Microscopic models of interacting Yang–Lee anyons

Abstract

Collective states of interacting non-Abelian anyons have recently been studied mostly in the context of certain fractional quantum Hall states, such as the Moore–Read state proposed to describe the physics of the quantum Hall plateau at filling fraction ν=5/2. In this paper, we further expand this line of research and present non-unitary generalizations of interacting anyon models. In particular, we introduce the notion of Yang–Lee anyons, discuss their relation to the so-called 'Gaffnian' quantum Hall wave function and describe an elementary model for their interactions. A one-dimensional (1D) version of this model—a non-unitary generalization of the original golden chain model—can be fully understood in terms of an exact algebraic solution and numerical diagonalization. We discuss the gapless theories of these chain models for general su(2)k anyonic theories and their Galois conjugates. We further introduce and solve a 1D version of the Levin–Wen model for non-unitary Yang–Lee anyons.

Full text

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Microscopic models of interacting Yang-Lee anyons
E. Ardonne,1J. Gukelberger,2A.W.W. Ludwig,3S. Trebst,4and M. Troyer2
1Nordita, Roslagstullsbacken 23, 106-91 Stockholm, Sweden
2Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland
3Physics Department, University of California, Santa Barbara, CA 93106, USA
4Microsoft Research, Station Q, University of California, Santa Barbara, CA 93106, USA
(Dated: December 7, 2010)
Collective states of interacting non-Abelian anyons have recently been studied mostly in the context of certain
fractional quantum Hall states, such as the Moore-Read state proposed to describe the physics of the quantum
Hall plateau at filling fraction ν = 5/2. In this manuscript, we further expand this line of research and present
non-unitary generalizations of interacting anyon models. In particular, we introduce the notion of Yang-Lee
anyons, discuss their relation to the so-called ‘Gaffnian’ quantum Hall wave function, and describe an elemen-
tary model for their interactions. A one-dimensional version of this model – a non-unitary generalization of the
original golden chain model – can be fully understood in terms of an exact algebraic solution and numerical
diagonalization. We discuss the gapless theories of these chain models for general su(2)kanyonic theories and
their Galois conjugates. We further introduce and solve a one-dimensional version of the Levin-Wen model for
non-unitary Yang-Lee anyons.
PACS numbers: 05.30.Pr, 03.65.Vf, 73.43.Lp
I.INTRODUCTION
Over thirty years ago Leinaas and Myrheim pointed out1,
that in systems confined to two spatial dimensions parti-
cles with exotic exchange statistics, more general than those
of bosons and fermions, are possible. Such particles with
arbitrary exchange statistics were later coined anyons by
Wilzeck2. Today it is widely believed that this possibility is
indeed realized in the fractional quantum Hall effect. An even
more intriguing form of statistics has recently received con-
siderable attention, namely that of non-Abelian statistics, first
proposed in a seminal paper by Moore and Read3. This form
of statistics can occur in two-dimensional systems in which
introducing excitations gives rise to a macroscopic degener-
acy of states. Upon braiding the excitations, the wave func-
tion (or better, the vector of wave functions) describing the
system, does not merely acquire an overall phase, but can ac-
tually transform into one another, as described by a unitary
braid matrix acting within the degenerate manifold. In gen-
eral, these braid matrices do not commute, hence the name
non-Abelian statistics. While several systems have been theo-
retically proposed to exhibit quasiparticles with non-Abelian
statistics, such as unconventional px+ ipysuperconductors4,
rotating Bose-Einstein condensates5, or certain heterostruc-
tures involving a novel class of materials, so-called topologi-
cal insulators6, the system currently being cast under intense
experimental scrutiny is the fractional quantum Hall effect ob-
served at filling fraction ν = 5/2, with some evidence sug-
gesting that this state is indeed non-Abelian in nature7,8.
An early attempt to describe this ν = 5/2 quantum Hall
state came in the form of the so-called Haldane-Rezayi wave
function9, which can be thought of as a d-wave paired BCS
condensate of composite fermions – electrons with, in this
case, two flux quanta attached.
Haldane-Rezayi state is that its gapless edge modes are de-
scribed by a non-unitary conformal field theory10–14with cen-
tral charge c = −2. However, non-unitary dynamics can-
A peculiar feature of the
not describe a physical system as it would violate basic prin-
ciples of quantum mechanics. It has therefore been argued
that in general the non-unitary nature of an edge state indi-
cates that the underlying phase is not bulk gapped, but in fact
critical15–17in which case the edge states lose their identity as
they dissipate into the gapless bulk. For the Haldane-Rezayi
stateitindeedturnsoutthatitdoesnotdescribeagappedtopo-
logical phase, but rather the gapless state at a quantum phase
transition between two gapped states – the Moore-Read quan-
tum Hall state3and a so-called strong pairing quantum Hall
phase18. Since the work of Haldane and Rezayi many other
wave functions have been proposed, which appear to have
non-unitary edge state theories19–22. One particularly well-
known example is the so-called ‘Gaffnian state’17. These pro-
posed states typically arise from numerical studies of model
Hamiltonians, for which it is oftentimes hard to determine
whethertheydescribeagappedorgaplessphase. Theultimate
fate of these newly proposed states has therefore remained an
active field of research.
In this paper, we will study the excitations of the putative
quantum liquid described by such non-unitary wave functions
from an ‘anyon-model’ perspective. Thus, we explore the
physics that would occur if one were to assume that such
excitations indeed exist and study in detail their collective
behavior that would result. We refer to these excitations as
non-unitary anyons and model the interaction between these
anyons in the same way as it was done in the unitary case23.
It will turn out that the Hamiltonians one is led to consider for
the non-unitary anyons are, not surprisingly, non-Hermitian.
Nevertheless, the energy spectrum turns out to be completely
real. One can hope that the results of such a study would shed
some light on the questions surrounding the physical mean-
ing of non-unitary anyons. Of course, in the context of two-
dimensional classical statistical mechanics models having a
non-unitary conformal field theory (CFT) describing a critical
phase is a situation which often occurs – see, for instance, the
(restricted)heightmodelsdescribedinRef. 24, whicharegen-
arXiv:1012.1080v1 [cond-mat.str-el] 6 Dec 2010

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2
eralizations of the ‘restricted solid-on-solid’ models25– and is
not problematic.
Before closing this introduction with the outline of the pa-
per, we should mention one additional point, which moti-
vates the study on non-unitary anyon models: the concept of
‘Galois-conjugation’ of conformal field theories. In short, the
action of the Galois group relates different CFT’s which have
the same number of fields, obeying the same fusion rules. The
non-unitary models we encounter in this paper, are in fact
Galois-conjugates of unitary CFT’s, and thus, the study on
non-unitary anyons is closely related to the study of Galois
conjugation in conformal field theory26,27. For the purposes
of this paper we will however not need to use any details of
these connections.
This paper is organized as follows: In Sec. II we discuss the
Yang-Lee anyonic chains. We start by giving a review of the
derivation of the Hamiltonian of the (ordinary, i.e. unitary)
golden chain23of Fibonacci anyons in Sec. IIA, and a short
mathematical description of the Galois-conjugated model, the
so-called Yang-Lee chain in Sec. IIB, including an explicit
derivation of their non-Hermitian, microscopic Hamiltonians,
and discuss their relation to the ‘golden chain’ models of
Ref. 23 via Galois conjugation. We continue in Sec. IIC
with a discussion of the algebraic structure underlying these
microscopic Hamiltonians, which in turn allows for an analyt-
ical identification of their gapless theories in terms of certain
non-unitary minimal models of conformal field theory. We
next present a number of exact, numerical results in Sec. IID,
and discuss the topological symmetry protecting the critical
states in Sec. IIE before rounding off with a discussion of the
case of general level k in su(2)ktheories in Sec. IIF
In Sec. III we turn to ‘doubled Yang-Lee’ models and, in
particular, a non-unitary generalization of the (unitary) high-
genus ladder model which was previously studied for Fi-
bonacci anyons in Ref. 28, where we introduce the model in
Sec. IIIA. We then discuss the phase diagram in Sec. IIIB,
present the analytical solution at special critical points in Sec.
IIIC and finish with exact numerical results in Sec. IIID.
The final section summarizes the results and discusses their
relevance to hypothetical non-unitary topological phases. An
appendix contains a detailed discussion of the conformal en-
ergy spectra at the four critical points discussed in the two
models.
II.YANG-LEE CHAINS
At the focus of this manuscript are anyonic models that
are certain non-unitary generalizations of unitary non-Abelian
anyon models, which have been extensively studied in the re-
cent past23,28–35. The basic constituents of the generalizations
considered here are non-unitary, non-Abelian anyons. Like
their unitary counterparts they carry a quantum number that
corresponds to a generalized angular momentum in so-called
su(2)kanyonic theories, which are certain deformations36of
SU(2). We first concentrate on an elementary example where
there is only a single anyon type by explicitly considering the
anyon theory su(2)3. In the unitary version of this theory the
elementary degrees of freedom are often referred to as ‘Fi-
bonaccianyons’, anditistheirnon-unitarycounterpartswhich
we term ‘Yang-Lee anyons’. We will return to a discussion
of the non-unitary generalizations for general su(2)kanyonic
theories in Section IIF.
We start by quickly reviewing the basic construction of mi-
croscopic (chain) models of interacting non-Abelian anyons,
following the ideas of the ‘golden chain’ model of Ref. 23 and
the detailed exposition of Ref. 31. The construction of these
models proceeds in two steps. First, we describe the general
structure of the Hilbert space of these models in a particular
‘fusion chain’ representation, which is identical for the uni-
tary and non-unitary models. In a second step we turn to the
microscopic form of the Hamiltonian capturing interactions
between the anyons. While this second step is quite similar
for the unitary and non-unitary cases, the microscopic Hamil-
tonians for the two cases are distinct.
A.The golden chain
The elementary degrees of freedom in our microscopic
model are the particle types (or generalized angular momenta)
of the su(2)3anyonic theory. In its simplest form (considering
only integer momenta) this theory contains a trivial particle
(or vacuum state), which we denote by 1, and an anyonic par-
ticle, which we label as τ. These particles can form combined
states according to the fusion rules
1 × 1 = 1
The non-Abelian nature of the anyonic τ-particle reveals it-
self in the multiple fusion outcomes when combining two of
these particles. Our chain model then consists of L such τ-
particles in a one-dimensional arrangement as depicted on the
top of Fig. 1, where L denotes the number of sites of the
chain. Since pairs of τ-particles can be fused into more than
one state, such a system of L non-Abelian anyons spans a
macroscopic manifold of states, i.e. a vector space whose di-
mension grows exponentially in the number of anyons. It is
this manifold of states that constitutes the Hilbert space of our
microscopic model. To enumerate the states in the latter we
define a so-called ‘fusion chain’ as illustrated on the bottom
of Fig. 1. Here the original τ-particles constituting the chain
are denoted by the lines which are ‘incoming’ from above.
The links in the fusion chain carry labels {xi} which again
correspond to the particle types of the su(2)3theory. Reading
the labels from left to right a labeling is called admissible if
1 × τ = ττ × τ = 1 + τ .
(1)
x1 x2 x3
x0
...
τττττ
FIG. 1. A chain of Fibonacci or Yang-Lee anyons (denoted by the
τ’s in the upper row). The set of admissible labelings {xi} along
the fusion chain (lines) constitutes the Hilbert space of the Yang-Lee
(and Fibonacci) chains.

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3
at each vertex the fusion rules (1) of su(2)3are obeyed, i.e.
a τ label is followed by either a 1 or τ label, while a 1 la-
bel is always followed by a τ label. Every such admissible
labeling then constitutes one state in the Hilbert space of our
anyonic chain. Considering periodic boundary conditions, i.e.
xL= x0, it is straight forward to show that the dimension of
the Hilbert space is given in terms of Fibonacci numbers as
dimL= FibL−1+ FibL+1,
where Fibi denotes the i-th Fibonacci number, defined by
Fibi+1 = Fibi+ Fibi−1and the initial conditions Fib1 =
Fib2= 1.
We now proceed to the second step of our construction,
the derivation of a microscopic Hamiltonian. In doing so we
follow the perspective of the original ‘golden chain’ model23
in assuming that interactions between a pair of neighboring
τ particles – mediated, for instance, by topological charge
tunneling37–willresultinanenergysplittingofthetwopossi-
ble fusion outcomes in Eq. (1). Our Hamiltonian captures this
splitting by projecting the fusion outcome of two neighboring
τ particles onto the trivial fusion channel, i.e. assigning an
energy of E1= −1 to the fusion of two τ particles into the
trivial channel and an energy of Eτ= 0 to the fusion into the
τ channel. This anyonic Hamiltonian is thus reminiscent of
the common Heisenberg Hamiltonian for SU(2) spins, which,
for instance, projects two ordinary spin-1/2’s onto the singlet
channel and assigns a higher energy to the alternative triplet
channel.
To explicitly derive the Hamiltonian in the Hilbert space of
fusion chain labelings introduced above, we note that in this
basis the fusion of two neighboring τ particles is not explicit.
To get direct access to this fusion channel of two neighbor-
ing τ particles, we need to locally transform the basis as de-
picted in Fig. 2. The matrix describing this transformation
is typically called the F-symbol, which can be thought of as
an anyonic generalization of Wigner’s 6j-symbol for ordinary
SU(2) spins. Its general form (in the absence of fusion mul-
tiplicities) is given in Fig. 3.
Assuming that we know the explicit form of the F-symbols
(see the next section for more details), we can now explicitly
derive the microscopic Hamiltonian in the fusion chain basis.
After the basis transformation, the fusion channel of the two
neighboring anyons is manifest, so by means of a simple pro-
jection we can assign an energy to each of the fusion channels.
The final step left after this projection, is to transform back to
the original basis, which again employs the F-symbol.
To make the individual steps of this derivation more ex-
plicit, we consider the example of Fig. 2 in more detail. Let
us specify the five possible labelings of three neighboring fu-
sion chain labels xi−1,xi,xi+1, where in Fig. 2 we depicted
x1 x2 x3
x0
...
τττττ
x1
x3
x0
...
˜ x2
τττττ
F
FIG. 2. The F-symbol describing the local change of basis.
=
�
f
�
Fa,b,c
d
�e
f
a
bc
d
e
f
bc
ad
FIG. 3. The general form of the F-symbol.
the case where the site label is i = 2,
|xi−1,xi,xi+1? ∈
{|1,τ,1?,|1,τ,τ?,|τ,τ,1?,|τ,1,τ?,|τ,τ,τ?} .
After performing the basis transformation shown in Fig. 2, the
followinglabelssatisfy thefusionrulesateach vertexandthus
form the new basis
|xi−1, ˜ xi,xi+1? ∈
{|1,1,1?,|1,τ,τ?,|τ,τ,1?,|τ,1,τ?,|τ,τ,τ?} ,
where ˜ xiis the fusion channel of the two neighboring τ par-
ticles. In the transformed basis, we can project onto the trivial
channel, by means of a projection Pi,1, where the subscript i
denotes that we are acting on anyons i and i + 1, while the
label 1 denotes we are projecting onto the 1 channel. So, the
part of the Hamiltonian acting on anyons i and i + 1, which
we denote by Hi, acts on the Hilbert space as
Hi|xi−1,xi,xi+1? =
−
x?
?
i=1,τ
?
Fxi−1,τ,τ
xi+1
?xi
1
?
Fxi−1,τ,τ
xi+1
?1
x?
i
|xi−1,x?
i,xi+1? .
(2)
Here, we have used that for the su(2)kanyonic theories we are
considering, the F-symbols are their own inverses. Moreover,
we projected onto the 1 channel, which we favored, because
of the overall minus sign. The total Hamiltonian then simply
becomes the sum of (2) over all positions
H =
L
?
i=1
Hi,
(3)
where we assume periodic boundary conditions, i.e. xL= x0.
To describe the Hamiltonian of the various types of anyon
chains we consider in this paper, we only have to specify the
explicit form of the F-symbols (apart from the fusion rules,
which determine the Hilbert space). The explicit form of the
Hamiltonian then follows from Equation (2).
B. Galois conjugation and non-unitary models
NowthatwehaveexpressedtheHamiltonianintermsofthe
F-symbols, we should explain how to obtain the F-symbols
for a given anyon theory. As stated, the F-symbols transform

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4
Re q
Im q
q = e4πi/5
q = e2πi/5
Fibonacci anyons
Yang-Lee anyons
FIG. 4. The q-deformation parameters of Fibonacci and Yang-Lee
anyons correspond to different primitive roots of unity.
between two different fusion bases as illustrated in Fig. 3.
As such, the exact form of the these symbols can be deter-
mined self-consistently by identifying a circular sequence of
basis transformations, which yield a set of strongly overcon-
strained nonlinear equations called the ‘pentagon equations’
(for a more detailed exposition see, for instance, Refs. 31 and
38). While finding a solution to these pentagon equations is in
general a highly non-trivial task, it has been shown that they
allow only for a finite set of inequivalent solutions, a prop-
erty which goes under the name of ‘Ocneanu rigidity’, see for
instance Ref. 39. For the su(2)kanyonic theories of interest
here, the complete set of possible F-symbols can be found,
e.g., in Ref. 40 where they were obtained by using quantum
group techniques.
The different F-symbols are found to have a general form
that depends on a single, so-called ‘deformation parame-
ter’ q only. This deformation parameter has to be chosen
appropriately40and it turns out that for the su(2)kanyonic the-
ory it must be one of the (k + 2)ndprimitive roots of unity,
i.e. of the form
q = ep·2πi/(k+2),
(4)
where the integer index p runs from 1 ≤ p ≤ (k + 2)/2 (and
p and k + 2 are relative prime). The process of increasing
the index p by one, i.e. going from one root of unity to the
next, is what is usually referred to as Galois conjugation. For
our example theory, su(2)3, we can thus identify two possible
values for q, which are illustrated in Fig. 4. These two Ga-
lois conjugated theories corresponding to deformation param-
eters, q = e2πi/5and q = e4πi/5, then precisely correspond
to the cases of Fibonacci and Yang-Lee anyons, respectively.
The explicit form of the F-symbols and in particular the non-
diagonal 2×2 matrix for Fτ,τ,τ
of this deformation parameter q as

√
q−1+1+q
τ
can then be written40in terms
Fτ,τ,τ
τ
=

1
q−1+1+q
1
√
q−1−1+q
q−1+q
q−1+1+q
1

.
(5)
For Fibonacci anyons we set q = e2πi/5, in which case q−1+
1 + q = 1 + 2cos(2π/5) = (1 +√5)/2 = φ is the golden
ratio, and the F-symbol becomes the unitary matrix
?
FFibonacci=
φ−1
φ−1/2−φ−1
φ−1/2
?
.
(6)
The golden ratio, of course, is one solution of the equation
x2= 1+x, which is an algebraic analog of the fusion rule τ×
τ = 1+τ of the su(2)3anyonic theory. The process of taking
the Galois conjugate of the original Fibonacci anyon model
corresponds then simply to the substitution φ → −1/φ, where
−1/φistheothersolutiontotheequationx2= 1+x. Interms
of the deformation parameter q, this amounts to choosing the
other possible value of q = e4πi/5, which indeed yields q−1+
1+q1= 1+2cos(4π/5) = −1/φ. The F-symbol for Yang-
Lee anyons thus becomes the (invertible) non-unitary matrix
?
Having obtained the F-symbols in both the unitary as
well as the non-unitary case, we can now write down the
Hamiltonians for the Fibonacci and Yang-Lee chains. On
the states |xi−1,xi,xi+1? ∈ {|1,τ,1?,|1,τ,τ?,|τ,τ,1?}
both Hamiltonians act in the same diagonal way, Hi=
diag{−1,0,0}.
{|τ,1,τ?,|τ,τ,τ?}, the Hamiltonians take the following
forms41
?
Hi
Yang−Lee= −
FYang−Lee=
−φ
−iφ1/2
φ
−iφ1/2
?
.
(7)
Acting on the states |xi−1,xi,xi+1? ∈
Hi
Fibonacci= −
φ−2
φ−3/2
φ2
iφ3/2
φ−3/2
φ−1
iφ3/2
−φ
?
,
??
.
(8)
Before discussing these anyonic models in further detail,
we note that while Galois conjugation changes some aspects
of these models, i.e. the parameters in their respective Hamil-
tonians get ‘Galois conjugated’, this turns out to be a rather
mild change, since the underlying algebraic structure of these
models remains largely untouched. As a consequence, the
non-unitary Yang-Lee chains allow for an analytic solution
similar to their unitary counterparts as first obtained for the
‘golden chain’ model in Ref. 23. We will discuss the details
of this analytical solution and its resulting gapless theories in
the next Section.
C.Algebraic structure and analytical solution
In the original golden chain paper23, it was shown that the
Hamiltonian (3) based on the unitary F-symbols (6), can be
exactly solved. Following a similar sequence of steps as in the
unitary case we show here that all non-unitary models allow
for an exact analytical solution as well. We briefly outline
these steps in the following, for a more detailed discussion
see the references 23,31.
As a first step, it was noted that the operators Hi, i.e. the
summands of the Hamiltonian, form (upon suitable normal-
ization) a known representation42of the Temperley-Lieb al-
gebra43with “d-isotopy”- parameter d, namely ei= −dHi,
where d = φ. These operators satisfy the Temperley-Lieb
algeba
e2
i= d ei
eiei±1ei= ei
(9)
[ei,ej] = 0for |i − j| ≥ 2

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5
k
2
1
2
k − 1
2
k − 2
2
0
1
FIG. 5. The allowed ‘height’ configurations of the chain models.
This diagram is the Dynkin diagram of the Lie algebra Ak= su(k+
1). The Fibonacci and Yang-Lee anyons correspond to k = 3.
To show that this is the case, we will first map the Fibonacci
and Yang-Lee chains onto so-called restricted solid-on-solid
(RSOS) or ‘height’ models25. To do this, we first note that
one can relate the Fibonacci anyons to the anyonic theory
su(2)3, a theory with four particles (see section IIF for the
more general case of su(2)k), which can be labeled by their
’spin’ j = 0,1/2,1,3/2. The fusion rules of these particles
are given in table I. One immediately notices that the parti-
× 0 1/2
0 0 1/2
1/2
1
3/2
1
1
3/2
3/2
1
1/2
0
0 + 1 1/2 + 3/2
0 + 1
TABLE I. Fusion rules of the su(2)3theory.
cle with l = 1 has the same fusion rules as the Fibonacci
anyon. Indeed, the Fibonacci anyon model can be viewed as
the ‘integer subset’ of the su(2)3theory. In addition, one no-
tices that fusing a particle j with the particle 3/2, one finds
j × 3/2 = 3/2 − j, swapping integer ‘spin’ to half integer
‘spin’. This allows one to map the Fibonacci chain, consisting
of ‘spin’-1 particles, to a chain of ‘spin’-1/2 particles, by fus-
ing the labels x2i−1of the odd sites with the particle 3/2. In
this way, the new labels are constant labelings of a chain con-
sisting of ‘spin’-1/2 particles. Performing this map is advanta-
geous, because the transformed Fibonacci-chain can now di-
rectly be mapped onto a height model, in which the allowed
heights take the values 0,1/2,1,3/2, which can be seen as the
nodes of the Dynkin-diagram A4, which is given in Fig. 5.
The well-known Jones representation of the Temperley-
Lieb algebra, acting on the Hilbert space of the (transformed)
Fibonacci chain, can now be obtained by using the so-called
modular S-matrix.42Explicitly, this representation is known
to take on the following form:
ei|xi−1,xi,xi+1? =
?
x?
i
?
(ei)xi+1
xi−1
?x?
i
xi
??x?
?S0,xi
i−1,x?
i,x?
i+1
?
,
?
(ei)xi+1
xi−1
?x?
i
xi= δxi−1,x?
i−1δxi+1,x?
i+1
?S0,x?
i
?S0,xi−1
?S0,xi+1
(10)
where Sj1,j2is the modular S matrix of the su(2)3theory,
and the labels j1,j2are the ‘spins’ of the corresponding parti-
cles, which in general take the values 0,1/2,1,...,k/2, and
in terms of these, the S-matrix elements read
?
In equation (10), the row-label j1of the S-matrix elements
whichappearsheretakesonthevaluej1= 0, whichisthecor-
rect value in case of the (unitary) Fibonacci chain. In taking
the Galois conjugate, one has to replace this row-label j1= 0
by the label j1 = 1 (corresponding to a τ particle [see also
section IIF]).
The action of the ei’s on the height states, precisely cor-
responds to the action of the part of the Hamiltonian acting
on the ket |xi−1,xi,xi+1?, which shows that the operators of
Hamiltonian of the Fibonacci chain do indeed form a repre-
sentation of the Temperley-Lieb algebra.
We will now employ this observation, and map the Fi-
bonacci chain to an integrable, two-dimensional statistical
mechanics model, namely an RSOS model, which in the
present case is based on the Dynkin diagram A4– see Fig.
5. In an RSOS model, the degrees of freedom are the heights
(or nodes of the Dynkin diagram). These heights live on the
vertices of the lattice, with the constraint that heights of neigh-
boring vertices correspond to nodes of the graph which are
linked.
The two-row transfer matrix of the RSOS-model (depicted
in figure 6) can be written in terms of the plaquette weights of
the square lattice, namely T = T2T1, with
?
where the plaquette weights are of the form
?sin(pπ

j?=i
??
x�
2i−1
x�
2i
Sj1,j2=
2
k + 2sin
?(2j1+ 1)(2j2+ 1)π
k + 2
?
.
(11)
T1=
i
W[2i]T2=
?
i
W[2i + 1] ,
(12)
W[i]? x?
? x=
k+2− u)
sin(pπ
k+2)
1? x?
? x+
sin(u)
sin(pπ
k+2)e[i]? x?
? x
?
(13)
e[i]? x?
? x=
?
jδx?
δx?
j,xj


the identity operator.
?
(ei)xi+1
xi−1
?x?
i
xi
(14)
with 1? x?
? x=
j,xj
?
W[2i + 1]
x2i−1
x2i
x2i+1
x�
2i+1
W[2i]
FIG. 6. The two-row transfer matrix of the RSOS models.
The new feature of the non-unitary models consists in the
fact that equations (13) and (14) for the plaquette weight in
the transfer matrix now involve a general integer p = 2j1+ 1
labeling the primitive root of unity (4) beyond the unitary case

Page 6

6
of p = 1 (corresponding to j1 = 0) studied earlier. For a
general value of p, the Hamiltonian associated with this lattice
model can be obtained by taking the extreme anisotropic limit,
namely u → 0+(see for instance 44). One has,
T = exp?−a(H + c1) + O(a2)?
with a =
dsin(
which indeed gives that the Hamiltonian is of the form H =
−1
RSOS models.
We can now use the known results about the phases of
the RSOS models, to find the behavior of the chain models.
In the case of the Fibonacci anyons (i.e., when k = 3 and
p = 1, which appear in the plaquette weights W and the pa-
rameter a), the anti-ferromagnetic golden chain Hamiltonian
can be obtained by taking the limit u → 0 with u positive.
The corresponding (exactly integrable) RSOS model25is crit-
ical and known25,45to be in the universality class of the tri-
critical Ising model described by the minimal model M(4,5)
of conformal field theory. For the ferromagnetic golden chain,
which is obtained from the RSOS models with 0 < |u| ? 1
and u negative, the critical behavior then turns out25,45to be
in the universality class of the Z3parafermions corresponding
to the minimal model variant46 ?
chains (corresponding to k = 3 and p = 2), the same se-
quence of steps as above results in a mapping to another fam-
ily of exactly integrable RSOS models24and in this case the
gapless theories turn out be non-unitary minimal models47,48.
Inthisnon-unitarycase, theHamiltoniancanbeobtainedfrom
the two-row transfer matrices, but in this case, both the anti-
ferromagneticaswellastheferromagneticchainsareobtained
in the limit u → 0 with positive u. The difference in the
Hamiltonians stems from the sign-changes in the isotopy pa-
rameter d, namely from positive (but not necessarily bigger
than one) in the anti-ferromagnetic case, to negative in the
ferromagnetic case. The critical behavior in these two cases is
described by the (non-unitary) minimal models M(3,5) and
M(2,5) respectively.
,
(15)
u
pπ
k+2)? 1 and c1an unimportant constant,
d
?
iei, establishing the mapping from the chain to the
M(5,6).
In the case of the Galois-conjugated Yang-Lee anyonic
D.Numerical results
We have numerically studied the excitation spectra of the
Yang-Lee chains by exact diagonalization of systems with up
to L = 32 anyons, typically using periodic boundary condi-
tions. These excitation spectra not only allow for an indepen-
dent identification of the conformal field theory describing the
gapless collective state, as discussed in the previous section,
but also reveal further details about the correspondence be-
tween continuous fields and microscopic observables. In par-
ticular, the low-energy states of a conformally invariant sys-
tem can be identified with conformal fields and the excitation
spectrum is expected to take the form
E = E1L +2πv
L
·
?
−c
12+ h +¯h
?
,
(16)
0π / 4 π / 23π / 4
π
momentum K
00
11
22
33
44
55
rescaled energy E(K)
0
3/2
-1/10
2/5
1-1/10
2-1/10
2-1/10
2+0
1+2/5
2+2/5
3-1/10
1+3/2
σ τ-flux
no flux I
τ-flux ε
ψ no flux
L = 32
primary fields
descendants
FIG. 7.
Yang-Lee chain. The spectrum matches the non-unitary minimal
model M(3,5) with central charge c = −3/5, which is often re-
ferred to as ‘Gaffnian’ theory. Primary fields I,σ,?,ψ of this con-
formal field theory are indicated by squares, descendant fields by
circles. We also indicate the ‘topological flux’ of each energy eigen-
state, which indicates the topological symmetry sector.
Conformal excitation spectrum of the ‘antiferromagnetic’
where h and¯h are the (holomorphic and anti-holomorphic)
conformal weights of a given CFT with central charge c. E1
is a non-universal number, v a non-universal scale factor, and
L the length of the chain. To match the excitation spectra of
the Yang-Lee chains to these CFT predictions we consider the
family of so-called minimal models M(p,p?) (where p and p?
are mutually prime) with central charge
c = 1 −6(p − p?)2
pp?
,
and conformal weights
h(r,s) =(rp − sp?)2− (p − p?)2
4pp?
,
(17)
where the indices r and s are limited to 1 ≤ r < p?and
1 ≤ s < p. We note that the labels (r,s) and (p?− r,p − s)
correspond to the same field.
In the following, we will discuss our numerically obtained
excitation spectra for ‘antiferromagnetic’ and ‘ferromagnetic’
couplings, which are plotted in Figs. 7 and 8, respectively.
The antiferromagnetic chain.–
ferromagnetic’ chain, for which the pairwise anyon-anyon in-
teraction energetically favors the trivial fusion channel
We first turn to the ‘anti-
τ × τ → 1.
The conformal field theory describing the critical behavior of
this model is the non-unitary minimal model M(3,5) with
central charge c = −3/5, which is also referred to as the
‘Gaffnian’ theory17. The four primary fields of this CFT and
their respective scaling dimensions ∆ = h +¯h are
σI?ψ
∆ -1/10 0 2/5 3/2
(18)

Page 7

7
with the non-trivial fusion rules
σ × σ = I + ?σ × ? = σ + ψ
? × ? = I + ?
σ × ψ = ?
? × ψ = σ
ψ × ψ = I
For completeness, we give the conformal dimensions of the
fields (with minimal model labeling) in table II, and note that
this model is a particular Galois conjugate of the su(2)3CFT.
To identify the gapless theory numerically, we typically
perform the following procedure: We first look at the two
lowest energy eigenvalues in the spectrum, E0and E1, and
by identifying the energy gap ∆E = E1− E0with the dif-
ference of the two lowest scaling dimensions we can identify
the non-universal scale factor 2πv/L in (16), which we subse-
quently set to 1 thereby rescaling the entire energy spectrum.
This identification of the two lowest energy eigenvalues with
conformal operators also allows to identify an overall energy
shift, e.g. setting the energy of the trivial operator I with scal-
ing dimension h(1,1) +¯h(1,1) = 0 to zero. In the case at
hand, there is only one negative scaling dimension, so the low-
estenergycorrespondsto−2hmin= −1/10, whilethesecond
lowest state corresponds to the identity operator, with zero en-
ergy. At this point, all the energies are fixed, and indeed the
rescaled and shifted numerical spectrum is found to reproduce
the position of the (other) primary fields (indicated by green
squares in Fig. 7), as well as the descendants (indicated by red
circles in Fig. 7).
The ferromagnetic chain.–
netic’ chain, for which the pairwise anyon-anyon interaction
energetically favors the τ-fusion channel
We now turn to the ‘ferromag-
τ × τ → τ .
The critical theory is the non-unitary minimal model M(2,5)
with central charge c = −22/5, which is commonly referred
to as the ‘Yang-Lee’ theory49,50. The two primary fields of
this CFT and their respective scaling dimensions ∆ = h +¯h
are
I?
∆ 0 −2/5
(19)
with the non-trivial fusion rule
? × ? = I + ?
Again we give the conformal dimensions of the fields (with
minimal model labeling) in table II for completeness.
M(3,5)
h(r,s) s = 1
r = 1
3
2
03/4
-1/20 1/5
M(2,5)
h(r,s) s = 1
r = 1
3 -1/5
0
TABLE II. Kac table of conformal weights for the non-unitary mini-
mal model M(3,5) and M(2,5). We only displayed fields with odd
r labels, to avoid duplicates.
0 π / 4 π / 2 3π / 4
π
momentum K
00
22
44
66
88
1010
1212
rescaled energy E(K)
0
-2/5
1-2/5
I no flux
ε τ-flux
2-2/5
2-2/5
2+0
4+0
4-2/5
4+0
4-2/5
3-2/5
4-2/5
3+0
3-2/5
5-2/5
5-2/5
6-2/5 6-2/5
6+0
5+0
6+0
6-2/5
7-2/5
7+0
5-2/5
5+0
6-2/5
6+0
7+0
7-2/57-2/5
8+0
8-2/5
8-2/5
8+0
8-2/5
9+0
9-2/5 9-2/5
9+0
8-2/5
8+0
10-2/5
10+0
10+0
10-2/5
11-2/5
11+0
L = 32
primary fields
descendants
FIG. 8. Conformal excitation spectrum of the ‘ferromagnetic’ Yang-
Lee chain. The spectrum matches the non-unitary minimal model
M(2,5) with central charge c = −22/5, which is commonly re-
ferred to as ‘Yang-Lee’ theory. The primary fields I,? of this con-
formal field theory are indicated by squares, descendant fields by
circles. We also indicate the ‘topological flux’ of each energy eigen-
state, which indicates the topological symmetry sector.
The spectrum of this theory, after the appropriate shift and
rescaling of the energy is displayed in figure 8, which beauti-
fully reproduces the primary fields, as well as descendants to
a high level, constituting the spectrum of the Yang-Lee model.
E. Topological symmetry
Before considering further generalizations of the Yang-
Lee chains we mention another peculiarity of these anyonic
chains. Like their unitary counterparts the Yang-Lee chains
exhibit an unusual, non-local symmetry.
which was dubbed a ‘topological symmetry’ in the context
of the golden chain model23, corresponds to the operation of
commuting a τ particle through all particles of the chain. The
so-defined operator, which we denote by Y , is found to com-
mute with the Hamiltonian and for the su(2)3theory has two
distinct eigenvalues, thus defining two symmetry sectors. Its
matrix form is given by
This symmetry,
?x?
0,...,x?
L−1|Y |x0,...,xL−1? =
L−1
?
i=0
?
Fτxiτ
x?
i+1
?x?
i
xi+1
.
(20)
This definition solely in terms of the F-symbols immediately
suggests a generalization of this symmetry to the case of the
non-unitary Yang-Lee models studied above by simply replac-
ingtheF-symbolswiththeirnon-unitarycounterparts(7). For
both the unitary and non-unitary variants the two eigenvalues
of the respective topological symmetry operator are y1 = φ
and y2 = −1/φ. In the unitary case these are identified as
no-flux / τ-flux symmetry sectors. This assignment is simply
reversed in the Galois conjugated, non-unitary case.
For the unitary models it has been demonstrated that this

Page 8

8
topological symmetry protects the gapless ground state of the
interacting anyon chain model34: it was shown that all rel-
evant operators (in a renormalization group sense) that have
otherwise identical quantum numbers as the ground state, e.g.
the same momentum, fall into different topological symme-
try sectors. We have performed a similar symmetry analysis
for the Yang-Lee chains at hand. For chains with either anti-
ferromagnetic or ferromagnetic couplings, we have evaluated
the eigenvalue of the topological symmetry Y for all eigen-
vectors of the Hamiltonian and thereby assigned topological
symmetry sectors to the primary fields of the conformal field
theory describing their energy spectrum. These assignments
are given in Figs. 7 and 8 for antiferromagnetic and ferromag-
netic chain couplings, respectively. A situation similar to the
unitary models emerges: For both signs of the coupling the
ground state with conformal dimension h+¯h < 0 is found to
be in the topologically non-trivial (or τ-flux) sector, while all
other primary fields with the same momentum are found to be
in the topologically trivial (or no-flux) sector.
For the unitary anyon chains this topological protection
mechanism has subsequently been cast in a broader physi-
cal picture34interpreting the gapless modes of the anyonic
chains as edge states at the spatial interface of two topolog-
ical liquids, and the conclusion that anyon-anyon interactions
result in a splitting of the topological degeneracy for a set of
non-Abelian anyons and the nucleation of distinct topologi-
cal liquid within the parent liquid of which the anyons are
quasiparticle excitations34,35. The similarity of our results for
the topological symmetry properties of the non-unitary anyon
chains thus raises the question whether a similar interpretation
would also hold for the non-unitary systems at hand.
F.su(2)ktheories and generalized Galois conjugation
While we have focused our discussion in the preceding
sections on the Fibonacci / Yang-Lee anyonic theories for
which k = 3, our results can be readily generalized to a
much larger class of anyonic theories, so called su(2)kChern-
Simons theories. These theories, which are certain quantum
deformations36of SU(2), have a finite number of representa-
tions which are identified with the (anyonic) particle content
of the theory. For a given (integer) deformation parameter
k these representations/particles are labeled by a generalized
angular momentum
j = 0,1
2,1,...,k − 1
2
,k
2.
The fusion rules for combining two of these momenta (or their
associated particles) then resemble the tensor product rules of
ordinary SU(2) angular momenta, and explicitly read
j1× j2=
min(j1+j2,k−j1−j2)
?
j3=|j1−j2|
j3.
(21)
For all k ≤ 2 the first non-trivial fusion rule (i.e. a fusion rule
with multiple fusion outcomes) is encountered for the particle
carrying generalized angular momentum j = 1/2, for which
one finds
1
2×1
2= 0 + 1.
(22)
Upon further inspection of the fusion rules (see Table I for
an example) one notes that for anyonic theories with odd k
one can map all half-integer representations onto integer rep-
resentations byfusing thehalf-integer representationswith the
highest representation of the theory, the so-called ‘simple cur-
rent’ with angular momentumk
identifies the fusion rules of the particle carrying angular mo-
mentum j = 1/2 with those of the particle carrying angular
momentum j = (k − 1)/2 and Eq. (22) becomes
k − 1
22
2. In particular, this mapping
×k − 1
= 0 + 1.
(23)
Note that for odd k the fusion rules are now given entirely
in terms of integer representations51. For the su(2)k=3theory
thismappingthusturnsthefusionrulesofthej = 1/2particle
into those of the j = 1 particle, i.e. 1 × 1 = 0 + 1, which
is the momentum representation of the non-trivial fusion rule
in Eq. (1). One can thus equally identify the τ particle of
the Fibonacci / Yang-Lee theories with both the j = 1 and
j = 1/2 angular momenta in su(2)3.
The su(2)k generalizations of the Fibonacci / Yang-Lee
chains, which we want to consider in the following, are chains
of particles carrying angular momentum j = 1/2, which we
can alternatively identify with particles carrying angular mo-
mentum j = (k −1)/2. The microscopic chain model is then
constructed in an identical way to what we outlined for the
case of the Fibonacci / Yang-Lee theories in Section II, i.e.
we first build a Hilbert space by considering admissible label-
ings of the fusion chain basis for an incoming j = 1/2 (or
j = (k − 1)/2) particle and then derive a Hamiltonian via
the local projection onto one of the two fusion outcomes in
Eqs. (22) or (23), respectively. Our notion for the sign of the
interaction also remains unchanged, with ‘antiferromagnetic’
interactions favoring fusion of neighboring particles into a to-
tal j = 0 state and ‘ferromagnetic’ interactions favoring fu-
sion into a total j = 1 state.
Unitary models.– For the unitary case, such su(2)kgener-
alizations of the golden chain model were first discussed in
Ref. 23 and further explored in Ref. 34. Like their su(2)3
counterpart these generalized anyon chains are found to be
exactly solvable and their gapless collective states can be de-
scribedintermsofconformalfieldtheories. Tobespecific, the
critical theory for the generalized antiferromagnetic chains is
known to be the minimal model M(k+1,k+2) with central
charge c = 1 − 6/((k + 1)(k + 2)), while for ferromagnetic
chains it is known to be the Zkparafermion theory with cen-
tral charge c = 2(k − 1)/(k + 2). In both cases, the terms in
the Hamiltonian are found to form a Temperley-Lieb algebra
with isotopy parameter d = 2cos(π/(k + 2)).
Non-unitary models.–In turning to the Galois conjugated
models of these unitary su(2)kanyon chains, we first recall
thatforagivensu(2)kanyonictheorythereareseveralallowed

Page 9

9
su(2)kchains
central charge
c = 1 −
c = 2k−1
coupling
AFM
CFT
d-isotopy
?
2cos
M(k + 1,k + 2)
Zkparafermion
6
(k+1)(k+2)
2cos
π
k+2
?
FM
k+2
?
π
k+2
?
Yang-Lee chains
central charge
c = 1 −
c = 1 −
coupling
AFM
CFT
d-isotopy
?
2cos
M(k,k + 2)
M(2,k + 2)
Generalized Galois conjugates with 2 ≤ p < (k + 2)/2
CFTcentral charge
M(k + 2 − p,k + 2) c = 1 −
M(p,k + 2)
24
k(k+2)
3k2
k+2
2cos
2π
k+2
?
FM
?
kπ
k+2
?
coupling
AFM
d-isotopy
?
2cos
6p2
(k+2−p)(k+2)
6(k+2−p)2
p(k+2)
2cos
pπ
k+2
?
FM
c = 1 −
?
(k+2−p)π
k+2
?
TABLE III. Overview of the gapless theories for the various chain models.
q deformation parameters (see also Sec. IIB)
q = ep·2πi/(k+2)
(24)
as illustrated in Fig. 9. For each integer index 1 ≤ p ≤
(k+2)/2 (and the further restriction that p and k+2 are rela-
tive prime) the corresponding q deformation parameter estab-
lishes a solution of the pentagon equations of the underlying
su(2)ktheory and defines the explicit form of the F-symbols.
The unitary su(2)kmodels discussed above all correspond to
the first primitive root, i.e. p = 1, in this sequence. The
other primitive roots (with p ≥ 2) then give rise to the Ga-
lois conjugated models with non-unitary F-symbols and non-
Hermitian microscopic Hamiltonians constructed using these
F-symbols. In particular, we note that for k = 2 there is
no such Galois conjugated theory, for k = 3 there is exactly
one Galois conjugated theory (the Yang-Lee theory discussed
above), and for k ≥ 4 there are in general multiple Galois
conjugated theories.
Like their unitary counterparts all Galois conjugated su(2)k
chain models permit an analytical solution. Since the line
of argument closely follows the one given in Section IIC for
k = 3, we will be brief here, and merely state the main argu-
Re q
Im q
q = e2πi/11
q = e4πi/11
q = e6πi/11
q = e8πi/11
q = e10πi/11
FIG. 9. The q-deformation parameters for a general su(2)k theory
correspond to different primitive roots of unity. Only the first primi-
tive root in this sequence, e.g. q = e2πi/11in the illustrated example
of k = 9, gives rise to a unitary model.
mentsandresults. Inordertosimplifythepresentationwewill
further focus our arguments to anyonic theories with odd k
which allows to restrict the discussion to those which possess
only integer (not half-integer) values of “angular momenta” j.
(Even values of k work in an analgous manner.) The Galois
conjugated su(2)kchain models can then be directly mapped
onto the height representations of certain restricted solid on
solidmodelswheretheheightstakethevalues0,1/2,...,k/2
in the Dynkin diagram Akillustrated in Fig. 5. The Hamilto-
nian terms acting in this height representation then again form
a Temperley-Lieb algebra42with an isotopy parameter which
now turns out to depend on the sign of the interactions. For
antiferromagnetic coupling this generalized isotopy parameter
is found to be d = 2cos(pπ/(k + 2)) and the conformal the-
ory describing gapless interacting system is the non-unitary
minimal model M(k + 2 − p,k + 2) with central charge52
c = 1 −
topy parameter becomes d = 2cos
lesstheoryisthenon-unitaryminimalmodelM(p,k+2)with
central charge c = 1 −6(k+2−p)2
conjugated models, the central charges are related to the quan-
tum dimensions in the following way42,53,54Writing the quan-
tum dimension as d = 2cos(pπ/(k + 2)) = 2cos(π/δ)
(where δ is a fraction for the Galois-conjugated models), one
can obtain the central charge via c = 1 −
this is also the correct result for the anti-ferromagnetic unitary
chain with p = 1, but the central charge for the ferromagnetic
unitary chains is given by c = 2 − 6/δ.
6p2
(k+2)(k+2−p). For ferromagnetic couplings the iso-
?
(k+2−p)π
k+2
?
and the gap-
p(k+2). In the case of the Galois-
6
δ(δ−1). Note that
Our results for unitary and non-unitary models are summa-
rized in Table III. We note that for p = 2 and k = 3 we exactly
reproduce the results of the Yang-Lee chains discussed above.

Page 10

10
III.DOUBLED YANG-LEE MODELS
A. The ladder Hamiltonian
In this second part of the manuscript, we turn to ‘quan-
tum double’ variants of the anyonic chains discussed in the
first part. The unitary incarnations of these quantum double
models have been introduced in the context of exotic quan-
tum phase transitions in time-reversal invariant systems that
are driven by topology fluctuations28. The specific model is
defined on a ladder geometry, shown in figure 10.
The Hamiltonian
?
consists of two competing terms.
The first term favors the trivial label 1 on each rung of the
ladder, while the second term favors the no-flux state for all
plaquettes. As shown in Ref. 28, the projector onto the flux
through a square plaquette can be expressed in terms of the
unitary/non-unitary F-matrices (6)/(7). This termis equivalent
to the plaquette term in the Levin-Wen models55, which are
defined on a different lattice, the honeycomb lattice.
Explicitly, the plaquette term reads
Hladder= −Jr
rungs r
δl(r),1− Jp
?
plaquettes p
δφ(p),1
(25)
δφ(p),1
?????
α
β
γ
δ
a
b
c
d
?
=
?
s=1,τ
ds
D2
?
α?,β?,γ?,δ?
?
Fα?sδ
a
?δ?
α
?
α�
Fβ?sα
b
?α?
β
×
?
Fγ?sβ
c
?β?
γ
?
Fδ?sγ
d
?γ?
δ
?????
a
b
c
d
β�
γ�
δ�
?
,
(26)
where dsdenotes the quantum dimension of particle type s,
i.e. d1 = 1 and dτ = φ. D denotes the total quantum di-
mension, D =
as Yang-Lee) anyons. The latin and greek labels denote the
?d2
1+ d2
τ=√2 + φ for Fibonacci (as well
FIG. 10. The geometry of the Fibonacci and Yang-Lee ladders. To
discuss the physics, it is enlightening to ‘thicken’ the ladder, and
consider the two dimensional surface thus obtained.
solvable point
(c = 8/35)
gapped phase
gapped phase
Jp
Jr
solvable point
(c = -44/5)
critical phase
c = -44/5
FIG. 11. Phase diagram of the doubled Yang-Lee chain.
degrees of freedom, and any of these takes one of the values
{1,τ}. We note that the Hilbert space of the ladder mod-
els consists of all possible labelings of the rungs and the legs,
such that at each vertex, the Fibonacci fusion rules are obeyed.
With this description of the ladder models, we can easily go
back and forth between the Fibonacci anyon ladder, and the
Yang-Lee anyon ladder, simply by choosing the corresonding
set of F-symbols, namely equations (6) or (7) respectively.
B.The phase diagram
To discuss the phase diagram of both the original and Ga-
lois conjugated model, shown in figure 11, we parametrize the
couplings Jr= sinθ and Jp= cosθ in terms of an angle θ.
We start with the first gapped phase, for π/4 < θ < π, most
easily discussed at the special point θ = π/2, or Jr= 1 and
Jp = 0. In the ground state at this point, all rungs have the
trivial label. In the two-dimensional surface geometry, this
means that no τ-fluxes go through the rungs, implying that
the rungs can be completely pinched off, changing the geom-
etry to that of two, disconnected cylinders, one for each leg of
the ladder. Each of the cylinders accommodates two ground
states, either with or without τ-flux, leading to a four-fold
ground state degeneracy. The lowest (gapped) excited state
consists of configurations where one rung of the ladder ‘con-
tains’ a τ flux.
In the other extended gapped phase, for −π/2 < θ < π/4,
let us discuss the special point θ = 0, or Jp= 1 and Jr= 0.
Here the situation is reversed, and no τ-fluxes go through the
plaquettes. Thus they can be closed of, giving rise to a geome-
try consisting of a single cylinder. Again, this cylinder can ac-
commodate two ground states, with or without flux, resulting
in a two-fold degenerate ground state. The lowest (gapped)

Page 11

11
excited state consists of configurations with a single τ-flux
going through a plaquette, effectively piercing a hole through
the cylinder.
Precisely at the point where both couplings are equal in
strength, the gap closes, and the system is critical. At this
point, the geometry is fluctuating at all length scales, interpo-
lating between the two extremes of having one or two cylin-
ders, respectively. In addition, also precisely at this point, the
(critical) model is exactly solvable, as explained in the next
section.
Finally, for π < θ < 3π/2, there is an extended critical re-
gion, which is characterized by another exactly solvable point,
atθ = 5π/4, wherebothcouplingsareagainofequalstrength,
but negative.
C.Analytical solution
The analytic solution of the Fibonacci ladder model at the
two exactly solvable points has been described in detail in (the
supplementary material of) reference 28. Thus, we will be
rather brief here, and point out some of the crucial steps of the
mapping of the ladder model onto an exactly solved (restricted
solid-on-solid) model. We will then quickly mention in which
way this solution of the unitary Fibonacci case is changed, if
one takes the Galois conjugate, to go to the case of the Yang-
Lee ladder model. The most important step in the analytic
solution of the model, is to perform a basis transformation on
the Hilbert space. After this transformation, the model can
be mapped onto a restricted solid-on-solid model, where in
this case, the allowed height variables lie on the vertices of
the Dynkin diagram of the Lie algebra D6, as it will turn out.
This model is exactly solvable, both in the original, as well as
the Galois conjugated case.
The basis transformation: At each rung, one applies an F-transformation, changing the basis building blocks of the ladder:
=
�
c2,c4
�Fb1,a1,a3
b3
�c2
d2
�Fb3,a3,a5
b5
�c4
d4
b1
b3
b5
a1
a3
a5
b1
b3
b5
a1
a3
a5
c2
c4
d4
d2
FIG. 12. Basis transformation of the ladder model.
Thus, we can express the Hilbert space in terms of the vari-
ables {a2i−1,b2i−1,c2i}, or, equivalently, in terms of the vari-
ables {a2i−1,b2i−1,d2i}, as displayed in figure 12. The map-
ping to the restricted solid-on-solid model is performed in
terms of the latter variables. In particular, one can determine
which ‘values’ the links can take on the even and odd sublat-
tices. The variables on the even sublattice d2ican take the
values {1,τ}, while the combined variables (a2i−1,b2i−1)
can take the values {(1,1),(1,τ),(τ,1),(τ,τ)}, but there
are constraints on which values of (a2i−1,b2i−1) and d2ican
occur next to each other. From the fusion rules, it easily fol-
lows that the combination of variables which are consistent
with one another, have to be adjacent on the ‘height graph’, as
shown in Fig. 13.
(1,1)
1
(τ,τ)
(1,τ)
(τ,1)
τ
FIG. 13. The allowed ‘height’ configurations of the ladder models.
This diagram is the Dynkin diagram of the Lie algebra D6 = so(12).
From now on, we will call the allowed variables (or pairs
of variables on the odd sublattice), heights. The Hilbert space
of the ladder model consists of all height configurations on a
chain of length L, in such a way that two heights on neigh-
boring sites also have to be neighbors (i.e., be connected by
a solid line) in the height diagram of Fig. 13. In this way,
the model can be mapped on the D6version of the restricted
solid on solid model, and precisely when Jr = Jp = ±1,
the model is critical, and can be solved exactly. We will not
go into the details here (which can be found in Ref. 28), but
we will quickly state the result for this critical behavior in
the original, Fibonacci anyon setting, followed by the results
on the Galois conjugated Yang-Lee ladder. The details of the
phase diagram, which closely mimics the phase diagram of
the unitary (‘not Galois-conjugated’) ladder, and the numeri-
cal results will be presented in the next subsection.
As was the case for the chain models, also in the ladder case
do the parts of the Hamiltonian form a representation of the
Temperley-Lieb algebra, but in this case, the algebra is that42
basedontheDynkindiagramD6. Thepossibleassociatedval-
ues of the d-isotopy parameter are given by d = 2cos(mjπ
where h = 10, and the mjare the exponents of the Lie algebra
D6, namely mj= 1,3,5,7,9,5. In the original ladder model
mj= 1, for both cases Jr= Jp= ±1. The associated value
of the d-isotopy reads d = 2cos(π/10) =√2 + φ. In the an-
tiferromagnetic case, i.e. Jr= Jp= 1, or in the parametriza-
h),

Page 12

12
Fibonacci ladder
CFT
M(9,10)
coupling
AFM (θ = π/4)
FM (θ = 5π/4) Z8parafermion
central charge d-isotopy
c = 14/15
c = 7/5
2cos?π
10
?
2cos?π
10
?
Yang-Lee ladder
CFT
?
coupling
AFM (θ = π/4)
FM (θ = 5π/4)
central charge d-isotopy
c = 8/35
c = −44/5
M(7,10)
?
2cos?3π
10
?
M(3,10)2cos?7π
10
?
TABLE IV. Gapless theories for the ladder models.
tion Jp= cosθ, Jr = sinθ, the point θ = π/4, the critical
behavior is described by the minimal model?
that we are considering the (A8,D6) modular invariant56,57,
instead of the ‘usual’ (diagonal) (A8,A9) invariant (we note
that the (A8,D6) invariant minimal model?
denotes the Wess-Zumino-Witten CFT at level k based on the
Lie-algebra G2.
We will now switch our attention to the Galois conjugated
version of the ladder model. The phase diagram of the Yang-
Lee ladder is depicted in figure 11, and the location of the
gapped phases, as well as the critical behavior is exactly the
same as in the Fibonacci ladder model. Therefore, we will
be somewhat brief in the description of these phases (see
the introduction of this section for a brief description of the
gapped phases), and focus on the differences, which occur at
the exactly solvable points, namely θ = π/4,5π/4, where
the model is critical, but with a different critical behavior in
comparison to the Fibonacci ladder.
The antiferromagnetic (θ = π/4) critical point.–
the case in the Yang-Lee chains, the critical behavior of the
Galois conjugated model is obtained by picking a different
d-isotopy parameter, corresponding to the Galois conjugate
under consideration.For Fibonacci anyons, there is only
one Galois conjugate, but nevertheless, there seem to be dif-
ferent possible values for the d-isotopy parameter, namely
d = 2cos(mjπ
fact, the value we need is mj= 3, which corresponds to the
value obtained from the Fibonacci case, under Galois conju-
gation, namely d =?2 − 1/φ = 2cos(3π/10).
It is a simple matter to check that this value corresponds to
a model with central charge c = 8/35, by using the same ar-
guments as we gave in section IIF. This central charge corre-
sponds to the central charge of the non-unitary minimal model
M(7,10). Just as in the unitary case28, the critical CFT de-
scribing the Yang-Lee model at θ = π/4, is not the usual (‘di-
agonal’) modular invariant, but rather the (A,D) invariant, or,
to be more precise, (A6,D6). See57for information on the
different modular invariants of the minimal models. We will
denote this model by?
trum of this model will be discussed in the section IIID below.
M(9,10), which
has central charge c = 14/15. With the tilde, we denote
M(9,10) is equiv-
alent to the coset model (G2)1×(G2)1/(G2)2), where (G2)k
As was
h), with h = 10, and mj = 1,3,5,7,9,5. In
M(7,10). The field content of these
models is given in tables V and VI below. Details of the spec-
M(7,10)
h(r,s) s = 1
r = 1
2
3
4
5
6
7
8 301/40 667/280
910
35
013/7
247/280 1287/280
9/35
−1/56
2/35
27/56
44/35
46/7
1/40
2/5
9/8
11/5
29/8
27/5
104/35
95/56
27/35
11/56
−1/35
27/280
4/727/7
M(3,10)
h(r,s)
r = 1
2 −11/40
3
4
5
6
7
8 49/40
9
s = 1
0
−2/5
−3/8
−1/5
1/8
3/5
2
TABLE V. Kac table for the non-unitary minimal models M(7,10)
and M(3,10). Only odd values of r are displayed, so that each field
appears only once.
Theferromagnetic(θ = 5π/4)criticalpoint.–
unitary case, we can obtain the critical theory for the opposite
sign of the interaction, by simply swapping the sign of the
d-isotopy parameter, which hence corresponds to the value
d = 2cos(7π/10) = −?2 − 1/φ, from which it is a sim-
c = −44/5, corresponding to the minimal model M(3,10).
This minimal model also comes in two incarnations, one cor-
responding to the modular invariant (A2,A9), the other to
(A2,D6). (The field content of these models is also given
in tables V and VI below.) The latter is realized in the Yang-
Lee ladder at θ = 5π/4, and we will describe the spectrum
in more detail below. Before we do that, however, we will
first summarize the critical theories of both the Fibonacci and
Yang-Lee anyon ladders in table IV.
Inthenon-
ple matter to extract the central charge, which is given by
?
0
M(7,10)
h(r,s) s = 1
r = 1
3 2/5
5 11/5 2/35 27/35
35
13/7
9/35 −1/35
4/7
?
M(3,10)
h(r,s) s = 1
r = 1
3 −2/5
5 −1/5
0
TABLEVI.Kactableforthe‘(A,D)-modularinvariant’non-unitary
minimal models?
M(7,10) and?
M(3,10). The fields with the con-
formal dimensions in bold (i.e. those with label s = 5) appear twice.
D.Numerical results
We finally present the numerical spectra of the conjugated
ladder model at the two critical points discussed in the previ-
ous section. The spectrum for the critical point at θ = π/4, is
given in Fig. 14, where we indicated the locations of the pri-
mary fields of?
usual, there are only two free parameters to match the numer-
ically obtained spectrum with the result obtained from con-
formal field theory, so the fact that the six lowest primaries, as
M(7,10) (given in table VI) by green squares,
as well as some low-lying descendants with red circles. As

Page 13

13
0π / 2
π
3π / 22π
momentum K
00
11
22
33
44
rescaled energy E(K)
L = 12, ky = 0
L = 12, ky = π
primary fields
descendant fields
-2/35
4/35
0
18/35
54/35
2-2/35
2+4/35
2+18/35
1-2/35
1+4/35
1+18/35
4/5
8/7
1+4/5
1+8/7
2+4/5
-2/35
0
4/35
18/35
4/5
8/7
54/35
1+4/35
1-2/35
1+18/35
1+4/5
1+8/7
2-2/35
2+4/35
2+18/35
2+4/5
FIG. 14.
doubled ‘antiferromagnetic’ Yang-Lee chain. The spectrum matches
the non-unitary minimal model?
green squares, descendant fields by red circles.
Conformal energy spectrum of the critical points in the
M(7,10) with central charge c =
8/35. Primary fields of the conformal field theory are indicated by
0π / 2
π
3π / 22π
momentum K
-0.8-0.8
00
0.80.8
1.61.6
2.42.4
rescaled energy E(K)
L = 12, ky = 0
L = 12, ky = π
primary fields
descendant fields
-4/5
-2/5
0
1-4/5
1-2/5
2-4/5
2-2/5
-4/5
-2/5
0
1-4/5
1-2/5
2-4/5
2-2/5
FIG. 15.
doubled ‘ferromagnetic’ Yang-Lee chain. The spectrum matches
the non-unitary minimal model?
green squares, descendant fields by red circles.
Conformal energy spectrum of the critical points in the
M(3,10) with central charge c =
−44/5. Primary fields of the conformal field theory are indicated by
well as several descendants match to high precision (limited
by finite size effects) is a very non-trivial check on our results.
In Figure 15, we give the numerical spectrum of the ferro-
magnetic Yang-Lee ladder, at θ = 5π/4, which is character-
istic of the critical phase extending over θ ∈ (π,3π/2). In
this case the critical behavior is described by the?
magnetic case, we were able to identify the primary fields, as
well as several low-lying descendant fields, as indicated in the
figure. The fields of the CFT are given in table VI.
M(3,10)
non-unitary conformal field theory, and, as for the antiferro-
IV.DISCUSSION & SUMMARY
In this paper, we studied the collective states of Yang-Lee
anyons, a family of non-unitary, non-Abelian anyons which
are close cousins of the unitary Fibonacci anyons.
unitary anyons of this form have attracted interest in the con-
text of studies of certain quantum Hall wavefunctions, includ-
ingthe Gaffnianstate17. Both Yang-LeeandFibonacci anyons
arise from the same anyonic theory, su(2)3, and in particular
they share the same fusion rules. The key distinction between
the two anyon types is that Yang-Lee anyons are non-unitary
and relate to their unitary counterparts, the Fibonacci anyons,
via ‘Galois conjugation’. We have generalized this concept to
arbitrary su(2)kanyonic theories.
To characterize the collective states formed by a set of
anyons in the presence of pairwise interactions, we have
considered one-dimensional models of interacting Yang-Lee
anyons similar to the golden chain model of the unitary case23.
Analogous to the case of interacting Fibonacci anyons, the
collective states of such chains of Yang-Lee anyons are found
to be critical and the gapless theories are described by cer-
tain non-unitary conformal field theories (which depend on
the sign of the coupling, see Table III).
The non-unitary chain models are found to exhibit rather
peculiar features related to the presence of a non-local, topo-
logical symmetry first observed in the unitary models. For
the multicritical gapless theories of these chain models this
topological symmetry, whose related symmetry operator com-
mutes with the Hamiltonian, allows to classify all operators by
a topology symmetry sector. It turns out that all relevant oper-
ators corresponding to (uniform) perturbations of the gapless
system are in a different symmetry sector from the one of the
ground state. In the unitary case, this mechanism which ef-
fectively protects the gapless ground state from local pertur-
bations has led to an interpretation34of these gapless modes
of a chain of interacting anyons as edge states of the par-
ent topological liquid of which the anyons are excitations of.
Thishasfurtherledtotheconclusionthatinteractionsbetween
anyons (in two-dimensional arrangements) result in a splitting
of the macroscopic degeneracy of a set of non-Abelian anyons
and the nucleation of a new topological liquid inside and spa-
tially separated from the parent liquid of which the anyons are
excitations34,35.
For the non-unitary models, studied in this manuscript, a
crucial distinction comes in the symmetry sector assigned to
the ground state. While in the unitary case, the ground state
was found to be in the flux-free (topological trivial) sector, we
find that the ground state in the non-unitary case exhibits a
non-trivial topological flux (i.e. is in the non-trivial topolog-
ical symmetry sector). It remains an open question whether
such a spontaneous creation of topological flux can occur in
the ground state of a system of interacting anyonic quasipar-
ticles residing in a bulk-gapped topological liquid, which has
no topological flux associated with it. This may be an indica-
tion that the non-unitary anyons are not massive quasiparticles
of a gapped quantum liquid, but in fact excitations of a gapless
quantum liquid.
In summary, we have investigated one-dimensional models
Non-

Page 14

14
of non-unitary anyons based on su(2)kanyonic theories. The
collective ground states are found to display critical behavior,
similar to their unitary counterparts to which they are related
via a ‘Galois conjugation’. We described in detail the non-
unitary conformal field theories capturing their critical behav-
ior and commented on possible physical implications of our
results in connection with proposed quantum Hall states re-
lated to such non-unitary CFTs. In addition, we also studied
the phase diagram of an interacting ladder model, which is
a quasi one-dimensional version of the Levin-Wen model for
non-unitary Yang-Lee anyons.
ACKNOWLEDGMENTS
We acknowledge discussions with P. Bonderson, M. Freed-
man, J. Slingerland, and Z. Wang. A.W.W.L. was supported,
in part, by NSF DMR-0706140. We thank the Aspen Center
forPhysicsandtheKavliInstituteforTheoreticalPhysicssup-
ported by NSF PHY-0551164. Our numerical work used some
of the ALPS libraries,58,59see also http://alps.comp-phys.org.
Appendix A: Detailed description of the conformal energy
spectra.
In this section, we will describe in some detail the structure
of the descendant fields present in the numerically obtained
spectra. In the case of the Yang-Lee ladders, this structure
is somewhat different from the ‘usual’ structure. The object
containing the information about the number of states at a spe-
cific energy and momentum (in the thermodynamic limit) is
the partition function, which we will denote by Ztot.
Modular invariance of the partition function on the torus
constraints the possible partition functions of rational confor-
mal field theories at a particular central charge57. In this pa-
per, we will only be concerned with the partition functions of
minimal models, which are described in terms of the chiral
characters associated to the primary fields. An explicit ex-
pression for these chiral characters associated to the primary
fields φ(r,s)of the minimal models M(p,p?) is given in, for
instance, chapter 8 of reference 60.
ch(p,p?)
(r,s)(q) =qh(r,s)
(q)∞
?
k∈Z
?
qk(kpp?+rp−sp?)− q(kp+s)(kp?+r)?
,
(A1)
where (q)∞=?
Dropping the labels (p,p?), all the possible partition func-
tions can be written as
?
For the minimal models, there is always the so-called
‘diagonal modular invariant’, for which M is diagonal,
M(r,s);(r?,s?)= δr,r?δs,s?, and the sum in the partition func-
tion runs over all primary fields. However, in general, there
k>0(1−qk), and h(r,s)=(pr−p?s)2−(p−p?)2
the conformal dimension of the field φ(r,s).
4pp?
Ztot=
(r,s);(r?,s?)
M(r,s);(r?,s?)ch(r,s)ch∗
(r?,s?).
(A2)
exist other modular invariant partition functions. The criti-
cal points of the ladder models realize some of these, as we
pointed out in the main text.
In terms of the Virasoro generators L0and¯L0, the Hamilto-
nian and momentum operators can be written (after an appro-
priaterescalingandshift)asH = L0+¯L0andP = (L0−¯L0).
The partition function can be written as a trace over the
Hilbertspace, Ztot = tr qL0¯ q¯L0, which allows us to extract
both the energies and momenta for each state.
In particular, the energy of a state is given by the sum of
the powers of q and ¯ q. We note that this energy is measured
with respect to the energy corresponding to the energy of the
identity operator 1, which we have set to zero. In addition,
the momenta of a state is given by L0−¯L0, in units of 2π/L,
and measured with respect to the momenta of the correspond-
ing primary field, which is not determined by conformal field
theory, but rather by the specific Hamiltonian realization.
1.The ferromagnetic Yang-Lee ladder
The critical theory describing the ferromagnetic Yang-Lee
ladder is the (D6,A2) modular invariant?
(we will drop the labels (p,p?) = (3,10) from the characters
for convenience)
M(3,10). The total
partition function of this theory can be written as follows57
Z = |ch(1,1)|2+ |ch(3,1)|2+ 2|ch(5,1)|2+ |ch(7,1)|2
+ |ch(9,1)|2+ (ch(1,1)ch∗
By expanding the partition function, in terms of q and ¯ q,
we can completely explain the low-lying part of the spectrum
displayed in figure 15. Let us start with the vacuum sector
of the theory, which corresponds to the part of the partition
function with integer powers of q and ¯ q:
(A3)
(1,2)+ ch(3,1)ch∗
(3,2)+ c.c.) .
Z0= |ch(1,1)|2+ |ch(9,1)|2+ (ch(1,1)ch∗
1 + 2q2+ 2¯ q2+ ··· .
The total power of q and ¯ q gives the energy of the states,
while the difference between the powers gives the momentum.
Thus, the character of the vacuum sector implies the presence
of two states at energy E = 2 and Kx = 2 as well as two
states at energy E = 2 and momentum Kx= −2. Note that
we count the states irrespective of their Kysector.
We will continue with the descendants corresponding to the
primaryfieldwithenergyE = −2/5andmomentumKx= 0.
At this energy and momentum, there are in fact two states.
This sector of the partition function reads
(1,2)+ c.c.) =
(A4)
Z−2/5= 2|ch(5,1)|2=
2q−1/5¯ q−1/5?1 + q + ¯ q + 2q2+ 2¯ q2+ q¯ q + ···?
which implies two states at (E,Kx) = (1−2/5,1) and two at
(E,Kx) = (1−2/5,−1), as well as two states at (2−2/5,0)
and four at both (2 − 2/5,2) and (2 − 2/5,−2), all of which
is reproduced (modulo finite size effects) in the spectrum in
figure 15.
, (A5)

Page 15

15
Finally, we consider the descendants corresponding to the
primary field at E = −4/5, which also has momentum Kx=
0. This sector of the partition function reads
Z−4/5= |ch(3,1)|2+ |ch(7,1)|2+ (ch(3,1)ch∗
q−2/5¯ q−2/5?1 + 2q + 2¯ q + 3q2+ 3¯ q3+ 4q¯ q + ···?
This implies the presence of two states at both (E,Kx) =
(1 − 4/5,1) and (1 − 4/5,1), as well as three states at both
(2 − 4/5,2) and (2 − 4/5,−2). In addition, there should be
four states at (2 − 4/5,0). This seems to be at odds with the
figure 15, but it turns out that the state at Kx= 0 and Ky= π,
and energy of approximately E ≈ 2−4/5 (or, more precisely,
E ≈ 1.2144495) is in fact doubly degenerate, so indeed, there
are four states, as expected from the partition function.
(3,2)+ c.c.) =
.
(A6)
2.The antiferromagnetic Yang-Lee ladder
A similar analysis can be performed for the antiferromag-
netic critical point of the Yang-Lee ladder. We will be brief
here, because the analysis is identical to the one for the ferro-
magnetic critical point.
There are nine sectors in this particular case, corresponding
to the fields displayed in table VI, so the partition function can
be written as
Ztot= Z0+ Z4/5+ Z22/5+ Z26/7+ Z18/35+ Z4/35
+ Z8/7+ Z−2/35+ Z54/35
(A7)
Although we will not consider all of these sectors (some give
rise to states which are too high in energy to be unambigu-
ously identified, due to finite size effects), we will neverthe-
less give the form of the partition function restricted to each of
these sectors. We will drop the labels (p,p?) = (7,10) from
the characters
Z0= |ch(1,1)|2+ |ch(9,1)|2+ (ch(1,1)ch∗
Z4/5= |ch(3,1)|2+ |ch(7,1)|2+ (ch(3,1)ch∗
Z22/5= 2|ch(5,1)|2
Z26/7= |ch(1,3)|2+ |ch(9,3)|2+ (ch(1,3)ch∗
Z18/35= |ch(3,3)|2+ |ch(7,3)|2+ (ch(3,3)ch∗
Z4/35= 2|ch(5,3)|2
Z8/7= |ch(1,5)|2+ |ch(9,5)|2+ (ch(1,5)ch∗
Z−2/35= |ch(3,5)|2+ |ch(7,5)|2+ (ch(3,5)ch∗
Z54/35= 2|ch(5,5)|2
The parts of the partition function read explicitly
(1,6)+ c.c.)
(3,6)+ c.c.)
(1,4)+ c.c.)
(3,4)+ c.c.)
(1,2)+ c.c.)
(3,2)+ c.c.)
Z0= 1 + q2+ ¯ q2+ ...
Z4/5= q2/5¯ q2/5?1 + q + ¯ q + 2q2+ q¯ q + 2¯ q2+ ...?
Z22/5= 2q11/5¯ q11/5?1 + q + ¯ q + 2q2+ q¯ q + 2¯ q2+ ...?
Z26/7= q13/7¯ q13/7?1 + q + ¯ q + 3q2+ q¯ q + 3¯ q2+ ...?
Z4/35= 2q2/35¯ q2/35?1 + q + ¯ q + 2q2+ q¯ q + 2¯ q2+ ...?
Z−2/35= q−1/35¯ q−1/35?1 + q + ¯ q + 2q2+ q¯ q + 2¯ q2+ ...?
Z18/35= q9/35¯ q9/35?1 + 2q + 2¯ q + 3q2+ 4q¯ q + 3¯ q2+ ...?
Z8/7= q4/7¯ q4/7?1 + q + ¯ q + q2+ q¯ q + ¯ q2+ ...?
Z54/35= 2q27/35¯ q27/35?1 + q + ¯ q + 2q2+ q¯ q + 2¯ q2+ ...?
With this information, it is rather straight forward to check
that the primaries and descendant fields we indicated in figure
14 indeed come with the right multiplicities.
3.The ferromagnetic Yang-Lee chain
After having dealt with the (anti)ferromagnetic Yang-Lee
ladders in quite some detail, we will content ourselves here by
giving the (diagonal) partition functions describing the spec-
tra, and note that for all states where finite size effects allow
us to make an identification, we obtain full agreement with the
CFT prediction.
The partition function in case of the ferromagnetic Yang-
Lee chain, the critical model is given by the diagonal invariant
of the model M(2,5), whose partition function reads (drop-
ping the label (2,5) on the chiral character)
Z = |ch(1,1)|2+ |ch(3,1)|2= Z0+ Z−2/5,
(A8)
with
Z0= 1 + q2+ ¯ q2+ q3+ ¯ q3+ q4+ q2¯ q2+ ¯ q4+
q5+ q3¯ q2+ q2¯ q3+ ¯ q5+
2q6+ q4¯ q2+ q3¯ q3+ q2¯ q4+ 2¯ q6+ ···
Z−2/5= q−1/5¯ q−1/5?1 + q + ¯ q + q2+ q¯ q + ¯ q2+
q3+ q2¯ q + q¯ q2+ ¯ q3+
2q4+ q3¯ q + q2¯ q2+ q¯ q2+ 2¯ q4+
2q5+ 2q4¯ q + q2¯ q3+ q3¯ q2+ 2q¯ q4+ 2¯ q5+
3q6+ 2q5¯ q + 2q4¯ q2+ q3¯ q3+ 2q2¯ q4+ 2q¯ q5+ 3¯ q6
+ ···?
All these states, and in fact quite a few more which we did not
give here, are reproduced in the spectrum of the ferromagnetic
Yang-Lee chain, as shown in figure 8.
(A9)
(A10)
4.The antiferromagnetic Yang-Lee chain
Finally, we deal with the antiferromagnetic Yang-Lee
chain, the critical behavior of which is described by the (diag-
onal invariant of the) minimal model M(3,5), which has four
primary fields, as described in the main text. The relevant par-
tition function is given by (again, dropping the labels (3,5) on
the chiral characters)
Z = |ch(1,1)|2+ |ch(3,1)|2+ |ch(1,2)|2+ |ch(3,2)|2