. States with $U=1$ have no flux through the torus while states with
$U=-1$ have one unit of flux threading it.999In fact, for a genus $g$ Riemann
surface, we can put zero or one unit of flux through each of the 2g holes of
the surface, leading to a $2^{2g}$ degenerate ground state. The degeneracy is
very robust, as we can see if we treat the star term as a small perturbation
to the vortex term by assuming $g\gg t$ in eqn 7. Recall from degenerate
perturbation theory that to compute the splitting caused by a perturbation, we
need to identify a series of operators that takes the ground state up to an
excited state (or states) and then returns it back to another state in the
ground-state manifold. The only way this can occur here using the star
operator is for it to create excited stated by flipping spins all of the way
around the torus. For an $L\times L$ lattice, this requires $L$ excitations.
Perturbation theory tells us that any energy gap caused by such a perturbation
is of order $\Delta E\approx t^{L}/g^{L-1}$, but as we tend to the
thermodynamic limit of a system $L\rightarrow\infty$, we find $\Delta
E\rightarrow 0$. We conclude that the topological degeneracy is robust. In
fact, the degeneracies in our toric code are exact, and not split at all by
$t^{L}/g^{L-1}$ terms, although such a splitting would apply in a more generic
model.

###  6.2 Counting states in $Z_{2}$ theory

This subsection, and the following one, are a tutorial given in terms of a
simplified model theory to explicitly compute the degeneracy via a
demonstration of the gauge structure. You can skip to Section 7 at this stage
if desired.

Let’s simplify the toric code model further and consider only the plaquette
term [1] with spins acted on by operators $\hat{S}_{ij}=\hat{\sigma_{ij}^{z}}$
(that is, we set $t=0$ in eqn 7). The Hamiltonian is now simply
$\hat{H}=-g\sum_{p}\hat{P}_{p}$, where $\hat{P}_{p}=\prod_{ij\in
p}\hat{S}_{ij}$ now operates on the spins in a plaquette $p$ only. For
definiteness take the number of sites on the lattice to be $n=4$ so we have
only a single plaquette $p=\mathrm{I}$. In the absence of boundary conditions,
the spins on this plaquette can adopt $2^{n}=16$ configurations, half giving
$P_{\mathrm{I}}=1$; half giving $P_{\mathrm{I}}=-1$.

Now apply the all-important boundary conditions and get Fig 4(f) with $2n=8$
spins on the resulting bonds (labelled 1-8 in the figure), and therefore
$2^{2n}=256$ possible spin configurations. However, the description in the
Hamiltonian $\hat{H}=-g\sum_{p}\hat{P}_{p}$ is not given at the level of
spins, but at the higher level of plaquettes. By seeking to describe the
system’s states at this higher level, we’ll see that a description in terms of
$P_{p}$ can be visualised as a pattern of fluxes that thread the lattice.
Compared to the model without boundary conditions, the periodic boundaries
provide three additional plaquettes, making four in total [labelled I, II, III
and IV in Fig. 4(f)]. However, these four plaquettes are not independent,
since each spin is shared between two plaquettes. If we multiply all of the
plaquette operators together, each spin features twice in the product, and
since $P_{p}=\pm 1$, we have the constraint $\prod_{p}P_{p}=1$. A consequence
is that there are only $3$ possible values of energy
$E=\sum_{p=\mathrm{I}}^{\mathrm{IV}}P_{p}$, namely $4,0$ and$-4$. There is one
set of
$\\{P_{p}\\}=(P_{\mathrm{I}},P_{\mathrm{II}},P_{\mathrm{III}},P_{\mathrm{IV}})$
that leads to $E=4$, one set that gives $E=-4$, and six sets with $E=0$,
making 8 different allowed sets (called “patterns of flux” below) in total.
This is a lot less than the 256 possible spins states! In fact, consulting
Table 1, we find that there are $8\times 2\times 2=32$ ways of arranging spins
to get each allowed choice of $\\{P_{p}\\}$.
Takeaway: there are $2^{2n}=256$ configurations of spins in the model. There
are only 8 unique sets of $\\{P_{p}\\}$. That is, there are 8 distinct
patterns of flux through the lattice. It’s therefore impossible to uniquely
label spin configurations with a list of $P_{p}$.

Plaquette | Spins in | No. of free choices to
---|---|---
$p$ | the plaquette | obtain a given $P_{p}$
I | 1 2 3 4 | 8
II | 1 3 5 8 | 2
III | 2 4 6 7 | 2
IV | 5 6 7 8 | 0
Table 1: Decorate the periodic lattice in Fig. 4 with spins to realise a
pattern of
$(P_{\mathrm{I}},P_{\mathrm{II}},P_{\mathrm{III}},P_{\mathrm{IV}}$), in the
order $p=$I, II, III, IV. There are 8 choices to obtain $P_{\mathrm{I}}$, but
because spins are shared, these are constrained for the other plaquettes. The
table shows that there are $8\times 2\times 2=32$ spin configurations that
give each pattern of fluxes $\\{P_{p}\\}$. (We check that $32\times 8$
different $\\{P_{p}\\}$ gives our 256 possible spin configurations.)

Although there are 8 patterns of flux, we don’t actually know how many states
exist in the Hilbert space of the system when it is described in terms of
fluxes. This is because both degeneracy and gauge equivalence will cause there
to be several spin states that give an equivalent pattern of fluxes.

Evaluating the balance between degeneracy and gauge equivalence is the subject
of the next subsection. However, if formal gauge theories are not to your
taste, here is the direct argument: our simplified model has $t=0$ in eqn 7.
We might ask how the energies of these sets of states are split for $0<t/g\ll
1$. The answer is that all the exact eigenstates for $t,g>0$ can be labelled
by the positions of the $e$ and $m$ particles, plus the values of the fluxes
around the two loops that encircle the torus. Here we have 8 configurations of
$m$ particles (i.e. one with no plaquettes occupied, one with all four
plaquettes occupied, and six with two of the plaquettes occupied) , and for
each set of $m$ particles there are 8 possible configurations of $e$ particles
(i.e. one with no sites occupied, one with all four sites occupied, and six
with two of the sites occupied). For each of these 64 configuration of $m$ and
$e$ particles we’ve seen that there are 4 possible configurations of flux that
wrap the torus. So the total number of states is $8\times 8\times 4=256$ as
expected. For $t,g>0$ these states must exist as 4-fold degenerate levels,
since the values of the flux don’t influence the energy of the states.

###  6.3 Introducing gauges

Many of the 256 spin states are gauge equivalent. Recall that “gauge
equivalence” is a statement that several different configurations specified by
a gauge field can correspond to the same state in the Hilbert space of a
system. We see this if we make a local transformation: the transformed system
gives the same Hamiltonian and observables. For our problem, the Hamiltonian
is given in terms of the configuration of plaquettes, which therefore
determines the Hilbert space. We will call these states in the Hilbert space,
which can be visualised in terms of the patterns of fluxes through the
plaquettes, the physical states for this problem. Since many of the underlying
spin states correspond to the same pattern of fluxes, we can therefore treat
the specific description of spins as a gauge field. To summarise, on putting
the spins on bonds and specifying the configurations using the the plaquette
operators $\hat{P}_{p}$ in our Hamiltonian, rather than the spins, we obtain a
redundancy of description, and hence we say that a gauge structure
emerges.101010The situation here is subtly different to that of
electromagnetism, where there is no direct experimental access to the gauge
field. Here we could imagine making measurements of the spins in a real system
via magnetic susceptibility. However, by treating the physics at the
(Hamiltonian) level of plaquettes and, equivalently, fluxes, there’s a sense
in which we’re treating the spins as a microscopic description that we can’t
access, and working instead with what emerges at the more coarse-grained level
of fluxes.

In our model, $S_{ij}$ is the gauge field and we’ll see that several states
described by different $S_{ij}$ correspond the same state in the Hilbert space
of the model when described in terms of plaquette operators. Technically, we
can group spin states into collections called gauge-equivalent classes, where
all of the states in such a class are related by a local transformation and so
represent the same physical state. As a result, states in a Hilbert space are
in a 1-1 relation with the number of gauge-equivalent classes.

The gauge group for our model is $Z_{2}$, which is to say specifically that a
local transformation $W_{i}$ of the spin field $S_{ij}$ leaves the Hamiltonian
invariant. Here the local transformation $W_{i}$ is an arbitrary function that
takes values $\pm 1$ at the different lattice sites. Two spin fields $S_{ij}$
and $\tilde{S}_{ij}$ are gauge equivalent if they are related by a local
transformation $W_{i}$ via $\tilde{S}_{ij}=W_{i}S_{ij}W^{-1}_{j}$. You can see
an example of how this works by selecting some site $k$ and setting $W_{k}=-1$
and all of the other $W_{i}$ to be $+1$. This flips spins in a star shape
around site $k$, but since two of the flipped bonds feature in each plaquette
surrounding $k$, the Hamiltonian $\hat{H}=-g\sum_{p}\hat{P}_{p}$ is gauge
invariant: you get the same value of energy, no matter which $S_{ij}$ you
pick. Moreover, you can make $2^{n}=16$ different $W_{i}$ functions on our
lattice, so this is the number of possible local transformations.

In order to work out if two states on a lattice are really the same physical
state (i.e. if they are gauge equivalent), an important role is played by the
Wilson-loop operator $\hat{U}(\mathcal{C})$ from eqn 9. The key is that
$U(\mathcal{C})$ itself is gauge invariant, so states that give different
values of $U(\mathcal{C})$ cannot be related by an arbitrary local
transformation. Therefore if we know the pattern of fluxes provided by
evaluating $U(\mathcal{C})$ around each plaquette, then we have the physical
description of a state. Note that if $\mathcal{C}$ is a single plaquette $p$
then $\hat{U}(\mathcal{C})\equiv\hat{P}_{p}$, and so the sets $\\{P_{p}\\}$
pick up the flux through each plaquette and label states in a way that’s
invariant with respect to local transformations. In short: a different
$\\{P_{p}\\}$ means a different physical state. However, the same
$\\{P_{p}\\}$ doesn’t guarantee that all of the states are gauge equivalent
(i.e. the same physical state) because we don’t yet know how many states are
distinct states that are degenerate. Returning to our example, we have that a
number (32 in our case) of spins states will give the same pattern of fluxes
described by a given $\\{P_{p}\\}$.

To finally compute the degeneracy, we need a technical argument [1], which
goes as follows: There are $2^{n}=16$ different possible local
transformations, but there are two special transformations that don’t change
the spin configuration, namely $W_{i}=1$ and $W_{i}=-1$, for all $i$. A (non-
trivial) consequence of the existence of these two special transformations is
that the 16 gauge transformations can actually only create $2^{n}/2=8$ gauge-
equivalent configurations, where the 2 here is the number of these special
transformations.  With this key fact in hand, we can say that there are
$2^{2n}/(2^{n}/2)=2\times 2^{n}=32$ gauge-equivalent classes [i.e. total
number of spin states (256), divided by number of gauge-equivalent
configurations (8)] and therefore 32 different physical states in the Hilbert
space. So out of 256 total spin configurations, only 32 are distinct physical
states. To label the flux states we use the $Z_{2}$ flux through a plaquette
provided by the eigenstates of the $\hat{P}_{p}$ operator. We saw that there
are only $2^{n}/2=8$ different values of $\left\\{P_{p}\right\\}$ and so only
8 distinct patterns of flux through the system. However, if each state is
4-fold degenerate we are saved, since then the different flux patterns give
the $8\times 4=32$ physical states in the Hilbert space.

Takeaway: There are 32 physical states in the Hilbert space of the model. Each
of these states corresponds to 8 gauge-equivalent states (accounting for 256
spin configurations). There are 8 different patterns of flux encoded by
$\\{P_{p}\\}$. Each of the 8 flux patterns corresponds to 4 physically-
distinct degenerate states.111111Since each of the physically distinct states
can be expressed as 8 gauge equivalent configurations of spins, there are 32
spin configurations giving each pattern of fluxes, as we found above.

Let’s pause and consider what we’ve learnt from this technical discussion. The
simplified toric-code model has a spin-liquid ground state which respects
rotational spin symmetry and is highly entangled. The model is invariant with
respect to local $Z_{2}$ transformations, giving it a rich gauge structure.
The topology of the lattice, via the gauge structure, causes all states to be
4-fold degenerate. Finally, we might congratulate ourselves by getting this
far with a quotation from X-G. Wen [1]:

> If you feel the definition of $Z_{2}$ gauge theory is formal and the
> resulting theory is strange, then you get the point.

##  7 Spinons

An important excitation found in most spin-liquid models is the spinon. The
spinon is an $s=1/2$, neutral fermion and therefore an example of a fractional
excitation. The idea of fracturing of the quantum numbers of an electron is
easiest to see in one dimension [16], which demonstrates the remarkable
feature that the apparently fundamental properties of the electron: spin and
charge, can break into two. We imagine a set of electrons arranged along a
one-dimensional line as shown in Fig. 5(a), with electron spins aligned up-
down-up-down, so the overall spin of the system is zero. If we remove an
electron from this system then we leave behind a hole, which can be thought of
as an excited state of charge (known as a holon in this context). If we move
an electron along the line into the empty space without changing its spin,
then we see that the hole is mobile. This has a consequence: it leaves two
like spins as neighbours forming a $s=1/2$ spinon excitation. As we slide
electrons around we see that the holon can move independently of the spinon.
Similarly by flipping a pair of spins we can move the spinon around. The spin
excitation and the hole excitation are independent: we have spin-charge
separation.

Figure 5: (a) Spin and charge separate in a one-dimensional electron system. A
spin up electron is removed, leaving a hole behind. As the hole moves down the
chain it leaves behind it a spin excitation (circled). (b) In a spin chain we
create a $s=1$ magnon by flipping a spin. This is not a stable excitation but
can split into a pair of $s=1/2$ spinons (circled) that can propagate via
flips of spin pairs. (c) Spinons pairs give rise to a continuum of excitations
in the chain (grey area) lying between the magnon dispersion (upper curve) and
the single spinon dispersion (lower curve).

In the purely magnetic case (i.e. without the charge excitation), we attempt
to measure the dispersion by flipping a spin [Fig. 5(b)] to create a $s=1$
magnon excitation with wavevector $q$. This is not stable on the 1D chain and
so falls apart into a pair of $s=1/2$ spinons that propagate via pairs of spin
flips with wavevectors $q_{1}$ and $q_{2}$, where $q=q_{1}+q_{2}$. We
therefore expect to measure (in an inelastic neutron scattering measurement) a
continuum of possible excitations [the grey shading in Fig. 5(c)] lying
between the dispersion curves of a single spinon and the single magnon. It is
the characteristic continuum of excitations that tells us we have fractional
excitations.

###  7.1 Representing spins as fermions

It is rather surprising that we can form spinons from the spins in a magnet.
In this section we discuss how this is possible within a mean-field picture of
magnetism.121212Although our discussion is given in terms of fractionalising
spins to form fermionic spinons, it is also possible to choose to form bosonic
spinons via the use of the Schwinger boson representation of spin [17]. Such
an approach has the appealing feature that some states can be described in
terms of the Bose-Einstein condensation of such spinons [18]. The idea will be
to fractionalise the spins into parts or “partons”. If we guess how to make
the split correctly, the partons might represent the true excitations of the
system. As has been commented [1, 11], some practitioners are uncomfortable
with this approach, but we shall willingly suspend disbelief for the moment.

Let’s return to two dimensions and to the Heisenberg antiferromagnet with
Hamiltonian $\hat{H}=-\sum_{\langle
ij\rangle}J_{ij}\hat{\boldsymbol{S}}_{i}\cdot\hat{\boldsymbol{S}}_{j}$ and
make a mean field approximation [4]. This essentially causes a spin to sit in
the constant average magnetic field of its ordered neighbours, allowing us to
compute its dynamics. The usual recipe to do this replaces the original
Hamiltonian by

$\hat{H}_{\mathrm{mf}}=-\sum_{\langle
ij\rangle}J_{ij}\left[\langle\hat{\boldsymbol{S}}_{i}\rangle\cdot\hat{\boldsymbol{S}}_{j}+\hat{\boldsymbol{S}}_{i}\cdot\langle\hat{\boldsymbol{S}}_{j}\rangle-\langle\hat{\boldsymbol{S}}_{i}\rangle\cdot\langle\hat{\boldsymbol{S}}_{j}\rangle\right].$
(10)

where
$\langle\hat{\boldsymbol{S}}_{i}\rangle=\langle\Phi_{\mathrm{mf}}|\hat{\boldsymbol{S}}_{i}|\Phi_{\mathrm{mf}}\rangle$
and $|\Phi_{\mathrm{mf}}\rangle$ is the mean field ground state of the system.
Although this approximation represents the spirit of this section, this
prescription won’t actually work for a QSL, since the states with which we’re
interested have $\langle\boldsymbol{S}_{i}\rangle=0$, as we expect them to be
disordered.

To make progress, we perform a transformation trick that allows us to
represent the spin as a chargeless fermion.131313It’s worth noting that such
transformations have a rich history in many-body physics. Jordan and Wigner
spotted that a single spin state can be thought of as an empty, or singly-
occupied fermion state using a mapping $|\uparrow\rangle\equiv
f^{\dagger}|0\rangle$ and $|\downarrow\rangle\equiv|0\rangle$. However, to
deal with more than one spin they must add a phase factor, called a string, to
each fermion. [17] We introduce spinon operators $\hat{f}_{i\alpha}$ where $i$
is a site on the lattice and $\alpha=1,2$ [1, 13]. These are two-component,
fermionic $s=1/2$, charge-neutral operators (sometimes called Abrikosov
fermions). There are two sorts of these ($f_{i1}$ spinons and $f_{i2}$
spinons). A spin operator is represented by spinons as

$\hat{\boldsymbol{S}}_{i}=\frac{1}{2}\hat{f}^{\dagger}_{i\alpha}\boldsymbol{\sigma}_{\alpha\beta}\hat{f}_{i\beta}=\left(\begin{array}[]{cc}\hat{f}^{\dagger}_{i1}&\hat{f}^{\dagger}_{i2}\end{array}\right)\left(\begin{array}[]{cc}\boldsymbol{\sigma}_{11}&\boldsymbol{\sigma}_{12}\\\
\boldsymbol{\sigma}_{21}&\boldsymbol{\sigma}_{22}\\\
\end{array}\right)\left(\begin{array}[]{c}\hat{f}_{i1}\\\
\hat{f}_{i2}\end{array}\right).$ (11)

In these equations, sums over the Greek indices, which take the values 1 and
2, are implied.

The result of the spinon transformation is that the magnetic Hamiltonian
becomes $\hat{H}=\frac{1}{2}\sum_{\langle
ij\rangle}J_{ij}\hat{f}^{\dagger}_{i\alpha}\hat{f}_{j\alpha}\hat{f}^{\dagger}_{j\beta}\hat{f}_{i\beta}+\mathrm{const}.$
To recap, we have recast the magnetic Hamiltonian in terms of chargeless
spinons which yields an expression in the spirit of the tight-binding model of
eqn 1, that can be interpreted in terms of fermions hopping between sites. By
making this transformation we’ve allowed there to be four possible spinon
states per site ($|f_{i1}f_{i2}\rangle=|00\rangle$, $|01\rangle$,
$|10\rangle$, $|11\rangle$), where previously there were only two spin states
($|\uparrow\rangle$ and $|\downarrow\rangle$). To make sure there’s one
fermion per site we should, strictly speaking, set the constraint
$\hat{f}^{\dagger}_{i1}\hat{f}_{i1}+\hat{f}^{\dagger}_{i2}\hat{f}_{i2}=1$,
which reduces the possible states down to $|10\rangle\equiv|\uparrow\rangle$
and $|01\rangle\equiv|\downarrow\rangle$ only.

We can now make progress by imposing a mean-field approximation on the spinon
model [1, 13]. Recall that this involves taking an average of combinations of
operators. This is carried out here by relaxing the constraint on the number
of fermions per site, such that we set
$\langle\hat{f}^{\dagger}_{i\alpha}\hat{f}_{i\alpha}\rangle=1$, which says
that the average number of fermions per site is now one. In practice this is
done using a Lagrange multiplier $V_{i}$, leading to a new term
$V_{i}(f^{\dagger}_{i\alpha}f_{i\alpha}-1)$ in the Hamiltonian. We enact the
mean-field approximation, which now amounts to taking averages of combinations
of pairs of spinon operators, and throws up the answer

$\hat{H}_{\mathrm{mf}}=\frac{1}{2}\sum_{\langle
ij\rangle}J_{ij}\left[\hat{f}^{\dagger}_{i\alpha}\hat{f}_{j\alpha}\chi_{ji}+h.c.-|\chi_{ij}|^{2}+\sum_{i}V_{i}(\hat{f}^{\dagger}_{i\alpha}\hat{f}_{i\alpha}-1)\right],$
(12)

where $h.c.$ is the Hermitian conjugate of the preceding term. The new
quantity
$\chi_{ij}=\langle\hat{f}^{\dagger}_{i\alpha}\hat{f}_{j\alpha}\rangle=\langle\hat{f}^{\dagger}_{i1}\hat{f}_{j1}\rangle+\langle\hat{f}^{\dagger}_{i2}\hat{f}_{j2}\rangle$
is key here. It measures the amplitude for either sort of fermion to hop from
site $j$ to site $i$. The quantities $\chi_{ij}$ and $V_{i}$ don’t change if
we rotate spins, so we can conclude that the ground state also has this
rotational invariance. The model can now yield spin-liquid ground states.

From the previous discussion, we would conclude that there is a disordered,
correlated ground state whose excitations are spinons. However, to get a
reliable picture that maps to the Hilbert space of the original spin model, we
must also include the possibility of low-energy fluctuations in $\chi_{ij}$ by
saying $\chi_{ij}=\bar{\chi}_{ij}{\rm e}^{-\mathrm{i}\theta_{ij}}$ and obtain

$\hat{H}_{\mathrm{mf}}=\frac{1}{2}\sum_{\langle
ij\rangle}J_{ij}\left[\hat{f}^{\dagger}_{i\alpha}\hat{f}_{j\alpha}\bar{\chi}_{ji}{\rm
e}^{-\mathrm{i}\theta_{ij}}+h.c.+\sum_{i}V_{i}(\hat{f}^{\dagger}_{i\alpha}\hat{f}_{i\alpha}-1),\right]$
(13)

We see that if we make the local phase transformation
$\theta_{ij}\rightarrow\theta_{ij}+\phi_{i}-\phi_{j}$ and
$\hat{f}_{i}\rightarrow\hat{f}_{i}{\rm e}^{\mathrm{i}\phi_{i}}$ the
Hamiltonian is unchanged. We have discovered the $U(1)$ gauge structure of the
model. We identify the $V_{i}$ and $\theta_{ij}$, which interact with the
spinons in the same form, as components of a gauge field, and conclude that
the excitations of the theory are spinons coupled to this $U(1)$ gauge field.
To make further progress we need to specify the form of $\chi_{ij}$, which
then allows us to derive the dispersion of the excitations for specific QSL
models. This can result in a range of states including RVB-type models and a
several $Z_{2}$ spin liquids [1].

For now, we can recap the main features of this approach to finding QSLs: in
order to use a mean-field theory to describe magnetically-disordered states we
have needed to fractionalise the spins before taking the averages on which the
mean-field technique relies. The result is a theory that predicts magnetically
disordered ground states with neutral $s=1/2$ fermion excitations. On allowing
fluctuations in the averaged quantities we obtain a gauge structure, where we
interpret these fluctuations quantum mechanically as bosonic excitations in
the gauge field. In short, we split up the spins at the start, and we use
bosonic glue (via the gauge-field component $V_{i}$) to stick them back
together! This results in a Hamiltonian model that can support a spin liquid
ground state, with predictions of its excitations.

This general approach can be extended to more complicated gauge structures to
make a range of spin liquids states. We expect stable mean-field QSLs to have
a non-zero energy gap against gauge field fluctuations. The stable QSLs always
contain neutral $s=1/2$ spinons with short range interactions between them.
The mean field approach predicts four families of spin liquid in (2+1)
dimensions, characterised by the nature of the energy gap between the ground
state and the spinon and gauge excitations. (See Ref. [1] for the full story.)

Rigid spin liquids are topologically ordered, which means both spinon
excitations and the excitations in the gauge field have non-zero energy gaps.
The gaps lead the QSLs to be stable since there are no low-energy excitations
to destabilise the states. Examples include $Z_{2}$-gapped liquids and chiral
liquids.

Bose spin liquids are a class characterised by spinon excitations with an
energy gap, and gapless $U(1)$ gauge-boson excitations. These are not stable
in (2+1) dimensions.

Fermi spin liquids have gapless $s=1/2$ spinon excitations with short-range
interactions between them. The name derives from Fermi liquids, which are
characterised by their gapless electronic excitations. Examples include
$Z_{2}$-linear-, $Z_{2}$-quadratic-, $Z_{2}$-gapless liquids, where names
refer to features of the spinon dispersion.

Algebraic spin liquids have gapless excitations, but these are neither free
bosons nor free fermions. Excitations are massless fermions coupled to $U(1)$
gauge field. Examples include the $U(1)$-linear liquid.

If you’ve succeeded in suspending disbelief until now, disbelief is surely
flooding back! Should we trust any of these manipulations? Arguably, the jury
is still out, but the approach does suggest the features of some of the
possible states that might be out there for discovery. Ultimately experiment
will be the arbiter and we shall discuss the status of experiments at the end
of the article. For now we shall turn to a different means of realising a QSL
that, unlike the mean-field picture, does not rely on approximations.

##  8 Kitaev model

For many of the formative years of the subject, the search for QSLs was
inspired by the successes such as the FQH fluid and BCS superconductivity,
that suggested that identifying wavefunctions, rather than Hamiltonians, was
the way to discover new states of matter. Although this approach has indeed
been transformative, there was also a great deal of value in the later
identification of Hamiltonian models of QSLs. A major step forward has been
the formulation of the Kitaev model [20]. This is not least because it is
solvable, based on a simple magnetic Hamiltonian that has a spin-liquid ground
state. The solution, which involves the use of operators representing Majorana
fermions (a hypothetical fermion that is its own antiparticle), looks
potentially rather unfamiliar, but the result is less so: it leads to a
$Z_{2}$ gauge field and vortex excitations, just like we saw in Section 6.
Even better, in one limit of interactions the system gives us back the toric
code model exactly.

Figure 6: The Kitaev model. (a) Honeycomb lattice with directional bonds. (b)
To solve the model, we define a set of particle operators attached to each
vertex of the lattice. (c) A plaquette in the lattice leads to the definition
of the operator $\hat{W}_{p}$. (d) Dispersion for $J_{x}=J_{y}=J_{z}$ showing
Dirac cones. (e) The triangular phase diagram given in terms of
$(J_{x},J_{y},J_{z})$. (f) For $J_{z}\gg J_{x},J_{y}$, we create superspins
along the $z$ bonds and recreate the toric code lattice. A sample plaquette
and star is shown.

The Kitaev model is based on a planar honeycomb lattice with spins on each
vertex [Fig. 6(a)]. The links are labelled $x$, $y$, $z$ and the Hamiltonian
is written

$\hat{H}=-J_{x}\sum_{x}\hat{\sigma}^{x}_{i}\hat{\sigma}^{x}_{j}-J_{y}\sum_{y}\hat{\sigma}^{y}_{i}\hat{\sigma}^{y}_{j}-J_{z}\sum_{z}\hat{\sigma}^{z}_{i}\hat{\sigma}^{z}_{j}.$
(14)

To solve the model we introduce an operator that operates on hexagonal
plaquette $p$ shown in Fig. 6(c) via
$\hat{W}_{p}=\hat{\sigma}^{x}_{1}\hat{\sigma}^{y}_{2}\hat{\sigma}^{z}_{3}\hat{\sigma}^{x}_{4}\hat{\sigma}^{y}_{5}\hat{\sigma}^{z}_{6},$
with eigenvalues $W_{p}=\pm 1$. The $\hat{W}_{p}$ for each $p$ commutes so we
can describe the Hilbert space in terms of a set $\\{W_{p}\\}$. It will turn
out that the ground states has $W_{p}=1$ for all plaquettes.

Although this is progress, the full solution requires swapping spins for a new
set of operators as we did in the last section. Fascinatingly, these are the
operators describing Majorana fermions. Conventional fermions can be created
(with operator $\hat{d}^{\dagger}$) an annihilated ($\hat{d}$), with the
constraints $\hat{d}^{\dagger}\hat{d}^{\dagger}=0$ and $\hat{d}\hat{d}=0$, and
anticommutator $\hat{d}^{\dagger}\hat{d}+\hat{d}\hat{d}^{\dagger}=0$. A
Majorana fermion is its own antiparticle, which means that conjugating the
charge of a Majorana has no effect. In terms of operators this means we
represent Majoranas as real-valued. Therefore, if we have two species of
Majorana ($a$ and $b$ particles) then we could split up a fermion as
$\hat{d}=(\hat{a}+\mathrm{i}\hat{b})/2$, where the Majorana operators for this
representation are $\hat{a}$ and $\hat{b}$. It follows that the Majoranas have
the properties $\hat{a}=\hat{a}^{\dagger}$ and $\hat{b}=\hat{b}^{\dagger}$
(i.e. the creation and annihilation operators are the same),
$\hat{a}^{2}=\hat{b}^{2}=1$ (i.e. applying the operator twice has no effect)
and $\hat{a}\hat{b}+\hat{b}\hat{a}=0$ (the operators anticommute with each
other).

###  8.1 The Kitaev model solution and its gauge structure

For our problem [20], we will need to introduce four Majorana particles at
each vertex of the lattice [Fig. 6(b)]. At each vertex we therefore define a
set of Majorana operators $\hat{b}^{x},\hat{b}^{y},\hat{b}^{z},\hat{c}$, and
we write spin operators in terms of these as
$\hat{\sigma}^{x}=\mathrm{i}\hat{b}^{x}\hat{c}$,
$\hat{\sigma}^{y}=\mathrm{i}\hat{b}^{y}\hat{c}$, and
$\hat{\sigma}^{z}=\mathrm{i}\hat{b}^{z}\hat{c}$. The new operators square to
unity and they anticommute in the same way that the $\sigma$ matrices do.
However, we find that we also need an extra constraint: identify operator
$\hat{D}=\hat{b}^{x}\hat{b}^{y}\hat{b}^{z}\hat{c}$ (which commutes with the
spin operators) and, with the stipulation that $D=\pm 1$, we guarantee a
consistent theory.141414The point here is that the Hilbert space dimension for
a single spin is 2 but for four majoranas it is 4, so we need to project out
two spurious states per site. So why do we need four Majoranas? We’ll give the
$c$-field a local $Z_{2}$ invariance, and the $b$-fields will be the gauge
fields that guarantee this. This will give us a QSL ground state with fermion
excitations that derives from the $c$-field, along with vortex excitations
that will arise from the flux of the gauge field.

With the proposed set of transformations, the Kitaev Hamiltonian becomes

$\hat{H}=\frac{\mathrm{\mathrm{i}}}{4}\sum_{jk}2J_{\alpha}\left(\mathrm{i}\hat{b}^{\alpha}_{j}\hat{b}^{\alpha}_{k}\right)\hat{c}_{j}\hat{c}_{k}.$
(15)

The first thing to notice is that this looks like a tight-binding model for
the $c$-particles. Moreover, the shape of this Hamiltonian has the $c$-type
Majoranas interacting with an effective field $\mathrm{i}\hat{b}\hat{b}$. In
this case, the combination
$\mathrm{i}\hat{b}^{\alpha}_{j}\hat{b}^{\alpha}_{k}=\hat{u}_{jk}$, with
$\alpha$ determined by the direction of the bond [i.e. $\alpha=\alpha(i,j)$],
is a $Z_{2}$ gauge field, since $u_{jk}=\pm 1$.151515To see this, note that
the local transformation in this case would send $c_{j}\rightarrow W_{j}c_{j}$
and then a change in the gauge field of $u_{jk}\rightarrow
W_{j}u_{jk}W^{-1}_{k}$ cancels out this effect. The operator $\hat{u}_{jk}$
commutes with the Hamiltonian, so we can split up the solutions according to
the values of $u_{jk}$ using the product around a plaquette
$\hat{W}_{p}=\prod\hat{u}_{jk}$, which we now see evaluates the flux of the
gauge field through a plaquette.

The ground state has $W_{p}=1$ for all plaquettes. This state is a
magnetically-disordered (i.e. spin-rotation invariant), spin-liquid phase. The
Hamiltonian can then be diagonalised to reveal the dispersion for the
$c$-field excitations of $E(\boldsymbol{k})=\pm 2\left|J_{x}{\rm
e}^{\mathrm{i}\boldsymbol{k}\cdot\boldsymbol{n}_{1}}+J_{y}{\rm
e}^{\mathrm{i}\boldsymbol{k}\cdot\boldsymbol{n}_{2}}+J_{z}\right|,$ where
$\boldsymbol{n}_{1,2}=\left(\pm\sqrt{3}/2,3/2\right)$. As usual, we
characterise the material in terms of the gap against excitations. An example
of the spectrum for $J_{x}=J_{y}=J_{z}$ is shown in Fig. 6(d) and has several
sets of excitations existing from $E=0$, each with a striking linear
dispersion. The points in $k$-space from which they emerge are known as Dirac
points, where the dispersion resembles that of highly relativistic particles
(i.e. with linear dispersion $E=|p|c$, where $c$ is the speed of light).

We can always find some fermion solution with $E=0$ as long as the exchange
constants obey triangle inequalities: $|J_{x}|\leq|J_{y}|+|J_{z}|$ or
$|J_{y}|\leq|J_{z}|+|J_{x}|$ or $|J_{z}|\leq|J_{x}|+|J_{y}|$. When these
aren’t obeyed, the spectrum has a gap. We can draw a triangular phase diagram
of gapped and gapless phases as shown in Fig. 6(e). The phases $A_{i}$ are
completely gapped, while the phase $B$ is gapless. In addition to the fermion
excitations, we have $W_{p}=-1$ excitations in the gauge field. These are
bosonic vortices.

Finally, as promised, in the limit of strong $J_{z}$-coupling we recover the
toric code Hamiltonian. Physically, this is because the strong $z$-links tie
the spins together to form effective spins
$|1\rangle=|\uparrow\uparrow\rangle$ and
$|2\rangle=|\downarrow\downarrow\rangle$ which decorate the edges of a square
lattice. This is shown in Fig. 6(f), where a sample plaquette and star are
indicated.

##  9 Realising the a QSL in materials

We’ve seen several ways of describing different putative QSLs, but the
question remains whether the state is realised in real materials. As we said
at the start of this article, there is a wide-ranging literature claiming QSL
behaviour in a large number of materials. However, it’s probably fair to say
that we do not have a case where a consensus has been reached in favour of the
occurrence of the QSL state. Despite this we have reason to be hopeful. We
have dealt with (2+1) dimensions in our discussion so far, but the analogous
one-dimensional (1D) case of the spin-Luttinger liquid is agreed to be
realised in some spin-chain materials, in which direct measurements of spinons
have been made, most directly using inelastic neutron scattering (INS). The
spin-Luttinger regime is characterised by a lack of long-range magnetic order
(LRO), algebraically-decaying spin correlations and, most vividly, a continuum
of spinon excitations [Fig. 5(c)]. Here it is unwise to say a material is a
spin-Luttinger liquid, rather that it behaves as a spin-Luttinger liquid under
certain conditions. The 1D chain will be characterised by an intrachain
interaction $J$. If the chains have some small interchain interaction
$J^{\prime}$ then the material might well order at a very low temperature
$T_{\mathrm{N}}$ determined by $J^{\prime}$. In order to observe 1D physics we
seek to be in a regime that $T_{\mathrm{N}}\ll T\ll J$. This means we stay
away from magnetic order, but the temperature is low enough that the
collective behaviour is promoted by $J$. (For $T\gg J$ excitations will start
to resemble single uncorrelated spin flips.)

A QSL will be realised in more than one dimension. Although we have dealt with
2D in our discussion, there are also proposals for 3D QSL states. To realise a
QSL we seek to suppress magnetic order in materials with strong spin-spin
interactions. Low dimensionality is good for this, although many 2D materials
enjoy enough coupling in the third dimension to promote order at low
temperature. Magnetic order can also be suppressed by frustrating magnetic
interactions. (When this occurs, small interactions can tip the system over
into LRO at a low-enough $T$, so these must be borne in mind.) We also want to
promote quantum fluctuations and for this $s=1/2$ spins are most effective.
Other possibilities involve promoting higher-order exchange interactions, such
as next-nearest neighbour couplings.

The experimental identification of a QSL is a knotty problem, as discussed in
Refs [21, 19]. Clearly we seek a system without magnetic order down to low
temperature. A lack of order can be difficult to establish, but sensitive
local probes such as NMR and muon-spin spectroscopy ($\mu^{+}$SR) are often
used to search for weak signs of order, or spin dynamics characteristic of
disordered spin-liquid states. We have stressed that the excitation spectrum
is key to the characterisation of a QSL system. Perhaps the most important
question is whether there is a gap against excitations and there are several
ways of investigating this that are sensitive to the density of states,
including thermodynamic, scattering and spectroscopic probes. Beyond this, a
determination of the excitation spectrum might be expected to supply the
smoking gun evidence. INS remains a key method here, but requires samples that
are relatively large compared to those required for other measurements. There
is also the problem that some nuclei have large neutron-capture cross
sections, or the material contains hydrogen or other nuclei that cause a large
degree of incoherent scattering. Nevertheless, we look for fractionalised
quasiparticles which, like spinons in 1D, give broad responses in scattering
as they correspond to the creation of several particles. In the absence of
momentum-resolved scattering measurements, there are predictions for the
temperature and magnetic field dependence of properties such as heat capacity,
spin relaxation or transport. In the case of the latter, although QSLs should
strictly be insulators, there are possibilities for the state to couple to
charges, allowing an additional handle to probe the system.

We have stressed that the key to the QSL problem is the special pattern of
quantum entanglements that underlie the physics. Finding a means of evaluating
these could prove the key to this subject. The entanglement entropy $S$ of the
many-particle ground state is useful here. For a gapped system it follows the
form $S=cL-\Gamma+...$, where $L$ in the first term is the size of the system,
while the second term quantifies long-range entanglements [21]. This latter
part is only non-zero in a topological phase (for a $Z_{2}$ QSL, $\Gamma=\ln
2$). The bad news, however, is that there isn’t yet a known way of measuring
this quantity [21].

Below we briefly introduce four different candidate systems that have been
much discussed as possible approximate realisations of a QSL state. Further
discussion, including the many relevant references, can be found in the
detailed reviews of Refs. [11, 12, 13, 22, 23].

Herbertsmithite or ZnCu3(OH)6Cl2 is a mineral, originally discovered in a
Chilean mine, but since synthesised in the laboratory. It comprises $s=1/2$
spins on a frustrated 2D kagome lattice interacting with an exchange of
$J=180$ K. The kagome lattice is formed from corner sharing triangles (as
distinct from edge-sharing triangles of the triangular lattice), resulting in
a larger degeneracy and potential for realising QSL states. No evidence of
long-range magnetic order has been found in Herbertsmithite down to 50 mK.
Specific heat and some NMR measurements suggest gapless behaviour, although a
spin gap has also been identified on the basis of Knight-shift measurements.
Large crystals have enabled neutron scattering, but the broad continuum of
magnetic excitations found was not one predicted by spin liquid theory. In
fact, the low-energy excitation spectrum appears dominated by impurity spins
that originate from sites between the kagome planes. Although these sites
should host non-magnetic Zn2+, they appear to contain enough magnetic Cu2+ to
form a significant population of orphaned impurity moments (estimated to be
5-10%). This raises the question of how robust a QSL might be with respect to
disorder, which is a question that is pertinent to all of the cases discussed
here. The presence of perturbing anisotropic (Dzyaloshinskii-Moriya)
interactions between intrinsic kagome spins has also been found using electron
spin resonance, which could have a large effect on the ground state [27].

$\kappa$-(ET)2Cu2(CN)3 is formed from a layered structure built from
structural dimers of planar BEDT-TTF molecules, sandwiched between layers of
Cu2(CN)3. Each dimer gives rise to a magnetic $s=1/2$ spin that form a 2D
triangular lattice with a large exchange $J>200$ K. This material is part of a
family of materials that show superconductivity and magnetic order, along with
a number of spin-liquid candidates. For $\kappa$-(ET)2Cu2(CN)3 there is no
evidence of magnetic order from $\mu$SR or NMR. The combination of constant
low-temperature magnetic susceptibility $\chi$, linear specific heat (i.e.
like a metal $C=\gamma T$) and a resulting Fermi-gas like Wilson ratio was
suggestive of a gapless phase with spin-carrying excitations. However
conductivity and $\mu$SR both suggested the presence of an energy gap. Owing
to the tunability of molecular materials, several other systems based on
similar building blocks also show some promise.

$R_{2}M_{2}$O7 ($R=$rare earth, $M=$transition metal such as Ti or Sn) are a
well know frustrated series of 3D materials based on spins in a 3D arrangement
of corner-sharing tetrahedra, called the pyrochlore lattice, which is known to
exhibit a high degree of frustration. Strong spin-orbit interactions and
crystal field splitting can result in systems described by an effective
$s=1/2$ theory. Some of these materials are well-described by classical
theories (e.g. Ho2Ti2O7 and Dy2Ti2O7). For example, in Dy2Ti2O7 the crystal-
field anisotropy constrains the magnetic moments to lie along the local
$\langle 111\rangle$ axes (i.e. directly in our out of the tetrahedron) and
there is an effective local ferromagnetic coupling between these moments, as
well as long-range dipolar couplings, which turn out to be important. As a
result of this combination of interactions and the local anisotropy, below
about 1 K the system settles into a disordered spin-ice state, which is often
described as a classical spin liquid (i.e. a highly correlated magnetic system
that avoids magnetic order). Spin ice is characterised by a ‘2-in 2-out’ spin
configuration (meaning that two spins point in and two spins point out of each
tetrahedron), analogous to proton displacement vectors in Pauling’s model of
hydrogen disorder in water ice. The excitations in spin ice are created by
reversing a single spin, which produces a pair of effective magnetic monopoles
which can move independently through the lattice and interact with an emergent
gauge field, but remain connected by a topological Dirac string of flipped
spins between them. In contrast, strong quantum effects have been found in
$R_{2}$Ti2O7 with $R=$Yb, Er and Tb and in Pr2Zr2O7. Of these Yb is the most
studied, having been suggested to be a quantum version of spin ice which
features frustrated Ising interaction. Such a model is thought to support a
$U(1)$ QSL in some limits of its parameters. This picture has some
experimental support (e.g. it features a “pinch point” feature in its neutron
spectrum, predicted for classical spin ice owing to dipolar spin
correlations). However, there is also good evidence for a phase transition to
a ferromagnetic ground state in high-quality samples, and the unusual neutron
spectrum has been suggested to arise from an interplay of types of
conventional order [28]. This is another case where there is a degree of
sample dependence, suggesting a role for disorder.

Finally, there are several candidate materials that might be described by the
Kitaev Hamiltonian. Although the Kitaev Hamiltonian, with bond-dependent Ising
interactions, appears somewhat contrived, it was suggested that partially-
filled $t_{2g}$ levels in an octahedral environment with strong spin-orbit
coupling could realise the model. The idea was originally to look at materials
containing Ir4+, which has an effective $j=1/2$ moment, owing to the orbital
degeneracy being lifted in favour of a Kramers doublet. Studies of honeycomb
$\alpha$-Na2IrO3 and $\alpha$-Li2IrO3 revealed magnetic order, but also
indicated that the bond-directed Kitaev interactions were, to some extent,
realised. Iridium is well known for its high neutron capture cross section,
making INS difficult and so the most-studied Kitaev candidate is not an Ir-
hosting material, but $\alpha$-RuCl3. Here $j=1/2$ ruthenium ions form a
honeycomb lattice between Cl planes. This system also orders
($T_{\mathrm{N}}=15$ K), and seems to host conventional Heisenberg
interactions in addition to bond-directed ones. However, there is evidence
from Raman scattering for a fermionic continuum and magnetic order seems to be
fully suppressed by a strong applied field, suggesting that the interactions
might be tuned to QSL behaviour. As a result, the system is sometimes called a
proximate spin liquid. So is the honeycomb lattice really sufficient for the
Kitaev model? It is possible that the spin liquid regime might only be stable
in a tiny region and the interpretation of data on these materials has been
hotly debated. Owing to the large amount of work on this subject at the time
of writing, this remains an area to watch.

None of these systems uncontroversially show QSL behaviour, but all have
features suggesting they might be close. As with the 1D materials the key to
observation of this behavior will be looking at the realm of applicability of
the models to the materials. If the QSL states are realised in some form, and
it seems likely that they are, evidence will likely continue to accumulate
slowly as the vast parameter space of possible materials is surveyed. One
essential ingredient to consider is disorder. This is an inevitable feature of
condensed matter systems and will be of relevance here. It is perhaps worth
keeping in mind its dual role in the related field of quantum Hall physics:
the FQH fluid was only observed in relatively clean samples, but disorder is
also believed to be necessary in order to observe the Hall plateaux. It is
possible disorder might adopt a similar dual role in the story of the QSL, or
at least resolve some of the contradictory experimental evidence that is found
in the current literature.

##  10 Conclusion

The study of quantum magnetism started with doubts about the existence of the
antiferromagnet but, through careful experiment and thoughtful theoretical
description, the field now boasts a wealth of established results that
demonstrate the fractionalisation of excitations in 1D and the existence of
topological excitations related to vortices. Advances in materials
preparation, including those towards the ability to engineer magnets from
tuneable building blocks, also provide hope that hitherto purely-theoretical
models might be realised in material systems. The quest to realise a quantum
spin liquid remains a much sought-after goal that goes well beyond simply
finding a material that fails to order at low temperatures. The aim of this
article has been to give a sense of what is at stake here beyond a lack of
magnetic order: namely a macroscopic manifestation of a very specific type of
quantum entanglement that presents us with a zoology of elementary excitations
over and above those that fit comfortably within Landau’s “Standard Model” of
condensed matter physics. From this point of view, the investigation of QSLs
could help us in the next phase of determining the ordering principles that
quantum mechanics imposes on our material world.

## Acknowledgements

I am grateful the following colleagues for their many helpful comments and
corrections: Stephen Blundell, Theo Breeze, Claudio Castelnovo, Matjaž
Gomilšek, Thomas Hicken, Ben Huddart, Samuel Ladd and Francis Pratt.
Particular thanks are due to John Chalker who made detailed comments on two
early drafts of this article.

## Biography

Tom Lancaster is Professor of Physics at Durham University. His research
focuses on the physics of magnetism in one and two dimensions along, more
recently, with the physics of topological objects in magnetism. He is the
coauthor, with Stephen Blundell, of Quantum Field Theory for the Gifted
Amateur.

## References

  * [1] Wen, X-G. Quantum Field Theory of Many-body Systems, OUP, Oxford (2004).
  * [2] Landau, LD. and Lifshitz, EM. Statistical Physics, Pergamon, Oxford (1980).
  * [3] Anderson, PW. Basic Notions of Condensed Matter Physics, Westview, Boulder (1984).
  * [4] Lancaster, T. and Blundell, SJ. Quantum Field Theory for the Gifted Amateur, Oxford University Press, Oxford (2014).
  * [5] Singleton, J. Band Theory and Electronic Properties of Solids, OUP, Oxford (2001).
  * [6] Blundell, SJ. Magnetism in Condensed Matter, Oxford University Press, Oxford (2001).
  * [7] Feynman, RP., Leighton RB., and Sands M. The Feynman Lectures on Physics, Vol. I, Pearson Addison Wesley, San Francisco (2006).
  * [8] Anderson PW. Concepts in Solids, World Scientific, London (1997).
  * [9] Lhuillier C. Frustrated Quantum Magnets arXiv:cond-mat/0502464 (2002).
  * [10] Anderson PW. Mater. Res. Bull. 8, 153 (1973).
  * [11] Savary L. and Balents L. Rep. Prog. Phys. 80, 016502 (2017).
  * [12] Broholm C., Cava RJ., Kivelson SA., Nocera DG., Norman MR., Senthil T. Science 367, eaay0668 (2020).
  * [13] Zhou Y., Kanoda K. and Ng T-K. Rev. Mod. Phys. 89, 025003 (2017).
  * [14] Manousakis E. Rev. Mod. Phys. 63, 1 (1991).
  * [15] Lancaster T. Contemp Phys. 60, 246 (2019).
  * [16] Giamarchi T. Quantum Physics in One Dimension, OUP, Oxford (2003).
  * [17] Coleman P. Introduction to Many body Physics CUP, Cambridge (2015).
  * [18] Xu C. and Sachdev S. Phys. Rev. B 79, 064405 (2009).
  * [19] Khatua J., Sana B., Zorko A., Gomilsek M., Sethupathi K., Ramachandra Rao MS., Baenitz M., Schmidt B. and Khuntia P. arXiv:2310.15071 (2023).
  * [20] Kitaev A. Annals of Physics 321 2, (2006); Kitaev A. and Laumann C. arXiv:0904.2771v1 (2009).
  * [21] Knolle J. and Moessner R. Annu. Rev. Condens. Matter Phys. 10 451 (2019).
  * [22] Clark L. and Abdeldaim AH. Annu. Rev. Mater. Res. 51, 495 (2021).
  * [23] Wen J., Yu S-L., Li S., Yu W., Li J-X. npj Quantum Materials 4, 12 (2019).
  * [24] Khuntia P., Velazquez M., Barthélemy Q., Bert F., Kermarrec E., Legros A., Bernu B., Messio L., Zorko A. and Mendels P. Nature Phys. 16, 469 (2020).
  * [25] Elhajal M., Canals B., Sunyer R. and Lacroix C. Phys. Rev. B 71, 094420 (2005).
  * [26] Yokoi T., Ma S., Kasahara Y., Kasahara S., Shibauchi T., Kurita N., Tanaka H., Nasu J., Motome Y., Hickey C., Trebst S. and Matsuda Y. Science 373, 568 (2021).
  * [27] Zorko A., Nellutla S., van Tol J., Brunel LC., Bert F., Duc F., Trombe J-C., de Vries MA., Harrison A., and Mendels P. Phys. Rev. Lett. 101, 026405 (2008).
  * [28] Scheie A., Kindervater J., Zhang S., Changlani HJ., Sala G., Ehlers G., Heinemann A., Tucker GS., Koohpayeh SM. and Broholm C. PNAS 177, 27245 (2020).