# Operator fusion from wavefunction overlaps:
Universal finite-size corrections and application to Haagerup model

Yuhan Liu  Kadanoff Center for Theoretical Physics, University of Chicago,
Chicago, IL 60637, USA  These authors contributed equally to this work  Yijian
Zou  Stanford Institute for Theoretical Physics, Stanford University, Palo
Alto, CA 94305, USA  These authors contributed equally to this work  Shinsei
Ryu  Department of Physics, Princeton University, Princeton, New Jersey,
08540, USA

###### Abstract

Given a critical quantum spin chain described by a conformal field theory
(CFT) at long distances, it is crucial to understand the universal conformal
data. One most important ingredient is the operator product expansion (OPE)
coefficients, which describe how operators fuse into each other. It has been
proposed in [Zou, Vidal, Phys. Rev. B 105, 125125] that the OPE coefficients
can be computed from overlaps of low-energy wavefunctions of the spin chain.
In this work, we establish that all conformal data including central charge,
conformal dimensions, and OPE coefficients are encoded in the wavefunction
overlaps, with universal finite-size corrections that depend on the operator
content of the cyclic orbifold CFT. Thus this method allows us to numerically
compute all the conformal data based solely on the low-energy eigenstates. The
predictions are verified in the Ising and XXZ model. As an application, we
study the recently proposed Haagerup model built from the Haagerup fusion
category. We find that the CFT has central charge $c\approx 2.1$ and the
lowest spin-1 operator in the twisted sector has scaling dimension
$1<\Delta_{J}\leq 1.4$.

###### Contents

  1. I Introduction
  2. II Operator fusion from Wavefunction overlaps
    1. II.1 CFT on a cylinder
    2. II.2 Path integral for wavefunction overlaps
  3. III Universal finite-size corrections
    1. III.1 Cyclic orbifold on the cylinder
    2. III.2 Operator content of cyclic orbifold
    3. III.3 Universal finite-size corrections in wavefunction overlaps
    4. III.4 Finite-size corrections to Renyi entropy of CFT
  4. IV Applications to Ising and XXZ models
    1. IV.1 The Ising model
      1. IV.1.1 Orbifold operator content
      2. IV.1.2 Mapping to compactified boson orbifold
      3. IV.1.3 Universal finite-size correction
    2. IV.2 The XXZ model
      1. IV.2.1 Operator content
      2. IV.2.2 Universal finite-size correction
  5. V Applications to the Haagerup model
    1. V.1 Haagerup anyon chain and spectrum
      1. V.1.1 $\mathcal{H}_{3}$ fusion category
      2. V.1.2 Anyon chain construction and spectrum
      3. V.1.3 $\mathbb{Z}_{3}$ orbifold of sigma model
    2. V.2 Wavefunction overlaps
  6. VI Discussion
  7. A Fusion rules of cyclic orbifold from modular transformations
  8. B Wavefunction overlaps for antiferromagnetic Ising model
  9. C Periodic uniform matrix product states
  10. D Finite-size corrections for general overlaps
  11. E OPE from vertex state

##  I Introduction

Universality is one of the most prominent features of phase transitions [1].
Near a continuous phase transition, physics at low energies is determined by
the universal data. In 1+1 dimensions, a continuous phase transition is
usually described by a conformal field theory (CFT) [2]. A CFT is completely
determined by the conformal data, which consists of a set of primary operators
$\phi_{\alpha}$ with scaling dimensions $\Delta_{\alpha}$ and conformal spins
$s_{\alpha}$, the operator product expansion coefficients
$C_{\alpha\beta\gamma}$ and the central charge $c$ [3, 4].

Given a quantum critical spin chain, it is both important and challenging to
compute the universal data. One approach, initiated by Cardy, Affleck and
others in the 80’s [5, 6, 7], is based on the operator-state correspondence.
In this approach, one starts with the low-energy eigenstate
$|\phi_{\alpha}\rangle$ of the critical quantum spin chain with periodic
boundary conditions (PBC). Once the Hamiltonian is properly normalized, the
scaling dimensions $\Delta_{\alpha}$ and conformal spins $s_{\alpha}$ are
given by the energies $E_{\alpha}$ and momenta $p_{\alpha}$ of the states. In
order to compute the OPE coefficients, one has to first identify lattice
operators that correspond to CFT primary operators [8, 9], which is in general
hard to do without prior knowledge of the CFT.

To overcome this difficulty more recently, it has been proposed in Refs. [10,
11] that the OPE coefficients can be numerically extracted with the
eigenstates alone. The key is to start with three normalized eigenstates
$|\phi_{\alpha}\rangle$, $|\phi_{\beta}\rangle$, $|\phi_{\gamma}\rangle$ of
the critical quantum spin chain with PBC and sizes $N_{1},N_{2}$ and
$N_{3}=N_{1}+N_{2}$, respectively. It is then shown that the wavefunction
overlap

$\displaystyle
A_{\alpha\beta\gamma}=\langle\phi_{\gamma}|\phi_{\alpha}\phi_{\beta}\rangle$
(1)

is related to the OPE coefficient $C_{\alpha\beta\gamma}$. This provides a
numerically feasible way to extract the OPE coefficients by computing the
wavefunction overlaps.

However, two important problems remain unsolved in Ref. [11]. Firstly, the
paper only considered the case where $N_{1}=N_{2}$. It remains a question if
OPE coefficients can still be extracted when the sizes are not equal. Secondly
and more importantly, the relation between wavefunction overlap
$A_{\alpha\beta\gamma}$ and the OPE coefficient $C_{\alpha\beta\gamma}$ is
only expected to hold in the thermodynamic limit. At finite sizes, the finite-
size corrections to $A_{\alpha\beta\gamma}$ may be large. Unlike the finite-
size corrections to the energies $E_{\alpha}$, whose exponents are determined
by the scaling dimensions of irrelevant operators of the CFT through standard
conformal perturbation theory [12], an understanding of finite-size
corrections to $A_{\alpha\beta\gamma}$ is still missing. This makes it hard to
extrapolate the finite-size data to the thermodynamic limit.

In this work, we solve these two problems. First, using a generalized
conformal map from three-sided cylinder to the complex plane which appeared in
the string field theory literature [13, 14, 15], we establish a relation
between wavefunction overlaps $A_{\alpha\beta\gamma}$ and OPE coefficients
$C_{\alpha\beta\gamma}$ for general sizes. Second, we show that the exponents
in the finite-size correction to $A_{\alpha\beta\gamma}$ are universal by
mapping the wavefunction overlaps to three-point correlation functions of the
cyclic orbifold of the original CFT. The exponents are determined by the
operator content and fusion rules of the cyclic orbifold, which are in turn
determined by the original CFT. The result also implies similar universal
finite-size corrections in Renyi entropies of CFT, as observed in Ref. [16].
As a benchmark, we examine the finite-size corrections to
$A_{\alpha\beta\gamma}$ for the Ising and XXZ spin chain. All numerical data
agrees with the theoretical predictions in high precision.

Our result enables us to compute the conformal data from
$A_{\alpha\beta\gamma}$ in a systematic way, with a proper estimation of
finite-size corrections. For example, when $N_{1}=N_{2}=N$, the overlap
$\langle\phi_{\gamma}|\mathbbm{1}\mathbbm{1}\rangle$ scales as

$A_{\mathbbm{1}\mathbbm{1}\gamma}=O(N^{-\frac{c}{8}-\frac{\Delta_{\gamma}}{2}}),$
(2)

where $|\mathbbm{1}\rangle$ is the ground state. The exponent gives the
central charge and scaling dimensions without proper normalization of the
lattice Hamiltonian. Another example is that if the CFT has a spin-1 conserved
current $J$ and its conjugate $J^{*}$, then

$\frac{A_{JJ^{*}\mathbbm{1}}}{A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}}=\frac{1}{16}+O(N^{-2}).$
(3)

As an application, we compute the overlaps in a recently proposed Haagerup
model [17, 18], which is conjectured to have $c=2$ and a spin-1 current $J$.
Our numerical result, however, indicates that the CFT has $c\approx 2.1$ and
that the proposed operator $J$ cannot be a conserved current that saturates
the unitarity bound $\Delta_{J}\geq 1$. Rather, our numerical result supports
that the operator is a spin-1 primary operator close to the unitarity bound,
with an upper bound of the scaling dimension $\Delta_{J}\leq 1.4$.

Our work is both of interest from numerical and theoretical perspectives.
Numerically, we provide a simple method to compute the conformal data based
solely on the low-energy wavefunctions. Compared with the prior work, our
method does not rely on input such as identification of lattice operators with
CFT operators, symmetry of the model or correct normalization of the
Hamiltonian, which makes it practical to explore a less-known CFT.
Theoretically, we show that the wavefunction overlaps encode the information
of the corresponding cyclic orbifold CFT. It is therefore interesting to
investigate how to relate other orbifold CFT data with measurable quantities
in the original CFT model. Furthermore, as noted in Ref. [11], the overlaps
considered here can be regarded as wavefunctions of a toy three-sided wormhole
in holography [19]. Thus it may be desirable to find a holographic
interpretation of our results.

The rest of the paper is structured as follows. In Sec. II we review the
operator-state correspondence and derive the relation between wavefunction
overlaps $A_{\alpha\beta\gamma}$ and the OPE coefficients
$C_{\alpha\beta\gamma}$. In Sec. III we review the cyclic orbifold CFT and
relate its operator content to the exponents in the universal finite-size
corrections to $A_{\alpha\beta\gamma}$. As a corollary, we also derive finite-
size corrections to the second Renyi entropy of CFT. In Sec. IV we compute
$A_{\alpha\beta\gamma}$ for the Ising and XXZ model and verify that the
finite-size corrections all match the analytical predictions with good
accuracy. In Sec. V we apply our method to the Haagerup model and extract part
of the conformal data. Finally, we discuss possible applications and
extensions of the present work.

##  II Operator fusion from Wavefunction overlaps

In this section we derive the relation between wavefunction overlaps and OPE
coefficients of the CFT. We first represent the wavefunction overlaps as path
integral on a three-sided cylinder. Then we use a conformal transformation to
map the path integral to a three-point correlation function on the complex
plane. Our derivation generalizes that of Ref. [11] in that we do not restrict
two sides of the cylinder to have the same size.

###  II.1 CFT on a cylinder

Consider a CFT on a Euclidean cylinder with circumference $L$. The compact
dimension represents the space with coordinate $x\in[0,L)$, and the non-
compact dimension represents the imaginary time with coordinate
$\tau\in(-\infty,\infty)$. It is convenient to use the complex coordinates,
denoted as $z=\tau+ix$ and $\bar{z}=\tau-ix$. The Hilbert space is supported
on the equal-time slices. A CFT is composed of a set of local scaling
operators that are covariant under global rescaling of the spacetime, where
the eigenvalue is the scaling dimension. The scaling operators are organized
into conformal towers. For each tower, there is a primary operator
$\phi_{\alpha}$ with the lowest scaling dimension $\Delta_{\alpha}$, and other
descendant operators (later abbreviated as des.) with scaling dimensions that
differ $\Delta_{\alpha}$ by integers.

A crucial property of the CFT is the operator-state correspondence, which
states that each scaling operator corresponds to a state on the cylinder. The
energies of the states are related to the scaling dimensions by

$E_{\alpha}=\frac{2\pi}{L}\left(\Delta_{\alpha}-\frac{c}{12}\right),$ (4)

where $c$ is the central charge of the CFT. For a unitary CFT, the ground
state $|\mathbbm{1}\rangle$ corresponds to the identity operator $\mathbbm{1}$
with scaling dimension $\Delta_{\mathbbm{1}}=0$. Other states can be created
by acting with the scaling operator on the ground state at infinity,

$\displaystyle|\phi_{\alpha}\rangle$ $\displaystyle=$
$\displaystyle\phi_{\alpha}(-\infty)|\mathbbm{1}\rangle.$ (5)
$\displaystyle\langle\phi_{\alpha}|$ $\displaystyle=$
$\displaystyle\langle\mathbbm{1}|\phi^{*}_{\alpha}(+\infty),$ (6)

where

$\displaystyle\phi_{\alpha}(\pm\infty)\equiv\left(\frac{2\pi}{L}\right)^{-\Delta_{\alpha}}\lim_{\tau\rightarrow\pm\infty}e^{\pm\frac{2\pi}{L}\Delta_{\alpha}\tau}\phi_{\alpha}(\tau,0),$
(7)

is a shorthand notation and $\phi^{*}_{\alpha}$ is the conjugation of
$\phi_{\alpha}$. In many cases the primary operator is self-conjugate and we
will not distinguish the two. The states form an orthonormal basis,

$\langle\phi_{\alpha}|\phi_{\beta}\rangle=\delta_{\alpha\beta}.$ (8)

Given three primary operators $\phi_{\alpha},\phi_{\beta},\phi_{\gamma}$, the
three-point correlation function is determined by the operator product
expansion (OPE), where the only independent coefficient is given by

$C_{\alpha\beta\gamma}=\left(\frac{2\pi}{L}\right)^{-\Delta_{\alpha}}\langle\phi_{\gamma}|\phi_{\alpha}(0)|\phi_{\beta}\rangle.$
(9)

Other correlation functions, including those of descendant operators, are
completely determined by the OPE coefficients of primary operators. If
$C_{\alpha\beta\gamma}\neq 0$, there is a fusion channel
$\phi_{\alpha}\times\phi_{\beta}\rightarrow\phi_{\gamma}$, and we denote it by
$\mathcal{N}_{\alpha,\beta,\gamma}\neq 0$. Otherwise, the fusion channel is
said to be forbidden, and $\mathcal{N}_{\alpha,\beta,\gamma}=0$. In all cases
considered in this work, $\mathcal{N}_{\alpha,\beta,\gamma}$ is either $0$ or
$1$, depending on whether the fusion channel is forbidden or allowed.

###  II.2 Path integral for wavefunction overlaps

Figure 1: The path integral for the wavefunction overlap
$A_{\alpha\beta\gamma}=\langle\phi^{3*}_{\gamma}|\phi^{1}_{\alpha}\phi^{2}_{\beta}\rangle$.
The geometry is a three-sided cylinder $\Sigma$ with circumference
$L_{1}+L_{2}=L_{3}$.

(a)

(b)

Figure 2: Illustration of the conformal transformations that map three-sided
cylinder to the complex plane. (a) Eq. (15) for the case of $L_{1}=L_{2}$ and
(b) Eq. (17), for the case of $L_{2}=2L_{1}$. Colored lines represent equal
time slices with equal real part of $z$. Darker color corresponds to larger
$\tau$. Red lines correspond to equal time slices on cylinders $1$ and $2$,
and the blue lines correspond to equal time slices on cylinder $3$. For better
illustration purposes we show both $w$ and $1/w$ for the two conformal
transformations.

Let us consider three cylinders with circumference $L_{1}$,$L_{2}$ and $L_{3}$
with $L_{1}+L_{2}=L_{3}$. On each cylinder a primary operator is inserted at
infinity to create a state. Then the wavefunction overlaps can be represented
by a path integral on the three-sided cylinder $\Sigma$ with three insertions

$\displaystyle{A_{\alpha\beta\gamma}=}$
$\displaystyle\langle\phi^{3}_{\gamma}|\phi^{1}_{\alpha}\phi^{2}_{\beta}\rangle$
(10) $\displaystyle=$
$\displaystyle\int_{\Sigma}D\Phi\,\phi^{1}_{\alpha}(-\infty)\phi^{2}_{\beta}(-\infty)\phi^{3*}_{\gamma}(+\infty)e^{-S_{\mathrm{CFT}}[\Phi]},$

where $D\Phi$ is the functional measure and $S_{\mathrm{CFT}}$ is the action
of the CFT,

$S_{\mathrm{CFT}}[\Phi]=\int_{\Sigma}d\tau
dx\,\mathcal{L}_{\mathrm{CFT}}[\Phi],$ (11)

where $\mathcal{L}_{\mathrm{CFT}}[\Phi]$ is the Lagrangian density. An
illustration of the three-sided cylinder is shown in Fig. 1. The partition
function is the path integral with no insertions, thus

$\int_{\Sigma}D\Phi\,e^{-S_{\mathrm{CFT}}[\Phi]}=\langle\mathbbm{1}^{3}|\mathbbm{1}^{1}\mathbbm{1}^{2}\rangle.$
(12)

Taking the ratio gives

$\frac{\langle\phi^{3}_{\gamma}|\phi^{1}_{\alpha}\phi^{2}_{\beta}\rangle}{\langle\mathbbm{1}^{3}|\mathbbm{1}^{1}\mathbbm{1}^{2}\rangle}=\langle\phi^{1}_{\alpha}(-\infty)\phi^{2}_{\beta}(-\infty)\phi^{3*}_{\gamma}(+\infty)\rangle_{\Sigma}$
(13)

Note that the three cylinders meet at $z_{0}=iL_{1}$ on the $\tau=0$ time
slice.

Next, we use a conformal transformation $w(z)$ to map $\Sigma$ to the complex
plane ${\mathbbm{C}}$. Inspired by the string field theory literature [13,
14], the conformal map is defined implicitly by

$\displaystyle z+\lambda=$
$\displaystyle\frac{1}{2\pi}[L_{1}\log(w-w_{1})+L_{2}\log(w-w_{2})-L_{3}\log(w-w_{3})],$
(14)

where $w_{1},w_{2}$ and $w_{3}$ are arbitrary complex numbers and $\lambda$
depends on the $w$’s. The role of $\lambda$ will be evident shortly. It is
clear that the conformal transformation maps the three infinity points to
$w_{1}$, $w_{2}$ and $w_{3}$, respectively. One example considered in Ref.
[11] is when $L_{3}=2L_{1}=2L_{2}=L$, $w_{1}=i,w_{2}=-i,w_{3}=0$ and
$\lambda=0$, where the conformal transformation simplifies to an explicit form

$w=\frac{1}{\sqrt{e^{4\pi z/L}-1}}.$ (15)

There is a second-order algebraic branch point at $z_{0}=0$ where the three
cylinders meet. This is illustrated in Fig. 2 (a). For general choices of
parameters, the conformal transformation $w(z)$ is hard to express explicitly.
However, the conformal transformation still possesses a second-order algebraic
branch point, which can be shifted to $z_{0}=0$ by tuning $\lambda$. Let
$w(z_{0})=w_{0}$, then the branch-point condition demands that

$\frac{dz}{dw}(w_{0})=0.$ (16)

Solving the equation gives $w_{0}$. Substituting $w_{0}$ back to
$w(z_{0})=w_{0}$ then fixes $\lambda$. For example, if we choose
$L_{2}=2L_{1}=2L/3$ and $w_{1}=i,w_{2}=-i,w_{3}=0$, then $w_{0}=3i$ and
$\lambda=(L\log(32/27))/(6\pi)$. The conformal transformation then becomes

$\displaystyle\frac{32}{27}\cdot e^{\frac{6\pi
z}{L}}=\frac{(w-i)(w+i)^{2}}{w^{3}},$ (17)

which is illustrated in Fig. 2 (b).

Using conformal transformation of primary operators, the correlation function
on $\Sigma$ transforms into a three-point correlation function on the complex
plane $\mathbbm{C}$,

$\displaystyle\langle\phi^{1}_{\alpha}(-\infty)\phi^{2}_{\beta}(-\infty)\phi^{3*}_{\gamma}(+\infty)\rangle_{\Sigma}$
(18) $\displaystyle=$
$\displaystyle\prod_{i}\left|J_{i}\right|^{-\Delta_{i}}\langle\phi^{1}_{\alpha}(w_{1})\phi^{2}_{\beta}(w_{2})\phi^{3*}_{\gamma}(w_{3})\rangle_{{\mathbbm{C}}},$

where

$J_{i}=e^{\pm\frac{2\pi}{L_{i}}\lambda}\frac{dz}{dw}(w_{i})$ (19)

is the conformal factor. The prefactor $e^{\pm\frac{2\pi}{L_{i}}\lambda}$
comes from the definition Eq. (7), where the plus sign is taken for
$\phi^{3}_{\gamma}$ and the minus sign is taken for $\phi^{1}_{\alpha}$ and
$\phi^{2}_{\beta}$. The three-point correlation function on the complex plane
has the standard form

$\langle\phi^{1}_{\alpha}(w_{1})\phi^{2}_{\beta}(w_{2})\phi^{3*}_{\gamma}(w_{3})\rangle_{{\mathbbm{C}}}=C_{\alpha\beta\gamma}\prod_{i<j}|w_{i}-w_{j}|^{-d_{ij}},$
(20)

where $d_{12}=\Delta_{\alpha}+\Delta_{\beta}-\Delta_{\gamma}$, and others
follow from permutation symmetry.

Finally, combining Eqs. (18), (20) and the expression for $\lambda$, we obtain

$\displaystyle\frac{\langle\phi^{3}_{\gamma}|\phi^{1}_{\alpha}\phi^{2}_{\beta}\rangle}{\langle\mathbbm{1}^{3}|\mathbbm{1}^{1}\mathbbm{1}^{2}\rangle}$
(21) $\displaystyle=$
$\displaystyle\left[\left(\frac{L_{3}}{L_{1}}\right)^{\frac{L_{1}}{L_{3}}}\left(\frac{L_{3}}{L_{2}}\right)^{\frac{L_{2}}{L_{3}}}\right]^{-\frac{L_{3}}{L_{1}}\Delta_{\alpha}-\frac{L_{3}}{L_{2}}\Delta_{\beta}+\Delta_{\gamma}}C_{\alpha\beta\gamma}.$

The result is independent of the parameters $w_{1},w_{2}$ and $w_{3}$, as
expected. The special case in Ref. [11] corresponds to $L_{3}=2L_{1}=2L_{2}$,
where the expression simplifies to

$\frac{\langle\phi^{3}_{\gamma}|\phi^{1}_{\alpha}\phi^{2}_{\beta}\rangle}{\langle\mathbbm{1}^{3}|\mathbbm{1}^{1}\mathbbm{1}^{2}\rangle}=2^{-2\Delta_{\alpha}-2\Delta_{\beta}+\Delta_{\gamma}}C_{\alpha\beta\gamma}.$
(22)

On a finite lattice, Eqs. (21) and (22) are satisfied up to finite-size
corrections. Physically, the finite-size corrections come from the singular
point where the three cylinders meet. In order to define the geometry, one has
to put a UV cutoff $\epsilon$ around the singular point, analogous to the UV
cutoff in the calculation of entanglement entropy in CFT [20, 21, 22]. On the
lattice, the UV cutoff $\epsilon$ is inversely proportional to the number of
spins $N$. So far the conformal transformation works in the limit of
$\epsilon\rightarrow 0$. Thus we expect that Eqs. (21) and (22) strictly hold
in the thermodynamic limit $N\rightarrow\infty$. In what follows, we focus on
the special case with $L_{1}=L_{2}$ and study the finite-size corrections to
Eq. (22). To this end, it is useful to switch from a path integral of a CFT on
$\Sigma$ to an equivalent viewpoint, the $\mathbb{Z}_{2}$ orbifold of the CFT
defined on a cylinder.

##  III Universal finite-size corrections

In this section we consider finite-size correction to Eq. (22). Specifically,
we consider the normalized wavefunction overlap

$\displaystyle\tilde{A}_{\alpha\beta\gamma}\equiv\frac{\langle\phi^{3}_{\gamma}|\phi^{1}_{\alpha}\phi^{2}_{\beta}\rangle}{\langle\mathbbm{1}^{3}|\mathbbm{1}^{1}\mathbbm{1}^{2}\rangle}$
(23)

and its finite-size corrections

$\tilde{A}_{\alpha\beta\gamma}=\tilde{A}^{(0)}_{\alpha\beta\gamma}+\sum_{p_{\alpha\beta\gamma}>0}\tilde{A}^{(p)}_{\alpha\beta\gamma}N^{-p_{\alpha\beta\gamma}},$
(24)

where

$\tilde{A}^{(0)}_{\alpha\beta\gamma}=2^{-2\Delta_{\alpha}-2\Delta_{\beta}+\Delta_{\gamma}}C_{\alpha\beta\gamma}.$
(25)

We show that $p_{\alpha\beta\gamma}$ only take on specific values determined
by the scaling dimensions and fusion rules of the cyclic orbifold of the
original CFT. They are given explicitly in Eq. (51). Cyclic orbifold has
played an important role in the study of entanglement in quantum field
theories. It is defined by gauging the cyclic permutation group
$\mathbb{Z}_{\mathsf{N}}$ of $\mathsf{N}$ copies of the original CFT. In this
work we focus on $\mathsf{N}=2$, but generalizations to larger $\mathsf{N}$ is
straightforward.

In the following, we first review the definition and the operator content of
the cyclic orbifold. We then map the wavefunction overlaps to the OPEs of the
cyclic orbifold and derive the universal finite-size corrections. Finally, we
remark on the implication of our construction on the finite-size corrections
to entanglement in CFT.

###  III.1 Cyclic orbifold on the cylinder

Given any CFT with Lagrangian density $\mathcal{L}_{\mathrm{CFT}}[\Phi]$, we
can define the cyclic orbifold as follows [23, 24]. We start with two copies
of the original CFT known as the the mother CFT, whose Lagrangian density is
given by

$\mathcal{L}_{\mathrm{mother}}[\Phi_{1},\Phi_{2}]=\mathcal{L}_{\mathrm{CFT}}[\Phi_{1}]+\mathcal{L}_{\mathrm{CFT}}[\Phi_{2}].$
(26)

The mother CFT has a $\mathbb{Z}_{2}$ global symmetry that swaps the two
copies. Denote the generator of the $\mathbb{Z}_{2}$ symmetry as $\sigma$,
then

$\sigma\Phi_{i}=\Phi_{i+1},$ (27)

where the subscript is understood to have periodicity $2$. The cyclic orbifold
theory is obtained by promoting the global symmetry to a gauge symmetry. The
meaning of gauging a symmetry is twofold. Firstly, we enlarge the Hilbert
space by allowing twisted boundary conditions,

$\Phi_{i}(\tau,x+L)=\sigma\Phi_{i}(\tau,x).$ (28)

This plays the role of gauge fields in usual gauge theory such as
electrodynamics. Secondly, we project the Hilbert space onto subspace that is
invariant under $\sigma$ 111More precisely, we need to keep every inequivalent
irreducible representation once in order to have a modular invariant partition
function. This subtlety arises in cyclic orbifold.. This amounts to enforcing
gauge constraints in usual gauge theory.

To see the operator content of the cyclic orbifold, first note that there are
two sectors of operators, the untwisted sector and the twisted sector. The
untwisted sector contains eigenstates of the mother CFT with periodic boundary
conditions. The twisted sector contains eigenstates of the mother CFT with
twisted boundary conditions. Second, only the operators that are invariant
under the $\sigma$ swapping is kept. This is essential to obtain a consistent
CFT because any operator that is not invariant under $\sigma$ is non-local
with respect to the twist sector. In the following subsection, we review the
operator content of cyclic orbifold in more detail.

###  III.2 Operator content of cyclic orbifold

As has been well studied, the operator content of the cyclic orbifold is
completely determined by the original CFT. The operator content is composed of
the untwisted sector and the twisted sector. In the untwisted sector, the
states are obtained from the tensor product theory. For $\alpha\neq\beta$, the
symmetric states
$|\phi_{\alpha,i}\rangle|\phi_{\beta,j}\rangle+|\phi_{\beta,j}\rangle|\phi_{\alpha,i}\rangle$
and anti-symmetric states
$|\phi_{\alpha,i}\rangle|\phi_{\beta,j}\rangle-|\phi_{\beta,j}\rangle|\phi_{\alpha,i}\rangle$
in the Hilbert space are isomorphic. The labels $i,j$ in the above expressions
are the labels for descendants. Due to the isomorphism, we only keep one of
the copies, say, the symmetric states. We denote the orbifold primary operator
corresponds to this copy as:

$\displaystyle\phi_{(\alpha,\beta)},$ (29)

with scaling dimension:

$\displaystyle\Delta_{(\alpha,\beta)}=\Delta_{\alpha}+\Delta_{\beta}.$ (30)

For $\alpha=\beta$, the above isomorphism fails because when
$\alpha=\beta,i=j$, the antisymmetric state does not exist. For this reason,
both the copies should be kept, and we denote the two orbifold primary
operators as:

$\displaystyle\phi_{(\alpha,\alpha)_{s}},\quad\phi_{(\alpha,\alpha)_{a}},$
(31)

where $s$ stands for symmetric and $a$ stands for anti-symmetric. Their
scaling dimensions are:

$\displaystyle\Delta_{(\alpha,\alpha)_{s}}=2\Delta_{\alpha},$ (32)
$\displaystyle\Delta_{(\alpha,\alpha)_{a}}=\begin{cases}2\Delta_{\alpha}+2,&\text{if
$\phi_{\alpha}$ has level-1 des.}\\\
2\Delta_{\alpha}+4,&\text{otherwise},\end{cases}$

where ‘des.’ stands for descendant. Note that $\phi_{(\alpha,\alpha)_{a}}$ has
larger scaling dimension than $\phi_{(\alpha,\alpha)_{s}}$, because
antisymmetrization of the state with lowest $L_{0}$ eigenvalue
$|\phi_{\alpha}\rangle|\phi_{\alpha}\rangle$ simply vanishes. Hence, a primary
state $|\phi_{\alpha}\rangle$ must be combined with its descendant and then
antisymmetrized. As far as the chiral algebra involves only conformal
symmetry, only $\mathbbm{1}$ operator does not have a level-1 descendant.
(Namely, $|\mathbbm{1}\rangle$ is the conformal vacuum.) This is no longer the
case if the Kac-Moody algebra is taken into account, where the $\mathbbm{1}$
operator does have level-1 descendants which are current operators. The
identity operator in the orbifold theory is
$\phi_{(\mathbbm{1},\mathbbm{1})_{s}}$, which has $\Delta=0$.

Summarizing, if the original theory has $n$ primary operators, then the
untwisted sector of the orbifold theory has $n(n-1)/2+2n=n(n+3)/2$ primary
operators. In Appendix A we list the characters of each orbifold primary.

In the twisted sector, the operator with the lowest scaling dimension is
denoted as $\tau_{(\mathbbm{1},\hat{0})}$, whose scaling dimension is
completely determined by the central charge,

$\Delta_{(\mathbbm{1},\hat{0})}=\frac{c}{8}.$ (33)

This is known as the branch-point twist operator in the literature. However,
there are other primary operators in the twisted sector, which plays an
important role in this work. One may fuse $\tau_{(\mathbbm{1},\hat{0})}$ with
each primary operator $\phi_{(\alpha,\mathbbm{1})}$ in the untwisted sector to
generate other twist operators:

$\displaystyle\tau_{(\mathbbm{1},\hat{0})}\times\phi_{(\alpha,\mathbbm{1})}\rightarrow\tau_{(\alpha,\hat{0})}+\tau_{(\alpha,\hat{1})},\quad\alpha\neq\mathbbm{1}.$
(34)

For each $\alpha$, there are two primary operators in the twisted
sector222This is also true for $\alpha=\mathbbm{1}$, and
$\tau_{(\mathbbm{1},\hat{1})}$ can be generated using channel
$\tau_{(\beta,\hat{0})}\times\phi_{(\beta,\mathbbm{1})}$,
$\beta\neq\mathbbm{1}$.:

$\displaystyle\tau_{(\alpha,\hat{0})},\quad\tau_{(\alpha,\hat{1})},$ (35)

with scaling dimensions:

$\displaystyle\Delta_{(\alpha,\hat{0})}$
$\displaystyle=\frac{c}{8}+\frac{\Delta_{\alpha}}{2}$ (36)
$\displaystyle\Delta_{(\alpha,\hat{1})}$
$\displaystyle=\begin{cases}\frac{c}{8}+\frac{\Delta_{\alpha}}{2}+1,&\text{if
$\phi_{\alpha}$ has level-1 des.}\\\
\frac{c}{8}+\frac{\Delta_{\alpha}}{2}+3,&\text{otherwise}.\end{cases}$

Summarizing, if the original theory has $n$ primary operators, then the
orbifold theory has $2n$ primary operators in the twisted sector.

Finally, we need the fusion rules of the orbifold theory. It is intuitively
clear that an operator in the untwisted sector fusing with an operator in the
twisted sector only gives operators in the twisted sector. Furthermore, it has
been shown [23] that the fusion rules are completely determined by the modular
$\mathcal{S}$ and $\mathcal{T}$ matrices of the original CFT. For an
introduction to modular matrices and details of the derivation, see Appendix
A. We will need the following two fusion coefficients,

$\mathcal{N}_{(\alpha,\beta),{(\gamma,\hat{\psi})},{(\delta,\hat{\chi})}}=\sum_{\eta}\frac{\mathcal{S}_{\alpha\eta}\mathcal{S}_{\beta\eta}\mathcal{S}_{\gamma\eta}\mathcal{S}^{*}_{\eta\delta}}{\mathcal{S}_{\mathbbm{1}\eta}^{2}},$
(37)

and

$\displaystyle\mathcal{N}_{(\alpha,\alpha)_{s},{(\gamma,\hat{\psi})},{(\delta,\hat{\chi})}}=$
(38)
$\displaystyle\frac{1}{2}\sum_{\eta}\frac{\mathcal{S}_{\alpha\eta}^{2}\mathcal{S}_{\gamma\eta}\mathcal{S}^{*}_{\eta\delta}}{\mathcal{S}^{2}_{\mathbbm{1}\eta}}+\frac{1}{2}e^{i\pi(\psi+\chi)}\sum_{\eta}\frac{\mathcal{S}_{\alpha\eta}\mathcal{P}_{\gamma\eta}\mathcal{P}^{*}_{\eta,\delta}}{\mathcal{S}_{\mathbbm{1},\eta}},$

where
$\mathcal{P}=\mathcal{T}^{1/2}\mathcal{S}\mathcal{T}^{2}\mathcal{S}\mathcal{T}^{1/2}$.
In particular, Eq. (37) can be simplified to

$\mathcal{N}_{(\alpha,\beta),{(\gamma,\hat{\psi})},{(\delta,\hat{\chi})}}=\sum_{s}\mathcal{N}_{\alpha,\beta,s}\mathcal{N}_{\gamma,\delta,s^{*}}$
(39)

if all primary fields are self-conjugate.

Figure 3: Mapping the path integral for the wavefunction overlap
$A_{\alpha\beta\gamma}$ to the cyclic orbifold path integral. (a) The three-
sided cylinder is conformally equivalent to the double-sheeted Riemann sphere
with branch cut on $[1,\infty)$. The conformal transformation is achieved by
$z^{\prime}=e^{2\pi z/L_{1}}$. (b) The corresponding path integral of the
cyclic orbifold for $A_{\alpha\beta\gamma}$. The insertions of branch-point
twist operators at $\tau=0$ and $\tau=+\infty$ correspond to the branch cut in
(a).

###  III.3 Universal finite-size corrections in wavefunction overlaps

Now we relate the wavefunction overlaps to path integral of the cyclic
orbifold. Denote the field on $\Sigma$ as

$\displaystyle\Phi_{1}(\tau,x)$ $\displaystyle=$
$\displaystyle\Phi(\tau,x),~{}~{}~{}0\leq x<L_{1},$ (40)
$\displaystyle\Phi_{2}(\tau,x)$ $\displaystyle=$
$\displaystyle\Phi(\tau,x),~{}~{}~{}L_{1}\leq x<L_{3}.$ (41)

It is then clear (as illustrated in Fig. 3(b)) that the fields are subject to
the boundary conditions

$\displaystyle\Phi_{i}(\tau,x+L_{1})$ $\displaystyle=$
$\displaystyle\Phi_{i}(\tau,x),~{}~{}~{}\tau<0,$ (42)
$\displaystyle\Phi_{i}(\tau,x+L_{1})$ $\displaystyle=$
$\displaystyle\sigma\Phi_{i}(\tau,x),~{}~{}~{}\tau\geq 0,$ (43)

where $\sigma\Phi_{i}\equiv\Phi_{i+1}$ and the subscript is understood to have
periodicity $2$. Now we see the path integral on $\Sigma$ is exactly that of
the cyclic orbifold on a cylinder with circumference $L_{1}$, with two twist
operators $\tau$ inserting at $\tau=0$ and $\tau=+\infty$,

$\langle\mathbbm{1}^{3}|\mathbbm{1}^{1}\mathbbm{1}^{2}\rangle=\langle\mathbbm{1}|\tau(+\infty)\tau(0)|\mathbbm{1}\rangle_{\mathrm{orb}},$
(44)

where the twist operator changes the boundary condition. The twist operator
may contain all operators in the twisted sector, and it can be expanded as

$\tau=\sum_{\alpha,\psi}a_{(\alpha,\hat{\psi})}\epsilon^{\Delta_{(\alpha,\hat{\psi})}}\tau_{(\alpha,\hat{\psi})}+\mathrm{des.}$
(45)

where we have introduced the UV cutoff $\epsilon$, and
$a_{(\alpha,\hat{\psi})}$ is presumably non-universal constant that depends on
specific lattice realizations of the CFT. Some of the coefficients may also
vanish due to symmetry of the lattice model, as we later examine specific
models.

Using the operator-state correspondence, Eqs. (5) and (6), we obtain

$\langle\mathbbm{1}^{3}|\mathbbm{1}^{1}\mathbbm{1}^{2}\rangle=\langle\tau_{(\mathbbm{1},\hat{0})}|\tau(0)|\phi_{(\mathbbm{1},\mathbbm{1})_{s}}\rangle_{\text{orb}},$
(46)

where the bra is fixed to $\langle\tau_{(\mathbbm{1},\hat{0})}|$ because other
states in the twisted sector die off under the infinite imaginary time
evolution. Using Eq. (9) and the fact that $C_{\mathbbm{1}\phi\phi}=1$ for any
primary operator $\phi$ (we are using $\phi=\tau_{(\mathbbm{1},\hat{0})}$
here), we see

$A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}=\langle\mathbbm{1}^{3}|\mathbbm{1}^{1}\mathbbm{1}^{2}\rangle=a_{(\mathbbm{1},\hat{0})}N^{-c/8},$
(47)

where $N=L_{1}/(2\pi\epsilon)$ is the system size of the lattice model, and
$c/8$ is the scaling dimension of the leading twist operator
$\tau_{(\mathbbm{1},\hat{0})}$. This is remarkable, since it gives a way to
compute the central charge of the CFT from wavefunction overlaps.

Now we consider the general wavefunction overlap
$\langle\phi^{3}_{\gamma}|\phi^{1}_{\alpha}\phi^{2}_{\beta}\rangle$. The ket
state can be created by inserting $\phi_{(\alpha,\beta)}$ at $\tau=-\infty$.
Hereafter we take $\phi_{(\alpha,\alpha)}$ to mean
$\phi_{(\alpha,\alpha)_{s}}$. The bra state can be created by inserting
$\phi^{*}_{(\gamma,\mathbbm{1})}$ at $\tau=+\infty$333The reason that the
second subindex is $\mathbbm{1}$ can be understood in the following way. One
considers a four-sided cylinder where the third side of the three-sided
cylinder is further split into two cylinders. We insert primary operator
$\phi_{\alpha}$ and $\phi_{\beta}$ for the kets and $\phi^{*}_{\gamma}$ and
$\mathbbm{1}$ for the bras. On the bra side, the two insertions fuse into
$\phi^{*}_{\gamma}$ which gives the desired overlap
$\langle\phi_{\gamma}|\phi_{\alpha}\phi_{\beta}\rangle$. . Recall that at
$\tau=+\infty$ there is already a twist operator
$\tau_{(\mathbbm{1},\hat{0})}$ insertion, so the two operators fuse to give
$\tau_{(\mathbbm{1},\hat{0})}\times\phi^{*}_{(\gamma,\mathbbm{1})}\rightarrow\tau_{(\gamma^{*},\hat{0})}$+$\tau_{(\gamma^{*},\hat{1})}$,
and we only keep $\tau_{(\gamma^{*},\hat{0})}$ at $\tau\rightarrow+\infty$.
Therefore, we obtain

$\langle\phi^{3}_{\gamma}|\phi^{1}_{\alpha}\phi^{2}_{\beta}\rangle=\langle\tau_{(\gamma,\hat{0})}|\tau(0)|\phi_{(\alpha,\beta)}\rangle_{\text{orb}}.$
(48)

Using Eq. (45) and (9) we obtain the expansion

$\langle\phi^{3}_{\gamma}|\phi^{1}_{\alpha}\phi^{2}_{\beta}\rangle=\sum_{\delta,\chi}a_{(\delta,\hat{\chi})}N^{-\Delta_{(\delta,\hat{\chi})}}C_{(\alpha,\beta),(\delta,\hat{\chi}),(\gamma,\hat{0})}+\cdots,$
(49)

where $\cdots$ stands for contributions from descendant operators. The OPE
coefficients can be computed from the OPEs in the original CFT following Ref.
[25]. In particular,

$C_{(\alpha,\beta),(\mathbbm{1},\hat{0}),(\gamma,\hat{0})}=2^{-2\Delta_{\alpha}-2\Delta_{\beta}+\Delta_{\gamma}}C_{\alpha\beta\gamma},$
(50)

which recovers Eq. (22) in the thermodynamic limit. Furthermore, we obtain the
finite-size corrections of the form Eq. (24), where the exponents are

$p_{\alpha\beta\gamma}\in\\{\Delta_{(\delta,\hat{\chi})}-\Delta_{(\mathbbm{1},\hat{0})}|\,\mathcal{N}_{(\alpha,\beta),(\delta,\hat{\chi}),(\gamma,\hat{0})}\neq
0\\}.$ (51)

This is the central result of our work. We see that the exponents inside the
finite-size corrections are universal, as claimed. Later we will consider
various CFTs and their lattice realizations, where we numerically extract the
leading exponent $p_{\alpha\beta\gamma}$ in the finite-size corrections. To
compare it with Eq. (51), it is useful to use the crossing symmetry,

$\mathcal{N}_{(\alpha,\beta),(\delta,\hat{\chi}),(\gamma,\hat{0})}=\mathcal{N}_{(\alpha,\beta),(\gamma^{*},\hat{0}),(\delta^{*},\hat{\chi})}.$
(52)

The leading $\Delta_{(\delta,\hat{\chi})}$ in the set of Eq. (51) is the
lowest scaling dimension of the twist operators (except
$\tau_{(\mathbbm{1},\hat{0})}$) in the OPE of $\phi_{(\alpha,\beta)}$ and
$\tau_{(\gamma^{*},\hat{0})}$.

One simple example is
$\langle\phi^{3}_{\gamma}|\mathbbm{1}^{1}\mathbbm{1}^{2}\rangle$ when
$\phi_{\gamma}\neq\mathbbm{1}$. The fusion channel is forbidden, meaning that
$\tilde{A}^{(0)}_{\mathbbm{1}\mathbbm{1}\gamma}=0$. Since the fusion of
identity operator with $\tau_{(\gamma^{*},\hat{0})}$ only gives
$\tau_{(\gamma^{*},\hat{0})}$, the leading exponent in finite-size correction
is

$p_{\mathbbm{1}\mathbbm{1}\gamma}=\Delta_{(\gamma^{*},\hat{0})}-\Delta_{(\mathbbm{1},\hat{0})}=\frac{\Delta_{\gamma}}{2}.$
(53)

This provides another consistency check of scaling dimensions of primary
operators.

We stress that the result Eq. (51) only applies to wavefunction overlaps with
$N_{1}=N_{2}$. For $N_{1}\neq N_{2}$, the exponents $p_{\alpha\beta\gamma}$ in
finite-size corrections may be different. Nevertheless, the exponents are
still the difference between the scaling dimensions of an operator
$\tau_{(\delta,\hat{\chi})}$ in the twisted sector and
$\tau_{(\mathbbm{1},0)}$, while the operator $\tau_{(\delta,\hat{\chi})}$ does
not necessarily satisfy the fusion constraint in Eq. (51). This is examined
and further explored in Appendix D.

###  III.4 Finite-size corrections to Renyi entropy of CFT

Cyclic orbifold has also played an important role in computing the
entanglement of CFT. We expect that the second Renyi entropy of CFT has
similar finite-size corrections as the wavefunction overlaps, since the twist
operator insertion is essentially of the same nature as Eq. (45). Consider,
for example, the ground state of a CFT on the complex plane and the reduced
density matrix on $[0,l]$,

$\rho=\textrm{Tr}_{\mathbb{R}-[0,l]}\,|\mathbbm{1}\rangle\langle\mathbbm{1}|.$
(54)

The Renyi entropies are defined by

$S_{\mathsf{N}}=\frac{1}{1-\mathsf{N}}\log\frac{\textrm{Tr}\,\rho^{\mathsf{N}}}{(\textrm{Tr}\rho)^{\mathsf{N}}}.$
(55)

Specializing to $\mathsf{N}=2$, the purity can be computed as the path
integral of the cyclic orbifold on the complex plane with two twist operators
insertions,

$\frac{\textrm{Tr}\,\rho^{2}}{(\textrm{Tr}\rho)^{2}}=\langle\mathbbm{1}|\tau(0)\tau(l)|\mathbbm{1}\rangle_{\mathrm{orb}}.$
(56)

Using Eq. (45), the correlation function can be expanded as

$\langle\mathbbm{1}|\tau(0)\tau(l)|\mathbbm{1}\rangle_{\text{orb}}=\sum_{(\alpha,\hat{\chi})}a^{2}_{(\alpha,\hat{\chi})}l^{-2\Delta_{(\alpha,\hat{\chi})}}+\cdots,$
(57)

where $\cdots$ contains contributions from descendant operators. In the limit
where $l\gg\epsilon$, the second Renyi entropy also has universal finite-size
corrections

$\displaystyle S_{2}$ $\displaystyle=\frac{c}{4}\log\frac{l}{\epsilon}-2\log
a_{(\mathbbm{1},\hat{0})}$
$\displaystyle\quad+\sum_{(\alpha,\hat{\chi})\neq(\mathbbm{1},\hat{0})}\frac{a^{2}_{(\alpha,\hat{\chi})}}{a^{2}_{(\mathbbm{1},\hat{0})}}\,l^{-2(\Delta_{(\alpha,\hat{\chi})}-\Delta_{(\mathbbm{1},\hat{0})})},$
(58)

where the last term contains the universal finite-size corrections, which has
been observed in Ref. [16]. As a side remark, one may consider similar finite-
size corrections to other Renyi entropies on the same footing, and then
analytically continue the result to the case of entanglement entropy
($\mathsf{N}=1$). Finally, we note that only the twist operators
$\tau_{(\alpha,\hat{0})}$ have been considered in previous work that discusses
entanglement of CFT. As we will show numerically, in the context of
wavefunction overlaps the other twist operator $\tau_{(\alpha,\hat{1})}$ also
plays an important role. It remains to be explored whether operators such as
$\tau_{(\alpha,\hat{1})}$ can be observed in the finite-size corrections to
the entanglement entropy of CFT.

##  IV Applications to Ising and XXZ models

In this section, we apply the above general method to two specific lattice
models: the Ising model and the XXZ model. The Ising model is the lattice
realization of the $c=1/2$ Ising CFT, which is a minimal model. The XXZ model
is a lattice realization of the $c=1$ compactified boson theory at radius $R$,
which is a rational CFT at special radii. Both the Ising model and the XXZ
model at $R=2$ can be mapped to a free fermion problem. Thus we are able to
obtain low-energy eigenstates for up to thousands of spins using the
covariance matrix techniques [26].

We first review the operator content of each CFT and the $\mathsf{N}=2$ cyclic
orbifold, and then show how the orbifold fusion rule determines the universal
finite-size corrections in wavefunction overlaps. We then verify the leading
exponent in the finite-size corrections Eq. (24) numerically. All the
exponents agree with our analytical expression Eq. (51).

###  IV.1 The Ising model

In this subsection we study the ferromagnetic Ising model. For later
discussions, it will also be useful to study the antiferromagnetic Ising model
which is described by the same CFT. Since the computations are largely
identical, the result for the antiferromagnetic case is shown in Appendix B.
The ferromagnetic Ising spin chain Hamiltonian at the critical point is:

$H=-\sum_{i=1}^{N}X_{i}X_{i+1}-\sum_{i=1}^{N}Z_{i},$ (59)

where $X,Z$ are Pauli matrices. There are three primary operators in the Ising
CFT: $\mathbbm{1},\sigma,\epsilon$, with scaling dimension
$\Delta=0,\frac{1}{8},1$, respectively.

The non-trivial OPE coefficients in the Ising CFT are:

$\displaystyle C_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}=1,\quad
C_{\epsilon\epsilon\mathbbm{1}}=1,\quad C_{\sigma\sigma\mathbbm{1}}=1,\quad
C_{\sigma\sigma\epsilon}=\frac{1}{2},$ (60)

and the modular matrices are (in the basis $\mathbbm{1},\sigma,\epsilon$):

$\displaystyle\mathcal{T}$
$\displaystyle=\left(\begin{array}[]{ccc}e^{-\frac{i\pi}{24}}&0&0\\\
0&e^{\frac{i\pi}{12}}&0\\\ 0&0&e^{i\pi\frac{23}{24}}\end{array}\right),$ (61)
$\displaystyle\mathcal{S}$
$\displaystyle=\frac{1}{2}\left(\begin{array}[]{ccc}1&1&\sqrt{2}\\\
1&1&-\sqrt{2}\\\ \sqrt{2}&-\sqrt{2}&0\end{array}\right).$

In the following, we consider the cyclic orbifold theory of the Ising CFT. It
is straightforward to compute scaling dimensions of all primary operators and
the fusion rules using general framework presented in Sec. III.2. For the
Ising CFT, however, it is possible to map the cyclic orbifold to a more
familiar CFT, that is, a compactified boson $\mathbb{Z}_{2}$ orbifold at
radius $R=4$. The $U(1)$ charge conservation symmetry in the latter CFT makes
it more intuitive to understand the fusion rules. We have checked that the
fusion rules given by Eqs. (37) and (38) coincide with the those of the free
boson orbifold.

####  IV.1.1 Orbifold operator content

Following the general discussion in Sec. III.2, there exist 15 primary
operators in the cyclic orbifold CFT, 9 in the untwisted sector and 6 in the
twisted sector. The 15 primary operators are listed in the first column of
Table. 1, along with their characters $\chi$ in the last column.

##### Untwisted sector

The three primaries for $\alpha\neq\beta$ are

$(\mathbbm{1},\sigma),\quad(\mathbbm{1},\epsilon),\quad(\sigma,\epsilon),$
(62)

with scaling dimensions $\Delta=\frac{1}{8},1,\frac{9}{8}$. Additionally,
there are 6 primary operators,

$(\mathbbm{1},\mathbbm{1})_{s},\quad(\sigma,\sigma)_{s},\quad(\epsilon,\epsilon)_{s},$
(63)

with $\Delta=0,\frac{1}{4},2$ and

$(\mathbbm{1},\mathbbm{1})_{a},\quad(\sigma,\sigma)_{a},\quad(\epsilon,\epsilon)_{a},$
(64)

with $\Delta=4,\frac{9}{4},4$.

##### Twisted sector

In the twisted sector, twist operators $\tau_{(\alpha,\hat{0})}$ have scaling
dimension $\Delta=\frac{c}{8}+\frac{\Delta_{\alpha}}{2}$, which gives
$\Delta=\frac{1}{16},\frac{1}{8},\frac{9}{16}$. $\tau_{(\alpha,\hat{1})}$ have
conformal dimension $\Delta=\frac{c}{8}+\frac{\Delta_{\alpha}}{2}+1$, except
for $\tau_{(\mathbbm{1},\hat{1})}$, which has
$\Delta=\frac{c}{8}+\frac{\Delta_{i}}{2}+3$. This stems from the fact that
$\mathbbm{1}$ corresponds to conformal vacuum $|0\rangle$ and
$L_{-1}|0\rangle=0$. Thus, we have
$\Delta=\frac{49}{16},\frac{9}{8},\frac{25}{16}$ for the primary operators
$\tau_{(\alpha,\hat{1})}$.

####  IV.1.2 Mapping to compactified boson orbifold

It has been noted in Ref. [24] that the scaling dimensions of the Ising
orbifold listed above are identical to those of the compactified boson
$\mathbb{Z}_{2}$ orbifold at radius $R=4$. Indeed the two theories are dual to
each other. The free boson action is:

$S=\frac{1}{4\pi}\int d^{2}z\partial\varphi\bar{\partial}\varphi.$ (65)

The compactified boson with radius $R$ is obtained by identifying
$\varphi\equiv\varphi+2\pi nR,n\in\mathbb{Z}$. It has a $\mathbb{Z}_{2}$
reflection symmetry: $\varphi\rightarrow-\varphi$, and a $U(1)$ symmetry
$\varphi\rightarrow\varphi+\alpha$. After gauging the $\mathbb{Z}_{2}$
reflection symmetry, we obtain the compactified boson orbifold theory.

The compactified boson orbifold at radius $R=\sqrt{2k}$ with
$k\in\mathbb{Z}_{+}$ is a rational CFT with $k+7$ extended primary fields,
which consists of the following five families [27]:

  1. 1.

$\phi_{\lambda},\lambda=1,2,\cdots,k-1$ with dimension
$\Delta_{\lambda}=\frac{\lambda^{2}}{2k}$.

  2. 2.

$\Phi^{(1)}$ and $\Phi^{(2)}$, with dimension
$\Delta=\frac{k^{2}}{2k}=\frac{k}{2}$.

  3. 3.

$\sigma^{(1)},\sigma^{(2)}$ with dimension $\Delta=\frac{1}{8}$, and
$\tau^{(1)},\tau^{(2)}$ with dimension $\Delta=\frac{9}{8}$.

  4. 4.

Identity operator $\mathbbm{1}$ with dimension 0.

  5. 5.

Operator $\Theta$ with dimension $\Delta=2$.

When taking $k=8$, i.e., at radius $R=\sqrt{2k}=4$, the conformal dimensions
of these 15 extended primaries match those of the 15 primaries in the cyclic
Ising orbifold, as shown in Table 1. This correspondence of conformal
dimensions was first noticed in [24]. We took one step further and also
verified numerically that this correspondence holds at the level of the
characters $\chi$ and the modular matrices $\mathcal{T},\mathcal{S}$.

Mapping to the compactified boson orbifold at radius $R=4$ is useful in the
problem concerned, because the fusion rule in the compactified boson orbifold
is well understood, which we will now exploit to derive the universal finite-
size corrections. For example, the $U(1)$ charge conservation gives:

$[\phi_{\lambda}]\times[\phi_{\mu}]=[\phi_{\lambda+\mu}]+[\phi_{\lambda-\mu}],\quad\mu\neq\lambda,2k-\lambda,$
(66)

and

$\displaystyle[\Theta]\times[\phi_{i}]=[\phi_{i}].$ (67)

The latter corresponds to the fact that $\Theta$ is charge neutral.

Untwisted |  |  |
---|---|---|---
primary (Ising orb.) | $\Delta$ |  primary (boson orb.) | $\chi$
$(\mathbbm{1},\sigma)$ | $\frac{1}{8}$ | $\sigma^{(1)}$ | $\chi_{\mathbbm{1}}(\tau)\chi_{\sigma}(\tau)$
$(\mathbbm{1},\epsilon)$ | $1$ | $\phi_{4}$ | $\chi_{\mathbbm{1}}(\tau)\chi_{\epsilon}(\tau)$
$(\sigma,\epsilon)$ | $\frac{9}{8}$ | $\tau^{(1)}$ | $\chi_{\sigma}(\tau)\chi_{\epsilon}(\tau)$
$(\mathbbm{1},\mathbbm{1})_{s}$ | $0$ | $\mathbbm{1}$ | $\frac{1}{2}\chi_{\mathbbm{1}}^{2}(\tau)+\frac{1}{2}\chi_{\mathbbm{1}}(2\tau)$
$(\mathbbm{1},\mathbbm{1})_{a}$ | $4$ | $\Phi^{(1)}$ | $\frac{1}{2}\chi_{\mathbbm{1}}^{2}(\tau)-\frac{1}{2}\chi_{\mathbbm{1}}(2\tau)$
$(\sigma,\sigma)_{s}$ | $\frac{1}{4}$ | $\phi_{2}$ | $\frac{1}{2}\chi_{\sigma}^{2}(\tau)+\frac{1}{2}\chi_{\sigma}(2\tau)$
$(\sigma,\sigma)_{a}$ | $\frac{9}{4}$ | $\phi_{6}$ | $\frac{1}{2}\chi_{\sigma}^{2}(\tau)-\frac{1}{2}\chi_{\sigma}(2\tau)$
$(\epsilon,\epsilon)_{s}$ | $2$ | $\Theta$ | $\frac{1}{2}\chi_{\epsilon}^{2}(\tau)+\frac{1}{2}\chi_{\epsilon}(2\tau)$
$(\epsilon,\epsilon)_{a}$ | $4$ | $\Phi^{(2)}$ | $\frac{1}{2}\chi_{\epsilon}^{2}(\tau)-\frac{1}{2}\chi_{\epsilon}(2\tau)$
Twisted |  |  |
primary (Ising orb.) | $\Delta$ |  primary (boson orb.) | $\chi$
$\tau_{(\mathbbm{1},\hat{0})}$ | $\frac{1}{16}$ | $\phi_{1}$ | $\frac{1}{2}[\chi_{\mathbbm{1}}(\frac{\tau}{2})+T_{\mathbbm{1}}^{-\frac{1}{2}}\chi_{\mathbbm{1}}(\frac{\tau}{2}+\frac{1}{2})]$
$\tau_{(\mathbbm{1},\hat{1})}$ | $\frac{49}{16}$ | $\phi_{7}$ | $\frac{1}{2}[\chi_{\mathbbm{1}}(\frac{\tau}{2})-T_{\mathbbm{1}}^{-\frac{1}{2}}\chi_{\mathbbm{1}}(\frac{\tau}{2}+\frac{1}{2})]$
$\tau_{(\sigma,\hat{0})}$ | $\frac{1}{8}$ | $\sigma^{(2)}$ | $\frac{1}{2}[\chi_{\sigma}(\frac{\tau}{2})+T_{\sigma}^{-\frac{1}{2}}\chi_{\sigma}(\frac{\tau}{2}+\frac{1}{2})]$
$\tau_{(\sigma,\hat{1})}$ | $\frac{9}{8}$ | $\tau^{(2)}$ | $\frac{1}{2}[\chi_{\sigma}(\frac{\tau}{2})-T_{\sigma}^{-\frac{1}{2}}\chi_{\sigma}(\frac{\tau}{2}+\frac{1}{2})]$
$\tau_{(\epsilon,\hat{0})}$ | $\frac{9}{16}$ | $\phi_{3}$ | $\frac{1}{2}[\chi_{\epsilon}(\frac{\tau}{2})+T_{\epsilon}^{-\frac{1}{2}}\chi_{\epsilon}(\frac{\tau}{2}+\frac{1}{2})]$
$\tau_{(\epsilon,\hat{1})}$ | $\frac{25}{16}$ | $\phi_{5}$ | $\frac{1}{2}[\chi_{\epsilon}(\frac{\tau}{2})-T_{\epsilon}^{-\frac{1}{2}}\chi_{\epsilon}(\frac{\tau}{2}+\frac{1}{2})]$
Table 1: Correspondence between the primary fields in the Ising cyclic
$\mathbb{Z}_{2}$ orbifold (first column) and the free boson $\mathbb{Z}_{2}$
orbifold (third column) at $R=4$. $T_{i}=e^{2\pi i(h_{i}-\frac{c}{24})}$ is
the $i$-th diagonal element of the modular $\mathcal{T}$ matrix.

####  IV.1.3 Universal finite-size correction

Now we examine the universal finite-size corrections in Eq. (24). As the Ising
model has a $\mathbbm{Z}_{2}$ global symmetry, the expansion Eq. (45) does not
contain $\tau_{(\sigma,\hat{\chi})},\chi=0,1$. We further consider several
examples.

As a first example, let us compute the universal finite-size correction to the
wavefunction overlap $A_{\sigma\epsilon\sigma}$ for the following allowed OPE
channel:

$\sigma\times\epsilon\rightarrow\sigma.$ (68)

To find all the scaling terms in the wavefunction overlap, we would need to
find all operators $\tau_{(\alpha,\hat{\chi})}$ such that
$C_{(\sigma,\epsilon),(\alpha,\hat{\chi}),(\sigma,\hat{0})}$ is non-zero.
Using the crossing symmetry, we can compute
$C_{(\sigma,\epsilon),(\sigma,\hat{0}),(\alpha,\hat{\chi})}$ equivalently. By
mapping to compactified boson orbifold, $(\sigma,\epsilon)$ is mapped to
$\tau^{(1)}$ and $\tau_{(\sigma,\hat{0})}$ is mapped to $\sigma^{(2)}$. The
fusion rule reads:

$[\tau^{(1)}]\times[\sigma^{(2)}]=[\phi_{1}]+[\phi_{3}]+[\phi_{5}]+[\phi_{7}],$
(69)

which maps back to:

$[(\sigma,\epsilon)]\times[\tau_{(\sigma,\hat{0})}]=[\tau_{(\mathbbm{1},\hat{0})}]+[\tau_{(\epsilon,\hat{0})}]+[\tau_{(\epsilon,\hat{1})}]+[\tau_{(\mathbbm{1},\hat{1})}].$
(70)

The four resulting primaries on the r.h.s. have scaling dimensions
$\Delta=\frac{1}{16},\frac{9}{16},\frac{25}{16},\frac{49}{16}$, respectively,
and the scaling of the wavefunction overlap is thus:

$\displaystyle A_{\sigma\epsilon\sigma}$ $\displaystyle\approx
a_{(\mathbbm{1},\hat{0})}C_{(\sigma,\epsilon),(\mathbbm{1},\hat{0}),(\sigma,\hat{0})}N^{-\frac{1}{16}}$
(71)
$\displaystyle\quad+a_{(\epsilon,\hat{0})}C_{(\sigma,\epsilon),(\epsilon,\hat{0}),(\sigma,\hat{0})}N^{-\frac{9}{16}},$

where we only keep the leading and first subleading terms. The leading term
scales as $N^{-\frac{1}{16}}$ with power agrees with $-\frac{c}{8}$. This
aligns with the fact that the wavefunction overlap for an allowed fusion
process has a leading term that scales as $N^{-\frac{c}{8}}$. After dividing
by $A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}$ and subtracting the leading term,
the universal finite-size correction scales as $N^{-\frac{1}{2}}$, namely,
$p_{\sigma\epsilon\sigma}=\frac{9}{16}-\frac{1}{16}=\frac{1}{2}$. This is
exactly what we get from numerics in Fig. 4.

As a second example, let us study the channel:

$\displaystyle\sigma\times\sigma\rightarrow\epsilon,$ (72)

where we need to find non-zero
$C_{(\sigma,\sigma)_{s},\tau_{p},\tau_{(\epsilon,\hat{0})}}$. The fusion rule
$[\phi_{2}]\times[\phi_{3}]\rightarrow[\phi_{1}]+[\phi_{5}]$ is mapped to
$[(\sigma,\sigma)_{s}]\times[\tau_{(\epsilon,\hat{0})}]\rightarrow[\tau_{(\mathbbm{1},\hat{0})}]+[\tau_{(\epsilon,\hat{1})}]$.
This shows $\tau_{p}$ can take $\tau_{(\mathbbm{1},\hat{0})}$ or
$\tau_{(\epsilon,\hat{1})}$, and the difference of their scaling dimension
predicts $p_{\sigma\sigma\epsilon}=\frac{25}{16}-\frac{1}{16}=\frac{3}{2}$.

Following the same reasoning, for the OPE channel:

$\displaystyle\epsilon\times\epsilon\rightarrow\mathbbm{1},$ (73)

the corresponding orbifold fusion channel is
$[\Theta]\times[\phi_{1}]\rightarrow[\phi_{1}]$, which translates into
$[(\epsilon,\epsilon)_{s}]\times[\tau_{(\mathbbm{1},\hat{0})}]\rightarrow[\tau_{(\mathbbm{1},\hat{0})}]$.
This indicates $\tau_{p}$ lies within the conformal tower of
$[\tau_{(\mathbbm{1},\hat{0})}]$. Indeed, the leading exponent is controlled
by a level-two descendant, which gives $p_{\epsilon\epsilon\mathbbm{1}}=2$.

Finally, we consider the following forbidden channel:

$\mathbbm{1}\times\epsilon\rightarrow\mathbbm{1}.$ (74)

The fusion rule $[\phi_{4}]\times[\phi_{1}]=[\phi_{3}]+[\phi_{5}]$ is
translated to
$[(\mathbbm{1},\epsilon)]\times[\tau_{(\mathbbm{1},\hat{0})}]=[\tau_{(\epsilon,\hat{0})}]+[\tau_{(\epsilon,\hat{1})}]$,
which predicts the leading term in this forbidden channel scales as
$N^{-\frac{9}{16}}$, i.e.,
$p_{\mathbbm{1}\epsilon\mathbbm{1}}=\frac{9}{16}-\frac{1}{16}=\frac{1}{2}$.

We show in Fig. 4 the numerically obtained finite-size scaling
$F_{\alpha\beta\gamma}$ defined by:

$\displaystyle F_{\alpha\beta\gamma}$
$\displaystyle\equiv\frac{\langle\phi^{3}_{\gamma}|\phi^{1}_{\alpha}\phi^{2}_{\beta}\rangle}{\langle\mathbbm{1}^{3}|\mathbbm{1}^{1}\mathbbm{1}^{2}\rangle}-\tilde{A}^{(0)}_{\alpha\beta\gamma}$
(75)
$\displaystyle=\frac{A_{\alpha\beta\gamma}}{A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}}-2^{-2\Delta_{\alpha}-2\Delta_{\beta}+\Delta_{\gamma}}C_{\alpha\beta\gamma},$

which agrees exactly with our prediction that for the leading order,

$\displaystyle\log F_{\alpha\beta\gamma}=-p_{\alpha\beta\gamma}\log
N+\text{const.}$ (76)

For readers’ reference, the finite-size corrections for all fusion channels
are listed in Table 2.

overlap |  Responsible operator $\tau_{p}$ |  Responsible fusion channel |  FS correction leading power $p$
---|---|---|---
$\mathbbm{1}\times\mathbbm{1}\rightarrow\epsilon$ | $[\tau_{(\epsilon,\hat{0})}]$ | $[(\mathbbm{1},\mathbbm{1})_{s}]\times[\tau_{(\epsilon,\hat{0})}]\rightarrow[\tau_{(\epsilon,\hat{0})}]$ | $\frac{1}{2}$
$\mathbbm{1}\times\epsilon\rightarrow\mathbbm{1}$ | $[\tau_{(\epsilon,\hat{0})}]$ | $[(\mathbbm{1},\epsilon)]\times[\tau_{(\epsilon,\hat{0})}]\rightarrow[\tau_{(\mathbbm{1},\hat{0})}]$ | $\frac{1}{2}$
$\epsilon\times\epsilon\rightarrow\mathbbm{1}$ |  $[\tau_{(\mathbbm{1},\hat{0})}]$ des. | $[(\epsilon,\epsilon)_{s}]\times[\tau_{(\mathbbm{1},\hat{0})}]\rightarrow[\tau_{(\mathbbm{1},\hat{0})}]$ | $2$
$\epsilon\times\epsilon\rightarrow\epsilon$ | $[\tau_{(\epsilon,\hat{0})}]$ | $[(\epsilon,\epsilon)_{s}]\times[\tau_{(\epsilon,\hat{0})}]\rightarrow[\tau_{(\epsilon,\hat{0})}]$ | $\frac{1}{2}$
$\mathbbm{1}\times\epsilon\rightarrow\epsilon$ | $[\tau_{(\mathbbm{1},\hat{1})}]$ | $[(\mathbbm{1},\epsilon)]\times[\tau_{(\mathbbm{1},\hat{1})}]\rightarrow[\tau_{(\epsilon,\hat{0})}]$ | $3$
$\mathbbm{1}\times\sigma\rightarrow\sigma$ | $[\tau_{(\epsilon,\hat{0})}]$ | $[(\mathbbm{1},\sigma)]\times[\tau_{(\epsilon,\hat{0})}]\rightarrow[\tau_{(\sigma,\hat{0})}]$ | $\frac{1}{2}$
$\sigma\times\epsilon\rightarrow\sigma$ | $[\tau_{(\epsilon,\hat{0})}]$ | $[(\sigma,\epsilon)]\times[\tau_{(\epsilon,\hat{0})}]\rightarrow[\tau_{(\sigma,\hat{0})}]$ | $\frac{1}{2}$
$\sigma\times\sigma\rightarrow\mathbbm{1}$ | $[\tau_{(\epsilon,\hat{0})}]$ | $[(\sigma,\sigma)_{s}]\times[\tau_{(\epsilon,\hat{0})}]\rightarrow[\tau_{(\mathbbm{1},\hat{0})}]$ | $\frac{1}{2}$
$\sigma\times\sigma\rightarrow\epsilon$ | $[\tau_{(\epsilon,\hat{1})}]$ | $[(\sigma,\sigma)_{s}]\times[\tau_{(\epsilon,\hat{1})}]\rightarrow[\tau_{(\epsilon,\hat{0})}]$ | $\frac{3}{2}$
Table 2: Universal finite-size corrections to wavefunction overlaps of the
ciritical Ising model for all fusion channels. The wavefunction overlaps for
$\mathbbm{1}\times\mathbbm{1}\rightarrow\sigma$ and
$\sigma\times\sigma\rightarrow\sigma$ are strictly zero due to the global
$\mathbb{Z}_{2}$ symmetry in the Ising model. Figure 4: Finite-size
corrections of wavefunction overlap in the critical Ising model, for the four
channels: $\sigma\times\epsilon\rightarrow\sigma$,
$\sigma\times\sigma\rightarrow\epsilon,\epsilon\times\epsilon\rightarrow\mathbbm{1},\mathbbm{1}\times\epsilon\rightarrow\mathbbm{1}$
under discussion. The powers of leading finite-size corrections are
$p=\frac{1}{2},\frac{3}{2},2,\frac{1}{2}$, respectively, from the orbifold
CFT. In the numerical computation we choose the system size
$N_{1}\in[100,500)$, and we see the discrepancy between the theory and
numerical value is less than $1\%$.

###  IV.2 The XXZ model

The XXZ model is a lattice realization of $c=1$ compactified boson with radius
$R$. The free boson has a $U(1)$ current $J=i\partial\varphi$. In the spin
basis, the Hamiltonian is:

$H=-\sum_{i=1}^{N}(X_{i}X_{i+1}+Y_{i}Y_{i+1}+\Delta Z_{i}Z_{i+1}),$ (77)

where the radius $R$ is related to $\Delta$ via $\Delta=\cos(2\pi/R^{2})$. At
$\Delta=0$ (namely, $k=2,R=\sqrt{2k}=2$), the spin Hamiltonian can be Jordan-
Wigner transformed into the free fermion Hamiltonian:

$H=-2\sum_{i=1}^{N}(c_{i}^{\dagger}c_{i+1}+c_{i+1}^{\dagger}c_{i}).$ (78)

In the following, we will focus on this free fermion point for its numerical
efficiency.

####  IV.2.1 Operator content

##### Original theory

At radius $R=\sqrt{2k}$, the valid chiral and anti-chiral momentum modes in
the compactified boson theory are:

$\displaystyle p=\frac{m}{R}+\frac{nR}{2}=\frac{m}{2}+n,$ (79)
$\displaystyle\bar{p}=\frac{m}{R}-\frac{nR}{2}=\frac{m}{2}-n,\quad
m,n\in\mathbb{Z},$

where $m$ is the quantized momentum and $n$ is the quantized winding. The
operators can be labeled by $\mathcal{O}_{(m,n)}$. The partition function is
obtained after summing over contribution from valid momentum modes:

$\displaystyle
Z=\frac{1}{|\eta|^{2}}\sum_{m,n}q^{\frac{1}{2}(\frac{m}{R}+\frac{nR}{2})^{2}}\bar{q}^{\frac{1}{2}(\frac{m}{R}-\frac{nR}{2})^{2}},$
(80)

where $\eta(\tau)$ is the Dedekind eta function. Equivalently, for integer
$k$, this $U(1)$ chiral algebra can be viewed as being extended by two vertex
operators $\\{e^{i\sqrt{2k}\varphi},e^{-i\sqrt{2k}\bar{\varphi}}\\}$, which
allows us to interpret it as $u(1)_{k}$ rational CFT and block diagonalize the
partition function:

$\displaystyle
Z=\frac{1}{|\eta|^{2}}\sum_{l=0}^{2k-1}|\Theta_{l,k}|^{2},\quad\Theta_{l,k}=\sum_{n}q^{k(n+\frac{l}{2k})^{2}}.$
(81)

From the block-diagonal partition function, we read off $2k$ extended
primaries. For the $u(1)_{2}$ rational CFT at $k=2$, there are $2k=4$ extended
primaries with scaling dimension
$\Delta_{l}=\frac{l^{2}}{2k}=\frac{l^{2}}{4},l=0,1,2,3$. The first several
modes in each extended primaries are:

| $(m,n)$
---|---
$\mathbbm{1}$ | (0,0), (0,2), (2,1), (4,0)
$\mathcal{O}_{1}$ | (1,0), $(-3,0)$
$\mathcal{O}_{2}$ | (2,0), $(-2,0)$, (0,1)
$\mathcal{O}_{3}$ | (3,0), $(-1,0)$, (1,1)
(82)

Namely, $\mathcal{O}_{(1,0)},\mathcal{O}_{(-3,0)}$ belong to the conformal
family $\mathcal{O}_{1}$, etc.

The modular matrix $\mathcal{T}$ can be obtained directly from the scaling
dimension: $\mathcal{T}_{ll}=\exp{(2\pi i(\Delta_{l}/2-c/24))}$, and the
modular matrix $\mathcal{S}$ is:

$\displaystyle\mathcal{S}_{l,l^{\prime}}=\frac{1}{\sqrt{2k}}\exp{(-i\pi\frac{ll^{\prime}}{k})}.$
(83)

The original theory modular matrices allow us to compute the modular matrices
in the orbifolded theory.

For later discussions, we focus on the wavefunction overlap associated with
current operator $J=i\partial\varphi$. Specifically, we consider the process:

$J\times J\rightarrow\mathbbm{1}.$ (84)

In the $U(1)$ Kac-Moody algebra, $J$ is the Kac-Moody descendant of
$\mathcal{O}_{(0,0)}=\mathbbm{1}$ (while not being the Virasoro descendant).

##### Orbifold theory

Following the general procedure in Sec. III.2, the untwisted sector has 14
primary operators, including operators like
$(\mathbbm{1},\mathcal{O}_{1}),(\mathcal{O}_{1},\mathcal{O}_{2}),(\mathcal{O}_{1},\mathcal{O}_{1})_{s/a}$,
etc.; and there are 8 primaries in the twisted sector:

| $\tau_{(\mathbbm{1},\hat{0})}$ | $\tau_{(\mathbbm{1},\hat{1})}$ | $\tau_{(\mathcal{O}_{1},\hat{0})}$ | $\tau_{(\mathcal{O}_{1},\hat{1})}$
---|---|---|---|---
$\Delta$ | $\frac{1}{8}$ | $\frac{9}{8}$ | $\frac{1}{4}$ | $\frac{5}{4}$
| $\tau_{(\mathcal{O}_{2},\hat{0})}$ | $\tau_{(\mathcal{O}_{2},\hat{1})}$ | $\tau_{(\mathcal{O}_{3},\hat{0})}$ | $\tau_{(\mathcal{O}_{3},\hat{1})}$
$\Delta$ | $\frac{5}{8}$ | $\frac{21}{8}$ | $\frac{5}{4}$ | $\frac{13}{4}$
(85)

Note that the difference between $\Delta$ for $\tau_{(\mathbbm{1},\hat{0})}$
and $\tau_{(\mathbbm{1},\hat{1})}$ is $1$ rather than $3$ as in the Ising
orbifold. This is because we are considering the Kac-Moody algebra and $J$ is
a level-1 descendant of $\mathbbm{1}$ with respected to the extended symmetry.

####  IV.2.2 Universal finite-size correction

Since the XXZ model has a $U(1)$ global symmetry (which corresponds to $m$
charge conservation in primary operators $\mathcal{O}_{(m,n)}$), the expansion
Eq. (45) does not contain
$\tau_{(\mathcal{O}_{i},\hat{\chi})},i=1,2,3,\chi=0,1$. As a first example,
consider the OPE channel:

$\displaystyle\mathcal{O}_{(1,0)}\times\mathcal{O}_{(-1,0)}\rightarrow\mathbbm{1}.$
(86)

From Eq. (82), $\mathcal{O}_{(1,0)}$ belongs to the conformal family
$\mathcal{O}_{1}$ and $\mathcal{O}_{(-1,0)}$ belongs to the conformal family
$\mathcal{O}_{3}$. We thus need to find $\tau_{p}$ such that
$[(\mathcal{O}_{1},\mathcal{O}_{3})]\times[\tau_{p}]\rightarrow[\tau_{(\mathbbm{1},\hat{0})}]$
is valid in the orbifold theory, namely,
$C_{(\mathcal{O}_{1},\mathcal{O}_{3}),\tau_{p},(\mathbbm{1},\hat{0})}$ is non-
zero. Using the crossing symmetry
$C_{(\mathcal{O}_{1},\mathcal{O}_{3}),\tau_{p},(\mathbbm{1},\hat{0})}=C_{(\mathcal{O}_{1},\mathcal{O}_{3}),(\mathbbm{1},\hat{0}),\tau_{p^{*}}}$
and orbifold fusion rule:

$[(\mathcal{O}_{1},\mathcal{O}_{3})]\times[\tau_{(\mathbbm{1},\hat{0})}]=[\tau_{(\mathbbm{1},\hat{0})}]+[\tau_{(\mathbbm{1},\hat{1})}],$
(87)

we see $\tau_{p}$ can be either $\tau_{(\mathbbm{1},\hat{0})}$ or
$\tau_{(\mathbbm{1},\hat{1})}$. The scaling dimension of the leading term
$\tau_{(\mathbbm{1},\hat{0})}$ gives the leading scaling $\Delta=\frac{1}{8}$,
which agrees with $c/8$. The difference between the scaling dimensions of
$\tau_{(\mathbbm{1},\hat{0})}$ and $\tau_{(\mathbbm{1},\hat{1})}$ gives the
finite-size correction with power $p=\frac{9}{8}-\frac{1}{8}=1$.

We may also consider the channel involving current operator $J$, which belongs
to the conformal family $\mathbbm{1}$. Consider the OPE channel:

$\displaystyle J\times\mathcal{O}_{(1,0)}\rightarrow\mathcal{O}_{(1,0)},$ (88)

we need to find non-zero
$C_{(\mathbbm{1},\mathcal{O}_{1}),\tau_{p},(\mathcal{O}_{1},\hat{0})}$, which
– by crossing symmetry – is,
$C_{(\mathbbm{1},\mathcal{O}_{1}),(\mathcal{O}_{3},\hat{0}),\tau_{p^{*}}}$.
The relevant orbifold fusion rule is:

$\displaystyle[(\mathbbm{1},\mathcal{O}_{1})]\times[\tau_{(\mathcal{O}_{3},\hat{0})}]=[\tau_{(\mathbbm{1},\hat{0})}]+[\tau_{(\mathbbm{1},\hat{1})}].$
(89)

Again, this gives the finite-size correction with power $p=1$ as the previous
example.

In the third example, let us consider:

$\displaystyle J\times J\rightarrow\mathbbm{1},$ (90)

where we need to find non-zero
$C_{(\mathbbm{1},\mathbbm{1})_{s},\tau_{p},(\mathbbm{1},\hat{0})}$. There is
only one valid $\tau_{p}=[\tau_{(\mathbbm{1},\hat{0})}]$ from:

$[(\mathbbm{1},\mathbbm{1})_{s}]\times[\tau_{(\mathbbm{1},\hat{0})}]=[\tau_{(\mathbbm{1},\hat{0})}].$
(91)

In this case, we observe $p=2$, which means a level-2 descendant in the
expansion Eq. (45) is responsible. Similarly, the process
$J\times\mathcal{O}_{(2,1)}\rightarrow\mathcal{O}_{(2,1)}$ also has correction
$p=2$ using the same orbifold fusion rule.

Figure 5: Finite-size correction of wavefunction overlap of the XXZ model at
$\Delta=0$, for the three channels under discussion. The system size is chosen
in the range $N_{1}\in[100,500)$. The discrepancy between the theory and
numerical value is less than $0.1\%$.

For other radius $R$, we have to resort to exact diagonalization or tensor
network methods. We have also examined the case of $R=\sqrt{6}$ and checked
that the finite-size correction exponents for various channels, in particular,
the channel involving $J$, agree with analytical predictions.

##  V Applications to the Haagerup model

In the previous section, we have examined the universal finite-size
corrections in the Ising model and XXZ model. We now move on to the Haagerup
model, which is a recently proposed [17, 18] anyonic chain that is believed to
be described by a CFT with $c\sim 2$. The input data for the lattice
Hamiltonian is the $\mathcal{H}_{3}$ fusion category. The fusion rule in this
category is non-commutative thus does not admit braiding. The operator content
of the Haagerup CFT is still obscure due to strong finite-size corrections as
only very small systems are numerically accessible. Due to its
$\mathbb{Z}_{3}$ global symmetry, one of the candidate theory is the
$\mathbb{Z}_{3}$ orbifold of toroidal compactified sigma model, with central
charge $c=2$. The candidate has two current operators $J$ and $J^{*}$ with
scaling dimension and conformal spin $\Delta=s=1$. The holomorphic nature of
the current operators strongly constrain possible fusion channels involving
these operators, which enables us to check it numerically.

In what follows, we use the wavefunction overlaps to compute central charge,
conformal dimensions of certain primary states. We present strong evidence
that the lowest operator with $s=1$ is not a holomorphic field, thus ruling
out the possibility of this $\mathbb{Z}_{3}$ orbifold. Nevertheless, we will
denote the operators as $J$ and $J^{*}$. Our results give an upper bound of
the scaling dimension $\Delta_{J}\leq 1.4$, and indicate that the central
charge is roughly $c\approx 2.1$. Our code for numerical calculation is
available in GitHub
444https://github.com/YuhanLiuSYSU/Haagerup_wave_function_overlap.

###  V.1 Haagerup anyon chain and spectrum

####  V.1.1 $\mathcal{H}_{3}$ fusion category

In the following we give a short review on $\mathcal{H}_{3}$ fusion category
[28, 29]. There are six simple objects (anyons) in this category:

$1,a,a^{2},\rho,a\rho,a^{2}\rho.$ (92)

We will use $a^{2}$ and $a^{*}$ interchangeably. $1,a,a^{*}$ are invertible
objects and $\rho,a\rho,a^{*}\rho$ are non-invertible simple objects. Their
quantum dimensions are: $d_{1}=d_{a}=d_{a^{*}}=1$,
$d_{\rho}=d_{a\rho}=d_{a^{*}\rho}=\frac{3+\sqrt{13}}{2}$. The non-trivial
fusion rules read:

$a^{3}=1,\quad a\rho=\rho a^{*}\quad\rho^{2}=1+(1+a+a^{2})\rho.$ (93)

Note that for example, $a$ and $\rho$ does not commute, $a\rho\neq\rho a$.
Given the fusion rule, the $F$-symbols can be determined by the pentagon
equations (self-consistency requirement).

Figure 6: Energy-momentum spectrum of the Haagerup model at (a) $N=12$, and
(b) $N=13$.

####  V.1.2 Anyon chain construction and spectrum

Given a fusion category as input data, the anyon chain Hamiltonian is
constructed using the fusion rule and the $F$-symbol [30, 31, 17]. The Hilbert
space consists of the anyon chains $|a_{1},a_{2},\cdots,a_{N}\rangle$ that
satisfy the following fusion constraint:

(94)

Namely, the fusion $a_{i}\times\rho$ needs to contain $\rho$. We assume
periodic boundary condition for the anyon chain.

The Hamiltonian is then constructed as sum of projectors acting on three
subsequent anyon sites [17, 30]:

$\displaystyle H=-\sum_{i}P_{\rho}^{(i)},$ (95) $\displaystyle
P_{c}^{(i)}|a_{i-1}a_{i}a_{i+1}\rangle=$
$\displaystyle\quad\sum_{a_{i}^{\prime}}[F_{a_{i+1}}^{a_{i-1}\rho\rho}]_{a_{i}c}[F_{a_{i+1}}^{a_{i-1}\rho\rho}]^{*}_{a_{i}^{\prime}c}|a_{i-1}a_{i}^{\prime}a_{i+1}\rangle.$

The projector $P_{\rho}$ projects the fusion $\rho\times\rho$ into channel
$\rho$. The above anyonic chain construction is standard, and can be regarded
as a generalization of the spin-$1/2$ Heisenberg chain [31].

It is nontrivial and has to be verified case-by-case whether an anyonic chain
Hamiltonian corresponds to a lattice realization of a CFT. One should
distinguish the input fusion category with the Moore-Seiberg tensor category
of a rational CFT, where the object of the Moore-Seiberg tensor category are
in one-to-one correspondence with primary operators which is not true for the
input fusion category. For the Haagerup model, previous evidence of CFT comes
from ground state energy and entanglement entropy [17]. We will see in the
following that the ground state wavefunction overlap
$A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}$ provides another evidence for CFT with
consistent central charge.

By the Haagerup fusion rules, the Hamiltonian has a global $\mathbb{Z}_{3}$
symmetry:

$\displaystyle\gamma:1\rightarrow a\rightarrow a^{*}\rightarrow
1,\quad\rho\rightarrow a\rho\rightarrow a^{*}\rho\rightarrow\rho.$ (96)

This can also be seen at the level of $F$-symbols. There is another intrinsic
shift $\mathbb{Z}_{3}$ symmetry $s$, that is, translation by $3n,3n+1$ and
$3n+2$ sites. The shift symmetry is nontrivial because the energy spectrum
depends on $N$ mod $3$. Similar phenomena occur in the antiferromagnetic Ising
model, which we review in Appendix B. To gauge the shift $\mathbb{Z}_{3}$
symmetry $s$, we keep only the states near $P=0$ at system size
$N=3n,3n+1,3n+2$, which correspond to the original Hilbert space
$\mathcal{H}$, shift $\mathbb{Z}_{3}$ twisted Hilbert space $\mathcal{H}_{s}$
and $\mathcal{H}_{s^{2}}$. It is clear from the $\mathbb{Z}_{3}$ symmetry that
$\mathcal{H}_{s}=\mathcal{H}_{s^{2}}$.

The energy-momentum spectrum of $N=12$ ($\mathcal{H}$) and $N=13$
($\mathcal{H}_{s}$) anyon chain is shown in Fig. 6. To better understand the
conformal towers, we also use the Koo-Saleur lattice Virasoro generators as in
Refs. [32, 33, 9],

$H_{n}=\frac{N}{2\pi}\sum_{j=1}^{N}h_{j}e^{ijn\frac{2\pi}{N}},$ (97)

where $h_{j}$ is the Hamiltonian density operator at site $j$. These operators
correspond to linear combinations of the Virasoro generators,

$H_{n}=L_{n}+\bar{L}_{-n}-\frac{c}{12}\delta_{n,0}.$ (98)

There are several numerical observations with the Koo-Saleur generators:

  1. 1.

There is only one spin-2 low-energy state that has nonzero overlap with
$H_{-2}|\mathbbm{1}\rangle$, which we identify as the stress tensor state
$|T\rangle$. This is used to normalize the energies in the plot by utilizing
$|T\rangle$ has $\Delta=2$. $H_{-1}$ acting on $|\mathbbm{1}\rangle$ gives
zero overlap for every state in the spectrum, confirming this state is the
conformal vacuum.

  2. 2.

In the spectrum of $N=13$ chain (shift $\mathbb{Z}_{3}$ twisted sector
$\mathcal{H}_{s}$), there exists a state with $\mathbb{Z}_{3}=0$ being close
to $P=1,\Delta=1$. This is proposed to be the chiral current operator $J$.
There is a similar state in the $N=14$ chain, which is proposed to be $J^{*}$.

  3. 3.

In the $N=13$ chain, $H_{-1}$ acting on the lowest primary state in
$\mathbb{Z}_{3}=0,P=0$ gives the largest overlap with the second lowest state
in $\mathbb{Z}_{3}=0,P=1$. This overlap is significantly larger than that with
the first excited state with $\mathbb{Z}_{3}=0,P=1$. This indicates the lowest
state in $\mathbb{Z}_{3}=0,P=1$ is not a descendant of the primary state. Thus
it is a primary state.

####  V.1.3 $\mathbb{Z}_{3}$ orbifold of sigma model

The $\mathbb{Z}_{3}$ orbifold theory referred in this paper is the nonlinear
sigma model on torus target space modded out the $\mathbb{Z}_{3}$ rotational
symmetry on the target space. We will briefly review the the relevant features
of this CFT below.

Let us start with the toroidal compactified boson theory, namely, there are
two bosonic fields and the target space is $T^{2}$ [34]. Recall that in the
familiar example of compactified boson, the target space is $S^{1}$, which is
equivalent to saying that the boson field $\Phi$ can only take value in
$\Phi\in[0,2\pi R)$, and $\Phi\sim\Phi+2n\pi R,n\in\mathbb{Z}$ are identified.
Analogously, the toroidal compactification free boson theory has two massless
scalar fields $\Phi^{\mu},\mu\in\\{1,2\\}$. For the target space torus $T^{2}$
generated by $\bm{\lambda}_{1},\bm{\lambda}_{2}$, we need to identify:

$\left(\begin{array}[]{c}\Phi^{1}\\\
\Phi^{2}\end{array}\right)\sim\left(\begin{array}[]{c}\Phi^{1}\\\
\Phi^{2}\end{array}\right)+n_{1}\bm{\lambda}_{1}+n_{2}\bm{\lambda}_{2},\quad
n_{1}\in\mathbb{Z},n_{2}\in\mathbb{Z}.$ (99)

This compactification gives constraint on the momentum modes $p,\bar{p}$, from
which one can write down the partition function [34].

When the target space torus $T^{2}$ is generated by basis vectors proportional
to $\bm{\lambda}_{1}=(1,0)^{T}$,
$\bm{\lambda}_{2}=(-\frac{1}{2},\frac{\sqrt{3}}{2})^{T}$, the theory has an
additional $\mathbb{Z}_{3}$ rotational symmetry. Gauging (modding out) this
$\mathbb{Z}_{3}$ rotational symmetry gives the $\mathbb{Z}_{3}$ orbifold
theory. The partition function of this rational CFT can be written out
explicitly, from which we can read out the operator content. We list the
several important features that are relevant to our problem, and refer the
interested readers to Ref. [34] for more details. The features are:

  1. 1.

The spectrum includes two chiral current operators: $J_{1}=i\partial\Phi^{1}$
and $J_{2}=i\partial\Phi^{2}$. This is true for all toroidal compactified
theory, independent of the $\mathbb{Z}_{3}$ symmetry.

  2. 2.

In the $\mathbb{Z}_{3}$ twisted sector, the lowest energy state is three-fold
degenerate, with $(h,\bar{h})=(\frac{1}{9},\frac{1}{9})$. Similar feature
appears in $\mathbb{Z}_{3}^{2}$ twisted sector. The degeneracy comes from the
three target space fixed points under rotation.

The exact degeneracy in the second feature is hard to confirm numerically, as
the states suffer from strong finite-size corrections and the energies do not
exactly coincide. However, the wavefunction overlap provides a way to check
the first feature, with relatively small finite-size corrections. We will see
below that the overlap indicates that the lowest spin-1 operator is not a
current, which contradicts the proposal that Haagerup model is described by
this $\mathbb{Z}_{3}$ orbifold.

###  V.2 Wavefunction overlaps

We now turn to the wavefunction overlaps, from which we can extract conformal
data, such as central charge, scaling dimensions and OPE coefficients. We
perform exact diagonalization for small system sizes $N\leq 15$ and the
periodic uniform matrix product state (puMPS) techniques for up to $N=27$. The
latter is briefly reviewed in Appendix C.

Firstly, from the conformal vacuum state overlap, we can extract the central
charge via:

$\displaystyle A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}\propto N^{-\frac{c}{8}}.$
(100)

We obtain $c=2.06$, in a reasonable agreement with previous results, which
give $c=2.03$ [18] and $c=2.11$ [17].

Figure 7: Ground state overlap $A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}$ for the
Haagerup model with $N,N$ and $2N$ spins, where $N=3,6,9,12$. The first two
points are obtained by exact diagonalization, and the last two points are
obtained by puMPS with maximal bond dimension $D=66$.

If we discard the data from the smallest system size ($N=3$), then we obtain
$c=2.12$ by fitting the remaining three points. Given that current numerical
methods all give $c>2$, it is possible that the true central charge is not
exactly at $c=2$, but $c\approx 2.1$.

Secondly, we compute the scaling dimension of the first spinless
$\mathbb{Z}_{3}$ neutral excited state, which we denote as
$|\varepsilon\rangle$. The relevant overlap is

$\displaystyle\frac{A_{\mathbbm{1}\mathbbm{1}\varepsilon}}{A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}}\propto
N^{-\Delta_{\varepsilon}/2}.$ (101)

As shown in Fig. 8, we obtain $\Delta_{\varepsilon}\approx 0.26$. This agrees
with the energy spectrum with reasonable accuracy.

Figure 8:
$A_{\mathbbm{1}\mathbbm{1}\varepsilon}/A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}$
for the Haagerup model. The states are taken from the Haagerup model with
sizes $3n,3n$ and $N=6n$, where $n=2,3,4$.

Thirdly, we compute the $A_{JJ^{*}\mathbbm{1}}$ where $|J\rangle$ is the
lowest eigenstate with $s_{J}=1$ at size $N=3n+1$. Similarly, $J^{*}$ is the
lowest eigenstate with $s_{J^{*}}=1$ at size $N=3n+2$. Suppose the operator
$J$ and $J^{*}$ are chiral currents with scaling dimension $1$, then they
would be Kac-Moody descendants of $\mathbbm{1}$, and the wavefunction overlap
$A_{JJ^{*}\mathbbm{1}}$ would only involve one fusion channel in the cyclic
orbifold,

$\displaystyle[(\mathbbm{1},\mathbbm{1})_{s}]\times[\tau_{(\mathbbm{1},\hat{0})}]=[\tau_{(\mathbbm{1},\hat{0})}].$
(102)

This implies that the finite-size corrections to $A_{JJ^{*}\mathbbm{1}}$ would
come from descendants of $\tau_{(\mathbbm{1},\hat{0})}$, thus we would expect

$\frac{A_{JJ^{*}\mathbbm{1}}}{A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}}=\frac{1}{16}+O(N^{-2})$
(103)

similar to the XXZ model considered in Sec. IV. However, our numerical result
shown in Fig. 9 violates Eq. (103) as the extrapolation gives
$A_{JJ^{*}\mathbbm{1}}/{A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}}\approx 0.02$.
Therefore we can conclude that $J$ is not a current. The scaling dimension can
be estimated by

$\Delta_{J}=-\frac{1}{4}\log_{2}\frac{A_{JJ^{*}\mathbbm{1}}}{A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}},$
(104)

which gives $\Delta_{J}\approx 1.4$. Considering the error in finite size
extrapolations, we conclude that $\Delta_{J}\leq 1.4$.

Figure 9: $A_{JJ^{*}\mathbbm{1}}/A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}$ for
the Haagerup model. In $A_{JJ^{*}\mathbbm{1}}$ the three states are taken from
the Haagerup model with sizes $3n+1,3n+2$ and $N=6n+3$. In
$A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}$, the three states are taken from the
Haagerup model with sizes $3n,3n+3$ and $N=6n+3$. The three points in the plot
correspond to $n=2,3,4$.

##  VI Discussion

In this work we have studied wavefunction overlaps
$A_{\alpha\beta\gamma}=\langle\phi_{\gamma}|\phi_{\alpha}\phi_{\beta}\rangle$
of a critical quantum spin chain, where the three states are primary states of
the spin chain. Through a conformal mapping from the three-sided cylinder to
the complex plane, we derive the relation between OPE coefficients
$C_{\alpha\beta\gamma}$ and the ratio of wavefunction overlap
$A_{\alpha\beta\gamma}/A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}$. In order to
study the finite-size corrections, we rewrite $A_{\alpha\beta\gamma}$ as a
path integral of the cyclic orbifold on the cylinder, with twist operator
insertion at the branch points. We then show that finite-size correction
exponents $p_{\alpha\beta\gamma}$ are universal, and are completely determined
by the orbifold operator content. As a benchmark, we extracted OPE
coefficients $C_{\alpha\beta\gamma}$ and finite-size correction exponents
$p_{\alpha\beta\gamma}$ for the Ising model and the XXZ model and found that
they agree with the prediction using the cyclic orbifold.

We then illustrate an application of our method to a less-known CFT. We have
computed the wavefunction overlaps for the Haagerup model, and found that it
has central charge $c\approx 2.1$, lowest spinless primary operator with
dimension $\Delta_{\varepsilon}\approx 0.3$ and lowest spin-1 primary operator
with dimension $\Delta_{J}\leq 1.4$. The fact that
$A_{JJ^{*}\mathbbm{1}}/A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}$ is not close to
$1/16$ provides negative evidence against the conjecture that $J$ is a current
operator, so the Haagerup CFT is likely more complicated than the free boson
$\mathbb{Z}_{3}$ orbifold.

As noted in Ref. [11], the overlap $A_{\alpha\beta\gamma}$ can be viewed as a
wavefunction of a multiboundary wormhole state. In this work we interpret the
same overlap as a three-point correlation function involving twist operators
in the cyclic orbifold. It would be interesting for future work to explore the
holographic interpretation of twist operators and cyclic orbifold.

Given three primary states, it is possible to construct other overlaps that
are inequivalent to the three-sided cylinder, but nevertheless also contain
similar universal information of the CFT. One useful example is given in Ref.
[35], where the authors consider the vertex state that describes a trisection
of a two-dimensional chiral topological phase (see Appendix E for details). It
is shown in the appendix that conformal data, such as central charge and OPE
coefficients, can be extracted from the wavefunction of the vertex state, with
finite-size corrections that appear universal. It is an open question whether
the finite-size corrections to these wavefunction overlaps are indeed
universal, and if so, what operators are responsible. More generally, given a
bunch of primary states, it is possible to construct many different overlaps,
which may correspond to multi-point correlation functions of the CFT. The
finite-size corrections remain elusive at this point.

In conclusion, we have developed a method based on wavefunction overlaps that
compute conformal data solely from low-energy eigenstates of a critical
quantum spin chain. We have shown that the finite-size corrections are
universal, which provides consistency conditions on the wavefunction overlaps
at finite sizes. It is interesting both practically and conceptually.
Practically, the method is useful for identifying new CFTs realized in lattice
models without too much prior knowledge. Conceptually, the method relates
wavefunction overlaps to operator fusion in the cyclic orbifold, which may be
easily generalized.

###### Acknowledgements.

We acknowledge useful discussions with Ramanjit Sohal, Yuji Tachikawa, Masaki
Tezuka, and Kantaro Ohmori. YZ acknowledges useful discussions with Qi Hu and
Guifre Vidal. YZ is supported by the Q-FARM fellowship at Stanford University.
SR is supported by the National Science Foundation under Award No.
DMR-2001181, and by a Simons Investigator Grant from the Simons Foundation
(Award No. 566116).

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##  Appendix A Fusion rules of cyclic orbifold from modular transformations

We first recall basic aspects of modular invariance which are crucial to the
consistency of a CFT defined on a torus. A torus is characterized by two
period vectors $w_{1}$ and $w_{2}$, which can be represented as complex
numbers. Scale invariance of CFT implies that the partition function only
depend on the modular parameter $\tau=w_{2}/w_{1}$. There are two basic
operations $\mathcal{S}:\tau\rightarrow-1/\tau$ and
$\mathcal{T}:\tau\rightarrow\tau+1$ that generate equivalent descriptions of
the same torus. Therefore, the two operations generate the modular group under
which the CFT partition function is invariant.

Modular invariance puts strong constraints on the operator content of the CFT.
To see this, we first recall that the partition function is bilinear in the
character of conformal towers,

$\chi_{a}(\tau)=\textrm{Tr}_{a}\,q^{L_{0}-c/24},$ (105)

where $q=\exp(2\pi i\tau)$, and $a$ is the label of primary operator.
Expanding the character in the power of $q$ gives degeneracy of the conformal
tower at each level of descendants. In a diagonal theory, the partition
function is simply

$Z(\tau)=\sum_{a}|\chi_{a}(\tau)|^{2},$ (106)

where the summation is finite for minimal model 555For rational CFT, the
partition function can be organized into finite summation of extended
characters. See the example of compactified boson (the XXZ model in the main
text)., for example, the Ising CFT. Modular invariance then demands that

$\displaystyle\chi_{a}(-1/\tau)$ $\displaystyle=$
$\displaystyle\sum_{b}\mathcal{S}_{ab}\chi_{b}(\tau),$ (107)
$\displaystyle\chi_{b}(\tau+1)$ $\displaystyle=$
$\displaystyle\sum_{b}\mathcal{T}_{ab}\chi_{b}(\tau),$ (108)

where $\mathcal{S}$ and $\mathcal{T}$ are unitary matrices that form a
representation of the modular group. One can further show that

$\mathcal{T}_{ab}=e^{2\pi i(h_{a}-c/24)}\delta_{ab}$ (109)

is diagonal with $h_{a}=\Delta_{a}/2$ and

$\mathcal{S}_{ab}=\mathcal{S}_{ba}$ (110)

is symmetric. The celebrated Verlinde formula relates the fusion rule to the
modular matrices,

$\mathcal{N}_{i,j,k}=\sum_{m}\frac{\mathcal{S}_{im}\mathcal{S}_{jm}\mathcal{S}^{*}_{mk}}{S_{\mathbbm{1}m}}.$
(111)

Next, we review how characters of the orbifold theory can be constructed from
the original theory and use the Verlinde formula to derive the fusion rules.

The characters in the orbifold theory are related to the characters in the
original theory via:

$\displaystyle\chi_{(\alpha,\beta)}(\tau)=\chi_{\alpha}(\tau)\chi_{\beta}(\tau)$
(112)
$\displaystyle\chi_{(\alpha,\alpha)_{\phi}}(\tau)=\frac{1}{2}\left[\chi_{\alpha}^{2}(\tau)+e^{i\pi\phi}\chi_{\alpha}(2\tau)\right]$
$\displaystyle\chi_{(\alpha,\hat{\psi})}(\tau)=\frac{1}{2}\left[\chi_{\alpha}(\frac{\tau}{2})+e^{i\pi\psi}T_{\alpha}^{-1/2}\chi_{\alpha}(\frac{\tau}{2}+\frac{1}{2})\right].$

Here $T_{\alpha}=\exp{(2\pi i(h_{\alpha}-c/24))}$ is the diagonal matrix
element in $\mathcal{T}$ matrix, $\phi$ can take $0$ (symmetric sector,
denoted as $s$ in the main text) or $1$ (antisymmetric sector, denoted as $a$
in the main text), and $\psi$ can take $0$ or $1$. The partition function of
the orbifold theory is:

$\displaystyle Z$ $\displaystyle=Z_{un}+Z_{tw},$ (113) $\displaystyle Z_{un}$
$\displaystyle=\sum_{\alpha<\beta}|\chi_{(\alpha,\beta)}(\tau)|^{2}+\sum_{\alpha,\phi}|\chi_{(\alpha,\alpha)_{\phi}}(\tau)|^{2}$
$\displaystyle=\frac{1}{2}\sum_{\alpha\beta}|\chi_{\alpha}(\tau)|^{2}|\chi_{\beta}(\tau)|^{2}+\frac{1}{2}\sum_{\alpha}|\chi_{\alpha}(2\tau)|^{2},$
$\displaystyle Z_{tw}$
$\displaystyle=\sum_{\alpha,\psi}|\chi_{(\alpha,\hat{\psi})}(\tau)|^{2}$
$\displaystyle=\frac{1}{2}\sum_{\alpha}(|\chi_{\alpha}(\frac{\tau}{2})|^{2}+|\chi_{\alpha}(\frac{\tau}{2}+\frac{1}{2})|^{2}).$

We see that since the characters of $\phi_{(\alpha,\alpha)_{s/a}}$ include
$\chi_{\alpha}(2\tau)$, it is necessary to include the twisted sector with
characters $\chi_{\alpha}(\tau/2)$ and $\chi_{\alpha}(\tau/2+1/2)$ such that
the partition function is modular invariant.

For the purpose of computing wavefunction overlap, we need to use the
following two orbifold fusion coefficients:
$\mathcal{N}_{(\alpha,\beta),{(\gamma,\hat{\psi})},{(\delta,\hat{\chi})}}$,
$\alpha\neq\beta$ and
$\mathcal{N}_{(\alpha,\alpha)_{s},{(\gamma,\hat{\psi})},{(\delta,\hat{\chi})}}$,
where $\psi,\chi$ take value in $\\{0,1\\}$.

We now derive the Eq. (37), (38) in the main text. Using the Verlinde formula,
the first fusion coefficient can be expressed as:

$\displaystyle\mathcal{N}_{(\alpha,\beta),{(\gamma,\hat{\psi})},{(\delta,\hat{\chi})}}$
(114)
$\displaystyle=\sum_{m}\frac{\mathcal{S}_{(\alpha,\beta),m}\mathcal{S}_{(\gamma,\hat{\psi}),m}\mathcal{S}^{*}_{m,(\delta,\hat{\chi})}}{\mathcal{S}_{(\mathbbm{1},\mathbbm{1})_{s},m}}$
$\displaystyle=\sum_{\eta,\phi}\frac{\mathcal{S}_{(\alpha,\beta),(\eta,\eta)_{\phi}}\mathcal{S}_{(\gamma,\hat{\psi}),(\eta,\eta)_{\phi}}\mathcal{S}^{*}_{(\eta,\eta)_{\phi},(\delta,\hat{\chi})}}{\mathcal{S}_{(\mathbbm{1},\mathbbm{1})_{s},(\eta,\eta)_{\phi}}}$
$\displaystyle=\sum_{\eta,\phi}\frac{\left(\mathcal{S}_{\alpha\eta}\mathcal{S}_{\beta\eta}\right)\left(\frac{1}{2}e^{i\pi\phi}\mathcal{S}_{\gamma\eta}\right)\left(\frac{1}{2}e^{-i\pi\phi}\mathcal{S}^{*}_{\eta\delta}\right)}{\frac{1}{2}\mathcal{S}_{\mathbbm{1}\eta}^{2}}$
$\displaystyle=\sum_{\eta}\frac{\mathcal{S}_{\alpha\eta}\mathcal{S}_{\beta\eta}\mathcal{S}_{\gamma\eta}\mathcal{S}^{*}_{\eta\delta}}{\mathcal{S}_{\mathbbm{1}\eta}^{2}}.$

From the second to the third line, we use the relation between the orbifold
modular $\mathcal{S}$ matrix and the original $\mathcal{S}$ matrix, which can
be derived using the orbifold character Eq. (112) as in [23]666Note that there
is a typo in the fusion coefficient $\mathcal{N}$ derived in [23], where the
conjugate sign is missing.. For the case where $\mathcal{S}$ is real (e.g.,
the Ising CFT), this still gives the correct result; while for the general
case where $\mathcal{S}$ is complex (e.g., the compactified boson rational
CFT), this would change the final expression. Only when $\mathcal{S}$ is real
can we decompose
$\mathcal{N}_{(\alpha,\beta),{(\gamma,\hat{\psi})},{(\delta,\hat{\chi})}}$
into the product of $\mathcal{N}$ in the original CFT. In short, the fusion
rule of the orbifold CFT is fully determined by the modular matrices of the
original theory.

Similarly, the second fusion coefficient can be written as:

$\displaystyle\mathcal{N}_{(\alpha,\alpha)_{s},{(\gamma,\hat{\psi})},{(\delta,\hat{\chi})}}$
(115)
$\displaystyle=\sum_{m}\frac{\mathcal{S}_{(\alpha,\alpha)_{s},m}\mathcal{S}_{(\gamma,\hat{\psi}),m}\mathcal{S}^{*}_{m,(\delta,\hat{\chi})}}{\mathcal{S}_{(\mathbbm{1},\mathbbm{1})_{s},m}}$
$\displaystyle=\sum_{\eta,\phi}\frac{\mathcal{S}_{(\alpha,\alpha)_{s},(\eta,\eta)_{\phi}}\mathcal{S}_{(\gamma,\hat{\psi}),(\eta,\eta)_{\phi}}\mathcal{S}^{*}_{(\eta,\eta)_{\phi},(\delta,\hat{\chi})}}{\mathcal{S}_{(\mathbbm{1},\mathbbm{1})_{s},(\eta,\eta)_{\phi}}}+\sum_{\eta,\xi}\frac{\mathcal{S}_{(\alpha,\alpha)_{s},(\eta,\hat{\xi})}\mathcal{S}_{(\gamma,\hat{\psi}),(\eta,\hat{\xi})}\mathcal{S}^{*}_{(\eta,\hat{\xi}),(\delta,\hat{\chi})}}{\mathcal{S}_{(\mathbbm{1},\mathbbm{1})_{s},(\eta,\hat{\xi})}}$
$\displaystyle=\sum_{\eta,\phi}\frac{\left(\frac{1}{2}\mathcal{S}^{2}_{\alpha\eta}\right)\left(\frac{1}{2}e^{i\pi\phi}\mathcal{S}_{\gamma\eta}\right)\left(\frac{1}{2}e^{-i\pi\phi}\mathcal{S}^{*}_{\eta\delta}\right)}{\frac{1}{2}\mathcal{S}_{\mathbbm{1}\eta}^{2}}+\sum_{\beta,\xi}\frac{\left(\frac{1}{2}\mathcal{S}_{\alpha\eta}\right)\left(\frac{1}{2}e^{i\pi(\psi+\xi)}\mathcal{P}_{\gamma\eta}\right)\left(\frac{1}{2}e^{-i\pi(\xi+\chi)}\mathcal{P}^{*}_{\eta,\delta}\right)}{\frac{1}{2}\mathcal{S}_{\mathbbm{1}\eta}}$
$\displaystyle=\frac{1}{2}\sum_{\eta}\frac{\mathcal{S}_{\alpha\eta}^{2}\mathcal{S}_{\gamma\eta}\mathcal{S}^{*}_{\eta\delta}}{\mathcal{S}^{2}_{\mathbbm{1}\eta}}+\frac{1}{2}e^{i\pi(\psi+\chi)}\sum_{\eta}\frac{\mathcal{S}_{\alpha\eta}\mathcal{P}_{\gamma\eta}\mathcal{P}^{*}_{\eta,\delta}}{\mathcal{S}_{\mathbbm{1},\eta}},$

where
$\mathcal{P}=\mathcal{T}^{1/2}\mathcal{S}\mathcal{T}^{2}\mathcal{S}\mathcal{T}^{1/2}$.

##  Appendix B Wavefunction overlaps for antiferromagnetic Ising model

In the main text, we consider the ferromagnet Ising model, where
$\mathbbm{1},\sigma,\epsilon$ are the primary operators when taking periodic
boundary condition (PBC), and $\mu,\psi,\bar{\psi}$ appears in anti-periodic
boundary condition (APBC). There is no distinction of low-energy spectrum for
even and odd number of sites, and all the primary operators are inside the
light cone at $P=0$.

On the other hand, the low-energy spectrum of the antiferromagnetic Ising
model depends on even or odd number of sites. Furthermore, there exists
lightcones centered at $P=0$ as well as $P=\pi$. The Haagerup model is more
similar to the antiferromagnetic Ising model, in the sense that (1) the
operator content changes with different system sizes $N=3n,3n+1,3n+2$; and (2)
there appear three light cones which are located at $P=0,P=\pi/3,P=-\pi/3$ if
$N=3n$. For this reason, we study the wavefunction overlap for the
antiferromagnetic Ising model in this appendix, with an emphasis on the case
where low-energy eigenstates with odd number of sites fuse into low-energy
eigenstates with even number of sites. This provides an analogy to what we did
for the Haagerup CFT.

The operator content of the antiferromagnetic Ising model is summarized below,
for both boundary conditions (PBC/APBC) and for both types of lattice site
(even/odd):

| spin PBC | spin APBC | fermion
---|---|---|---
even | $\mathbb{1},\epsilon:P=0$ | $\psi,\bar{\psi}:P=0$ | NS
| $\sigma:P=\pi$ | $\mu:P=\pi$ | R
odd | $\psi,\bar{\psi}:P=\pi$ | $\mathbb{1},\epsilon:P=\pi$ | NS
| $\mu:P=0$ | $\sigma:P=0$ | R
(116)

A possible fusion may be, for example, obtaining $A_{\sigma\sigma\mathbbm{1}}$
by fusing $\sigma$ at size $N_{1}=2N+1$ and $\sigma$ at size $N_{2}=2N-1$ into
$\mathbbm{1}$ at size $N_{3}=4N$. This is analog to what we did for the
current operator in the Haagerup model. We then take the ratio
$A_{\sigma\sigma\mathbbm{1}}/A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}$ where the
$\mathbbm{1}$ in the denominator is taken from $N_{1}=N_{2}=2N$ and
$N_{3}=4N$. For a better comparison with Haagerup model, we use the small
systems, where $N$ takes value from 5 to 12, and show the results in Fig. 10.

In Fig. 10(a), we show the scaling of $A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}$
computed using $N_{1}=2N,N_{2}=2N,N_{3}=4N$, where the theoretical derivation
gives
$A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}=a_{(\mathbbm{1},\hat{0})}N^{-1/16}$.
The fit gives the power $-0.0623$, which is within $1\%$ error even for such a
small system. In Fig. 10(b-f), we show the finite-size correction
$F_{\alpha\beta\gamma}$ for the five different channels
$\sigma\sigma\epsilon,\mu\mu\epsilon,\sigma\sigma\mathbbm{1},\mu\mu\mathbbm{1},\epsilon\epsilon\mathbbm{1}$,
where $A_{\alpha\beta\gamma}$ is computed using
$N_{1}=2N-1,N_{2}=2N+1,N_{3}=4N$, and $A_{\alpha\beta\gamma}^{(0)}$ is
computed using Eq. (21) with different system size. From the value of $\log
F$, we see in (f) the numerical value approaches the theoretical value much
quicker that (b,c); and (b,c) are quicker than (d,e). This is expected because
$p=2$ for $\epsilon\epsilon\mathbbm{1}$, $p=\frac{3}{2}$ for
$\sigma\sigma\mathbbm{1}$ and $\mu\mu\mathbbm{1}$; and $p=\frac{1}{2}$ for
$\sigma\sigma\epsilon,\mu\mu\epsilon$. At this system size, the largest error
for $p$ is $8\%$, and the error for $p$ gets smaller with increasing $N$.

The above observation shows that the overlap result
$A_{\alpha\beta\gamma}/A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}$ is indeed more
accurate for larger $p_{\alpha\beta\gamma}$.

The extraction for $p$ matches with the value in Table 2 as we further
increase the system size $N$.

Figure 10: (a) Scaling of $A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}$, computed
for $N_{1}=2N$, $N_{2}=2N$, and $N_{3}=4N$. (b-f) Scaling of
$F_{\alpha\beta\gamma}$, where $A_{\alpha\beta\gamma}$ is computed for
$N_{1}=2N+1$,$N_{2}=2N-1$, and $N_{3}=4N$. To simulate small system, $N$ takes
value from 5 to 12.

##  Appendix C Periodic uniform matrix product states

For a quantum spin chains with $N$ spins and each spin with Hilbert space
dimension $d$, a periodic uniform matrix product state (puMPS) is

$|\psi(A)\rangle=\sum_{\vec{s}}\textrm{Tr}(A^{s_{1}}A^{s_{2}}\cdots
A^{s_{N}})|\vec{s}\rangle,$ (117)

where $\vec{s}\equiv s_{1}s_{2}\cdots s_{N}$, $s_{i}=1,2,\cdots d$ and $A^{s}$
is a $D\times D$ matrix, with $D$ being the bond dimension. For a critical
system, a puMPS can represent the ground state faithfully given that $D$ grows
polynomially with $N$. The tensor $A$ can be found by optimizing the energy.
An ansatz for excited states with momentum $p$ is given by the puMPS Bloch
state

$|\phi_{p}(B;A)\rangle=\sum_{j}e^{ipj}\mathcal{T}^{j}\sum_{\vec{s}}\textrm{Tr}(B^{s_{1}}A^{s_{2}}\cdots
A^{s_{N}})|\vec{s}\rangle,$ (118)

where $\mathcal{T}$ is the translation operator. It has been shown that this
ansatz can capture all low-energy excitations given large enough bond
dimensions. The tensor $B$ can be found by solving a generalized eigenvalue
problem in the tangent space. The total cost for the ground state optimization
and finding low-energy excitations is $O(ND^{5})$ and $O(ND^{6})$,
respectively. Moreover, it is not hard to show that an anyon chain model of
the form

$H=\sum_{j=1}^{N}\sum_{a,b=1}^{d}O^{a}_{j-1}h^{a,b}_{j}O^{b}_{j+1}$ (119)

can be represented as a matrix product operator (MPO) with bond dimension
$D_{\mathrm{MPO}}=2d+2$, where the MPO tensor is

$M_{\mathrm{MPO}}=\begin{bmatrix}\mathbbm{1}&&&\\\ O^{T}&&&\\\ &h&&\\\
&&O&\mathbbm{1}\end{bmatrix}$ (120)

where $O=(O_{1},O_{2},\cdots,O_{d})$ is a $1\times d$ block and $h$ is a
$d\times d$ block with entries $h^{a,b}$. For the Haagerup model $d=6$ and
$O^{a}$ is a projector onto a specific anyon subspace, and $h^{a,c}$ has
matrix elements
$h^{a,b}_{\alpha,\beta}=-F^{a\rho\rho}_{b\alpha\rho}F^{a\rho\rho*}_{b\beta\rho}$,
where $\alpha,\beta=1,2,\cdots,d$ label physical indices. This MPO is useful
for the algorithm of finding the excitations.

Finally, given three puMPS Bloch states $|\Phi_{p_{i}}(B_{i};A_{i})\rangle$
with system sizes $N_{i}$ ($i=1,2,3$) and $N_{1}+N_{2}=N_{3}$, the
wavefunction overlap
$\langle\Phi_{p_{3}}(B_{3};A_{3})|\Phi_{p_{1}}(B_{1};A_{1})\Phi_{p_{2}}(B_{2};A_{2})\rangle$
can be computed with cost $O(D^{2}_{3}(N_{1}D^{2}_{1}+N_{2}D^{2}_{2}))$.

##  Appendix D Finite-size corrections for general overlaps

In the main text, we consider the finite-size correction to Eq. (21), where
$L_{1}=L_{2}$. The finite-size correction in this case is solvable using the
cyclic orbifold. In the more general case where $L_{1}\neq L_{2}$, the path
integral on Riemann surface $\Sigma$ is not the simple $\mathsf{N}=2$ cyclic
orbifold.

We numerically compute finite-size correction to wavefunction overlap in
Table. 3, for the three examples $L_{2}=2L_{1}$, $L_{2}=3L_{1}$ and
$L_{2}=\frac{3}{2}L_{1}$. In all the cases,
$A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}$ scales as $N^{-\frac{1}{16}}$, and the
values of $p$ are close to either $\frac{1}{2}$ or $2$. It is worth noticing
that for a given channel, the values of $p$ are the same for all the three
choices in the $N\rightarrow\infty$ limit. Note that all powers are still
$p\in\\{\Delta_{(\alpha,\hat{\chi})}-\Delta_{(\mathbbm{1},\hat{0})}\\}$,
indicating that they are determined by the scaling dimension of the twist
operator insertion at the singular point. However, not all $p$’s are the same
as the $L_{1}=L_{2}$ case. For example, $p_{\sigma\sigma\epsilon}=1/2$ for
$L_{1}\neq L_{2}$, as opposed to $3/2$ for $L_{1}=L_{2}$. This indicates that
the wavefunction overlaps for $L_{1}\neq L_{2}$ are no longer three-point
correlation functions of the cyclic orbifold, but more complicated correlation
functions of the cyclic orbifold.

(The case $L_{2}=\frac{4}{3}L_{1}$ is also computed with the same trend. Not
listed in the table.)

| $p$
---|---
overlap | $L_{2}=2L_{1}$ | $L_{2}=3L_{1}$ | $L_{2}=\frac{3}{2}L_{1}$
$\mathbbm{1}\times\mathbbm{1}\rightarrow\epsilon$ | 0.4999 | $\hskip 1.99997pt\frac{1}{2}\hskip 1.99997pt$ | 0.4996 | $\hskip 1.99997pt\frac{1}{2}\hskip 1.99997pt$ | 0.4999 | $\hskip 1.99997pt\frac{1}{2}\hskip 1.99997pt$
$\mathbbm{1}\times\epsilon\rightarrow\mathbbm{1}$ | 0.4983 | $\frac{1}{2}$ | 0.4978 | $\frac{1}{2}$ | 0.4969 | $\frac{1}{2}$
$\epsilon\times\epsilon\rightarrow\mathbbm{1}$ | 2.0003 | 2 | 2.0010 | 2 | 2.0002 | 2
$\epsilon\times\epsilon\rightarrow\epsilon$ | 0.4991 | $\frac{1}{2}$ | 0.4974 | $\frac{1}{2}$ | 0.4994 | $\frac{1}{2}$
$\mathbbm{1}\times\epsilon\rightarrow\epsilon$ | 1.9946 | 2 | 1.9957 | 2 | 1.9828 | 2
$\mathbbm{1}\times\sigma\rightarrow\sigma$ | 0.4993 | $\frac{1}{2}$ | 0.4989 | $\frac{1}{2}$ | 0.4990 | $\frac{1}{2}$
$\sigma\times\epsilon\rightarrow\sigma$ | 0.4994 | $\frac{1}{2}$ | 0.4996 | $\frac{1}{2}$ | 0.4984 | $\frac{1}{2}$
$\sigma\times\sigma\rightarrow\mathbbm{1}$ | 0.5000 | $\frac{1}{2}$ | 0.4999 | $\frac{1}{2}$ | 0.5000 | $\frac{1}{2}$
$\sigma\times\sigma\rightarrow\epsilon$ | 0.4827 | $\frac{1}{2}$ | 0.4909 | $\frac{1}{2}$ | 0.4057 | $\frac{1}{2}$
Table 3: Finite-size correction to the wavefunction overlap for three more
general cases $L_{2}=2L_{1},L_{2}=3L_{1}$, and $L_{2}=\frac{3}{2}L_{1}$. The
value of system size $N_{1}$ is taken in the range $[200,500]$.
$A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}$ scales as $N^{-1/16}$. The values for
$p$ are close to $1/2$ and $2$. For $\sigma\times\sigma\rightarrow\epsilon$
channel in the case $L_{2}=\frac{3}{2}L_{1}$, the value will go to 0.5 if we
further increase the system size.

##  Appendix E OPE from vertex state

The extraction of OPE coefficient from wavefunction overlap can be applied to
even more general cases. In Sec. II.2, we discuss the overlap configuration of
three cylinders with circumference $L_{1},L_{2},L_{3}$ and
$L_{1}+L_{2}=L_{3}$, where two cylinders ($L_{1},L_{2}$) are joined into one
cylinder ($L_{3}$). In this appendix, we consider a different configuration,
where all three cylinders have the same circumference $L$, as shown in Fig.
12. This configuration was considered by some of the authors [35] in the
context of vertex state in CFT.

Figure 11: Wavefunction overlap of three cylinders with the same
circumference. The maximally-entangled state $|\Omega\rangle\\!\rangle$ acts
as a “glue” to form the path integral on the Riemann surface.

The wavefunction overlap in this configuration can be understood as follows.
Let us define the maximally entangled state $|\Omega\rangle\\!\rangle$ which
can be regarded as a “glue” (as illustrated in Fig. 11). This is nothing but
the vertex state studied in [35]. Using $|\Omega\rangle\\!\rangle$, we can
glue the three half infinite cylinders into a junction shape of surface
$\Sigma$. Denote the wavefunction overlap as:

$\displaystyle
A_{\alpha\beta\gamma}=\langle\\!\langle\Omega|(|\phi^{1}_{\alpha}\rangle|\phi^{2}_{\beta}\rangle|\phi^{3}_{\gamma}\rangle)$
(121)

Note that this is not the same overlap as appeared in the main text. After the
conformal transformation $\omega(z)$, the path integral on this surface
$\Sigma$ is related to the three-point function on the plane:

$\displaystyle\frac{A_{\alpha\beta\gamma}}{A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}}$
$\displaystyle=\langle\phi^{1}_{\alpha}(-\infty_{\alpha})\phi^{2}_{\beta}(-\infty_{\beta})\phi^{3}_{\gamma}(-\infty_{\gamma})\rangle_{\Sigma}$
(122)
$\displaystyle=\prod_{i}\left(\frac{d\omega_{i}}{dz}\right)^{\Delta_{i}}\langle\phi^{1}_{\alpha}(e^{i\pi/3})\phi^{2}_{\beta}(e^{-i\pi/3})\phi^{3}_{\gamma}(e^{-i\pi})\rangle_{\mathbb{C}},$

where the conformal transformation $\omega(z)$ maps $\Sigma$ to a complex
plane:

$\omega_{i}=\omega_{i,0}\left(\frac{1+z}{1-z}\right)^{2/3},$ (123)

with
$\omega_{1,0}=e^{i\pi/3},\omega_{2,0}=e^{-i\pi/3},\omega_{3,0}=e^{-i\pi}$,
$z=e^{\tau+i\sigma}$. Plug in $d\omega/dz$ and the three-point correlation
function on a plane, we obtain:

$\displaystyle\frac{A_{\alpha\beta\gamma}}{A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}}$
$\displaystyle=\left(\frac{4}{3\sqrt{3}}\right)^{\Delta_{\alpha}+\Delta_{\beta}+\Delta_{\gamma}}C_{\alpha\beta\gamma}.$
(124)

The above relation is checked numerically using ferromagnetic Ising model with
accurate agreement for extracted OPE. In this configuration, however, the form
of finite-size correction to Eq. (124) is different, because the cyclic
orbifold picture does not apply to this case. We show the numerical obtained
$p$ in Table 4 using $N\in[100,500)$ and the plot for three channels in Fig.
12, where:

$\displaystyle F_{\alpha\beta\gamma}$
$\displaystyle=\frac{A_{\alpha\beta\gamma}}{A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}}-\left(\frac{4}{3\sqrt{3}}\right)^{\Delta_{\alpha}+\Delta_{\beta}+\Delta_{\gamma}}C_{\alpha\beta\gamma}$
(125) $\displaystyle=\text{const}\cdot N^{-p_{\alpha\beta\gamma}}.$

The values of $p$ are close to three values: $0.66,1.38,2.00$. This indicates
that finite-size corrections for the overlap with the vertex state may also be
universal. Also, we note that the overlap
$A_{\mathbbm{1}\mathbbm{1}\mathbbm{1}}$ in the vertex configuration decays as
$N^{-5c/36}$, which can be shown in a similar way as Ref. [10].

Figure 12: Finite-size correction of wavefunction overlap of Ising model using the vertex state , for system size $N\in[100,500)$. overlap |  $p$
---|---
$\mathbbm{1}\times\mathbbm{1}\rightarrow\epsilon$ |  0.6656
$\mathbbm{1}\times\epsilon\rightarrow\mathbbm{1}$ |  0.6656
$\epsilon\times\epsilon\rightarrow\mathbbm{1}$ |  1.3838
$\epsilon\times\epsilon\rightarrow\epsilon$ |  2.0002
$\mathbbm{1}\times\epsilon\rightarrow\epsilon$ |  1.3838
$\mathbbm{1}\times\sigma\rightarrow\sigma$ |  0.6676
$\sigma\times\epsilon\rightarrow\sigma$ |  0.6608
$\sigma\times\sigma\rightarrow\mathbbm{1}$ |  0.6676
$\sigma\times\sigma\rightarrow\epsilon$ |  0.6608
Table 4: Finite-size correction of wavefunction overlap in vertex-state
configuration, for system size $N\in[100,500)$.