# Crosscap states in the XXX spin-1/2 spin chain

Christopher Ekman
<EMAIL_ADDRESS>Department of Physics, KTH Royal Institute of Technology,
SE-106 91 Stockholm, Sweden

(July 2022)

###### Abstract

We consider integrable boundary states in the XXX spin-1/2 spin chain. We
begin by briefly reviewing the algebraic Bethe Ansatz as well as integrable
boundary states in spin chains. Then a recently discovered class of integrable
states known as crosscap states is described and expanded. In these states
each spin is entangled with its antipodal spin. We present a novel proof of
the integrability of both a crosscap state that is known in the literature and
one that has not previously been studied. We then use the machinery of the
algebraic Bethe Ansatz to derive the overlaps between the crosscap states and
off-shell Bethe states.

##  1 Introduction

In the last decade there has been an interest in obtaining exact formulae for
overlaps between particular spin-chain states and eigenstates of integrable
spin chains. The reason for this is twofold. First, a lot of progress has
recently been made in the study of quantum quenches (see e.g. [1, 2, 3] and
references therein). The idea is to take a macroscopic system in the ground
state $\ket{\Psi_{0}}$ of some Hamiltonian $H_{0}$ and at $t=0$ change it to
some other Hamiltonian $H$ and thereafter allow the system to time evolve
unitarily. If $H$ has energy eigenvectors $\ket{n}$ with energies $E_{n}$ the
time evolution of the expectation value of some observable is given by

$\langle\mathcal{O}\rangle(t)=\sum_{m,n}\bra{\Psi_{0}}\ket{m}\bra{m}\mathcal{O}\ket{n}\bra{n}\ket{\Psi_{0}}e^{i(E_{n}-E_{m})t}.$
(1)

Clearly the overlaps $\bra{\Psi_{0}}\ket{m}$ are of some importance in the
time evolution. In 2014 it was found that for integrable spin chains the
overlaps can sometimes be concisely written as determinants [4, 5]. Because of
the simple determinant formula it was possible to take the thermodynamic
limit, so it could be used as the initial state in a quantum quench, as was
done in [6]. This sparked an interest in the computation of overlaps for
integrable spin chains in the condensed matter community, and it has since
been shown that a number of overlaps can be written as determinants [7, 8, 9,
10, 11]. More overlaps still have been conjectured and then checked for small
spin chains [12, 13, 14, 15, 16]. The initial states for which the overlaps
can be written as determinants are known as integrable states.

Second, it was found that spin chain overlaps arise in $3+1$ dimensional gauge
theories [13]. The connection between integrable spin chains and gauge theory
was first discovered for $\mathcal{N}=4\ \mathrm{SYM}$, a highly symmetric
relative of QCD, when the one-loop anomalous dimensions of traces of products
of scalars was found to correspond to the eigenvalues of the Heisenberg model
[17]. More generally, the trace of some combination of scalars corresponds to
a particular state in a spin chain. If a defect, such as a domain wall, is
added, then these operators develop non-zero vacuum expectation values.
Because of the underlying spin chain structure these vacuum expectation values
are proportional to overlaps between eigenstates of the spin chain and a class
of states called matrix product states (see [18] for a review).

Due to the combined interest of the high-energy theory and condensed matter
theory communities it is clear that it is important to find more states for
which the overlaps can be written in a simple determinant form. The most well-
studied classes of integrable states are the valence bond states and the
matrix product states. In both cases the overlaps generally take the form

$\frac{\bra{B}\ket{\\{\lambda\\}}}{\sqrt{\bra{\\{\lambda\\}}\ket{\\{\lambda\\}}}}=\prod_{k=1}F(\lambda_{k})\sqrt{\frac{\det
G^{+}}{\det G^{-}}},$ (2)

where $\bra{B}$ is the integrable state, $\ket{\\{\lambda\\}}$ is an on-shell
Bethe state with Bethe roots $\\{\lambda\\},$ $F(\lambda)$ is a function of
one variable, and the matrices $G^{+},G^{-}$ are related to the Gaudin matrix.
For valence bond states this formula has been proven for $\mathfrak{gl}(n)$
symmetric spin chains [8].

Recently, a new class of integrable states was found [15]. These are known as
crosscap states, and were proposed as analogues to the crosscap states that
appear in conformal field theory. The purpose of this paper is to investigate
this new class of states and in particular to investigate the overlaps with
eigenstates of the Heisenberg model.

The paper is structured as follows: first, in section 2 we establish our
conventions by briefly describing the Algebraic Bethe Ansatz as well as
integrable boundary states. Second, in section 3 two classes of integrable
crosscap states for the Heisenberg model are described, and in section 4 some
formulas for the overlaps between the crosscap states and off-shell Bethe
states. Finally, in section 5 we present our conclusions and suggest some
directions for future research.

##  2 Background material

###  2.1 The Algebraic Bethe Ansatz

We begin by briefly reviewing the Algebraic Bethe Ansatz, mostly using the
notation of [19]. For more extensive reivews, see for example [19, 20, 21].
The physical model that we will consider is the XXX spin-$1/2$ spin chain with
Hamiltonian

$H=\sum_{n=1}^{L}1-P_{n,n+1},$ (3)

which is also known as the Heisenberg model [22]. Here $L$ is the number of
sites, and $P_{n,n+1}$ exchanges the spins at site $n$ and site $n+1$. This
Hamiltonian may be diagonalized by using the Algebraic Bethe Ansatz, where we
derive the Hamiltonian from an object known as the Lax operator. Following
[21] we use the Lax operator

$L_{n,a}(\lambda,\nu)=(\lambda-\nu-\frac{i}{2})I_{n,a}+iP_{n,a}=\begin{bmatrix}\lambda-\nu+iS^{z}_{n}&iS^{-}_{n}\\\
iS^{+}_{n}&\lambda-\nu-iS^{z}_{N}\end{bmatrix},$ (4)

which acts on the tensor product of the quantum space corresponding to site
$n$ and an auxiliary space, as indicated by the subscripts. The matrix
elements are the usual $\mathrm{SU}(2)$ spin operators acting on the quantum
space corresponding to site $n$. In addition to the rapidity $\lambda$ we have
also introduced an inhomogeneity $\nu$. Another important object is the
$R$-matrix which we take to be

$R(\lambda)_{a_{1},a_{2}}=I_{a_{1},a_{2}}+\frac{i}{\lambda}P_{a_{1},a_{2}}$
(5)

which acts on two auxiliary spaces. Together the Lax operator and the
$R$-matrix satisfy the $RLL$ equation

$R_{a_{1},a_{2}}(\lambda-\mu)L_{n,a_{1}}(\lambda)L_{n,a_{2}}(\mu)=L_{n,a_{2}}(\mu)L_{n,a_{1}}(\lambda)R_{a_{1},a_{2}}(\lambda-\mu).$
(6)

The monodromy matrix is given by the product of all Lax operators:

$T_{a}(\lambda,\\{\nu\\})=L_{1,a}(\lambda-\nu_{1})L_{2,a}(\lambda-\nu_{2})...L_{L,a}(\lambda-\nu_{L}).$
(7)

Suppressing the inhomogeneities in the argument the monodromy matrix
$T_{a}(\lambda)$ is defined as

$T_{a}(\lambda)=\begin{bmatrix}A(\lambda)&B(\lambda)\\\
C(\lambda)&D(\lambda)\end{bmatrix},$ (8)

where each element acts on the tensor product of all the quantum spaces in the
spin chain. It follows from the $RLL$ equation that the monodromy matrix
satisfies the $RTT$ relation

$R_{a_{1},a_{2}}(\lambda-\mu)T_{a_{1}}(\lambda)T_{a_{2}}(\mu)=T_{a_{2}}(\mu)T_{a_{1}}(\lambda)R_{a_{1},a_{2}}(\lambda-\mu),$
(9)

which implies the following set of commutation relations for the elements of
the monodromy matrix:

$[T^{i}_{j}(\lambda),T^{k}_{l}(\mu)]=g(\lambda,\mu)\left(T^{k}_{j}(\mu)T^{i}_{l}(\lambda)-T^{k}_{j}(\lambda)T^{i}_{l}(\mu)\right),$
(10)

where we introduced the notation $g(\lambda,\mu)=\frac{i}{\lambda-\mu}$.

Now, the auxiliary space is unphysical, so we can get rid of it by tracing
over it. This leads us to defining the transfer matrix $\tau(\lambda)$:

$\tau(\lambda)=\Tr_{a}T_{a}(\lambda)=A(\lambda)+D(\lambda).$ (11)

One can show that the Hamiltonian (3) is contained in the transfer matrix, so
diagonalizing the Hamiltonian amounts to diagonalizing the transfer matrix.
This can be done by first defining a vacuum state $\ket{0}$ which we take to
be the spin state with all spins pointing upwards. We say that a spin pointing
downwards is excited. The vacuum state satisfies

$A(\lambda)\ket{0}=a(\lambda)\ket{0}\quad
D(\lambda)\ket{0}=d(\lambda)\ket{0}\quad C(\lambda)\ket{0}=0,$ (12)

where

$a(\lambda)=\prod_{k=1}^{L}(\lambda-\nu_{k}+\frac{i}{2}),\quad
d(\lambda)=\prod_{k=1}^{L}(\lambda-\nu_{k}-\frac{i}{2})$ (13)

are the vacuum eigenvalues of the $A$ and $D$ operators. This motivates us to
interpret the $C$-operator as an annihilation operator. Next we define Bethe
states as

$\ket{\\{\lambda\\}}=\prod_{n=1}^{N}B(\lambda_{n})\ket{0},$ (14)

where each $B$-operator creates a magnon in the spin chain. One can use the
commutation relations implied by the $RTT$ equation to show that the state
$\ket{\\{\lambda\\}}$ is an eigenstate of the transfer matrix if the
rapidities $\\{\lambda\\}$ satisfy the Bethe equations:

$\frac{a(\lambda_{n})}{d(\lambda_{n})}=\prod_{k\neq
n}\frac{f(\lambda_{n},\lambda_{k})}{f(\lambda_{k},\lambda_{n})}.$ (15)

where we have introduced the notation
$f(\lambda,\mu)=1+\frac{i}{\lambda-\mu}$. If the rapidities satisfy the Bethe
equations then the corresponding Bethe state is said to be on-shell.

In the discussion of crosscap states below the idea of splitting the spin
chain in two will be important. The monodromy matrix is a product of all the
Lax matrices over the length of the spin chain, but we can equally well
perform the multiplication over two connected partitions of the spin chain
separately. This leads to two monodromy matrices $T_{1},T_{2}$ which are
defined as

$T_{1}(\lambda)=\prod_{k=1}^{L^{\prime}}L_{n,1}(\lambda-\nu_{n}),\quad
T_{2}(\lambda)=\prod_{k=L^{\prime}+1}^{L}L_{n,1}(\lambda-\nu_{n}).$ (16)

Each partition of the spin chain will henceforth be referred to as spin chain
1 and spin chain 2, respectively. The monodromy matrix on the full spin chain
is obtained by multiplying $T_{1}$ and $T_{2}$ so that

$T(\lambda)=T_{1}(\lambda)T_{2}(\lambda).$ (17)

Each spin chain has its own vacuum and its own creation and annihilation
operators, which are distinguished by the subscripts 1 and 2. Note that the
elements of $T_{1}$ commute with the elements of $T_{2}$ since they operate on
different Hilbert spaces The creation operator for the full spin chain can be
obtained from the creation operators on the sub-chains by carrying out the
matrix multiplication:

$B(\lambda)=A_{1}(\lambda)B_{2}(\lambda)+B_{1}(\lambda)D_{2}(\lambda).$ (18)

The Bethe states can be written in terms of Bethe states on each sub-chain

$\prod_{n=1}^{N}B(\lambda_{n})\ket{0}=\prod_{n=1}^{N}(A_{1}(\lambda_{n})B_{2}(\lambda_{n})+B_{1}(\lambda_{n})D_{2}(\lambda_{n}))\ket{0}_{1}\ket{0}_{2}.$
(19)

When we expand the product each term will contain operators acting on either
spin chain 1 or spin chain 2 such that only one operator corresponding to each
rapidity $\lambda_{k}$ is present. For each term, we need to move all
$A_{1}$:s through all $B_{1}$:s, and all $D_{2}$:s through all $B_{2}$:s. In
order to represent the sum we obtain in a useful way we split the set of
rapiditied $\\{\lambda\\}$ into two partitions $\\{\lambda^{I}\\}$ and
$\\{\lambda^{II}\\}$ such that all rapidities in $\\{\lambda^{I}\\}$ appear in
operators acting on spin chain 1, and all rapidities in $\\{\lambda^{II}\\}$
appear in operators acting on spin chain 2. Performing these maneuvers we
obtain

$\displaystyle\prod_{n=1}^{N}B(\lambda_{n})\ket{0}=\sum_{\\{\lambda\\}=\\{\lambda^{I}\\}\cup\\{\lambda^{II}\\}}$
$\displaystyle\prod_{j\in{I}}\prod_{k\in{II}}[a_{2}(\lambda_{j}^{I})d_{1}(\lambda_{k}^{II})f(\lambda_{k}^{II},\lambda_{j}^{I})]$
$\displaystyle\crossproduct\prod_{j\in{I}}B_{1}(\lambda^{I}_{j})\ket{0}_{1}\prod_{k\in{II}}B_{2}(\lambda^{II}_{k})\ket{0}_{2}.$
(20)

Here the sum is over all partitions of $\\{\lambda\\}$ into
$\\{\lambda^{I}\\}$ and $\\{\lambda^{II}\\}$ and the functions $a_{2}$ and
$d_{1}$ are vacuum eigenvalues of the second and first part of the spin chain
respectively. The most general version of this formula may be found in [19].

###  2.2 Integrable boundary states

For simplicity we set all inhomogeneities to $0$ in this section. Integrable
boundary states are states that satisfy [11]

$\left[\tau(\lambda)-\tau(-\lambda)\right]\ket{B}=0,$ (21)

where $\ket{B}$ is the integrable state under consideration. This implies a
selection rule for its overlaps with on-shell Bethe states, namely the
overlaps are non-zero only if the rapidities are parity invariant, so

$\\{\lambda\\}=\\{\lambda_{1},\lambda_{2},...,\lambda_{\frac{N}{2}},-\lambda_{1},-\lambda_{2},...,-\lambda_{\frac{N}{2}}\\}.$
(22)

If one of the rapidities is $0$ then it will only appear once, and in that
case there is an odd number of rapidities. The parity invariance implies that
any integrable state $\ket{B}$ may have non-zero overlap only with on-shell
Bethe states that have momentum $0$ or $\pi$. To see this, we write the total
momentum of a Bethe state $\ket{\\{\lambda\\}}$ [21]

$P(\\{\lambda\\})=\frac{1}{i}\sum_{k=1}^{N}\ln\frac{\lambda_{k}+\frac{i}{2}}{\lambda_{k}-\frac{i}{2}}.$
(23)

Note that this expression is singular for $\lambda_{k}=\pm\frac{i}{2}$. But if
both $\frac{i}{2}$ and $-\frac{i}{2}$ are present among the rapidities the
singularities may be regularized, and one can show that the pair
$\pm\frac{i}{2}$ contribute $\pi$ to the momentum [23]. Furthermore, the
rapidity $\lambda_{0}=0$ also contributes $0,$ while all the other rapidities
do not contribute to the momentum. Thus any integrable state may have non-zero
overlap only with on-shell Bethe states that have momentum $0$ or $\pi$.

As we stated in the introduction the overlaps between integrable states and
on-shell Bethe states generally take the form

$\frac{\bra{B}\ket{\\{\lambda\\}}}{\sqrt{\bra{\\{\lambda\\}}\ket{\\{\lambda\\}}}}=\prod_{k=1}F(\lambda_{k})\sqrt{\frac{\det
G^{+}}{\det G^{-}}}.$ (24)

For the XXX spin-$1/2$ spin chain the matrices $G^{\pm}$ are given by

$\displaystyle
G^{\pm}_{kl}=\delta_{kl}\left[i\frac{\partial}{\partial\lambda_{k}}\ln\frac{a(\lambda_{l})}{d(\lambda_{l})}+\sum_{n=1}^{N}K^{+}(\lambda_{k},\lambda_{n})\right]-K^{\pm}(\lambda_{k},\lambda_{l})$
(25)

where

$K^{\pm}(\lambda,\mu)=\frac{2}{(\lambda-\mu)^{2}+1}\pm\frac{2}{(\lambda+\mu)^{2}+1}.$
(26)

To complete this section we introduce two quantities that will serve as
building blocks for the crosscap overlaps below. First we write the scalar
products between two Bethe states with $N$ magnons in a spin chain of length
$L$ as

$S_{N}^{L}(\\{\lambda^{C}\\},\\{\lambda^{B}\\})=\bra{0}\prod_{k=1}^{N}C(\lambda^{C}_{k})\prod_{l=1}^{N}B(\lambda^{B}_{l})\ket{0}.$
(27)

In general off-shell scalar products are quite complicated objects (see [19]
for an extensive discussion). Second, we define

$Z_{L}=\bra{0^{\prime}}\prod_{k=1}^{L}B(\lambda_{k})\ket{0}$ (28)

where $\ket{0^{\prime}}$ is the state with all spins pointing downwards. One
can show that [24]

$Z_{L}(\\{\lambda\\},\\{\nu\\})=\frac{\prod_{k=1}^{L}\prod_{l=1}^{L}(\lambda_{k}-\nu_{l}+\frac{i}{2})(\lambda_{k}-\nu_{l}-\frac{i}{2})}{\prod_{1\leq
k<l\leq L}(\nu_{k}-\nu_{l})(\lambda_{l}-\lambda_{k})}\mathrm{det}\mathcal{M},$
(29)

where $\mathcal{M}$ is a matrix whose elements are given by

$\mathcal{M}_{kl}=\frac{i}{(\lambda_{k}-\nu_{l}+\frac{i}{2})(\lambda_{k}-\nu_{l}-\frac{i}{2})}.$
(30)

##  3 Crosscap states

###  3.1 General properties

The crosscap states are states where antipodal spins are entangled. An
arbitrary crosscap state can be written as

$\ket{\Psi}=\prod_{n=1}^{\frac{L}{2}}\ket{\psi}_{n,n+\frac{L}{2}}^{\otimes},$
(31)

where the subscripts $n,n+\frac{L}{2}$ indicates which sites are contained in
each $\ket{\psi}$. The pieces $\ket{\psi}$ can be parametrized in the
following way:

$\ket{\psi}=(\cos\beta+\nu)\ket{\uparrow\downarrow}+(\cos\beta-\nu)\ket{\downarrow\uparrow}+ie^{i\alpha}\sin\beta\ket{\uparrow\uparrow}+e^{i\alpha}\sin\beta\ket{\uparrow\uparrow}.$
(32)

The crosscap state where $\nu=0,\beta=\frac{\pi}{2}$ was introduced and shown
to be integrable in [15]. Explicitly this state is given by

$\ket{C}=\prod_{n=1}^{\frac{L}{2}}\ket{c}_{n}^{\otimes},$ (33)

where

$\ket{c}_{n}=\ket{\uparrow\uparrow}_{n,n+\frac{L}{2}}+\ket{\downarrow\downarrow}_{n,n+\frac{L}{2}}$
(34)

so antipodal spins are identified. In order to characterize this state we can
note that it has momentum $0$. To see this one take an arbitrary spin chain
state where antipodal spins are identified. If one operates on this state with
the translation operator $U=e^{iP}$ we obtain a state where antipodal spins
are still identified. Hence $U\ket{C}=\ket{C}$ so the momentum is $0$. This
distinguishes $\ket{C}$ from valence bond states, since the valence bond
states necessarily contain components with both momentum $0$ and momentum
$\pi$.

Another integrable crosscap state that has not appeared in the literature
previously is the crosscap singlet, which is obtained by taking the limit
$\nu\to\infty$. Renormalizing by dividing by $\nu$ we obtain

$\ket{C_{S}}=\prod_{n=1}^{\frac{L}{2}}\ket{c_{s}}_{n}^{\otimes},$ (35)

where

$\ket{c_{s}}_{n}=(\ket{\uparrow\downarrow}_{n,n+\frac{L}{2}}-\ket{\downarrow\uparrow}_{n,n+\frac{L}{2}})$
(36)

The proof that it is integrable will be presented in section 3.2. The crosscap
singlet consists of a set of $\mathrm{SU}(2)$ singlets where the entangled
spins are antipodal. Exactly half of the spins are excited in $\ket{C_{S}}$,
with the excited spins being spread out over each half of the spin chain. Each
term in $\ket{C_{S}}$ has a sign $(-1)^{n_{1}}$ where $n_{1}$ is the number of
excited spins on the first half of the spin chain. If we operate on the
crosscap singlet with the translation operator the two sides of the spin chain
exchange one excited spin, so

$U\ket{C_{S}}=-\ket{C_{S}}.$ (37)

We see that similarly to $\ket{C}$ the state $\ket{C_{S}}$ is a momentum
eigenstate, with the difference that $\ket{C_{S}}$ has momentum $\pi$. This
makes it uniquely distinguished among all integrable states: the valence bond
states are not momentum eigenstates at all [11], and the matrix product states
all have momentum $0$ (see e.g. [13]), so the crosscap singlet is the only
known integrable boundary state that has momentum $\pi$.

###  3.2 Proof of integrability

In order to prove that the two crosscap states are integrable we use the Lax
operator given by equation (4). Then the following relations can be easily
checked

$\sigma^{2}L_{n,a}(\lambda)\sigma^{2}\ket{c}_{n}=-L_{n+\frac{L}{2},a}(-\lambda)\ket{c}_{n},\quad
L_{n,a}(\lambda)\ket{c_{s}}_{n}=-L_{n+\frac{L}{2},a}(-\lambda)\ket{c_{s}}_{n},$
(38)

where the Pauli matrix $\sigma^{2}$ acts on the auxiliary space. Suppose
$T_{1}(\lambda,\\{\nu\\}_{1})$ is the monodromy matrix acting on all the sites
$1$ to $\frac{L}{2}$ with inhomogeneities $\\{\nu\\}_{1}$ and
$T_{2}(\lambda,\\{\nu\\}_{2})$ is the monodromy matrix acting on all the spins
with $\frac{L}{2}+1$ to $L$ with inhomogeneities $\\{\nu\\}_{2}$. Then
equations (38) implies the equations

$\displaystyle\sigma^{2}T_{1}(\lambda,\\{\nu\\}_{1})\sigma^{2}\ket{C}$
$\displaystyle=(-1)^{\frac{L}{2}}T_{2}(-\lambda,\\{-\nu\\}_{1})\ket{C},$ (39)
$\displaystyle T_{1}(\lambda)\ket{C_{S}}$
$\displaystyle=(-1)^{\frac{L}{2}}T_{2}(-\lambda,\\{-\nu\\}_{1})\ket{C_{S}}.$
(40)

Note that when $\lambda\to-\lambda$ all the inhomogeneities also change sign.
We will suppress the inhomogeneity arguments below to improve readability. We
prove integrability for the state $\ket{C}$ using a different method than the
one that was used in [15]. Our argument can easily be adapted to $\ket{C_{S}}$
by replacing $\sigma^{2}$ with the identity below. In order to prove that
$\ket{C}$ is integrable we need to show that

$(\tau(\lambda)-\tau(-\lambda))\ket{C}=0\iff\Tr T(\lambda)\ket{C}=\Tr
T(-\lambda)\ket{C}.$ (41)

The trace is over the auxiliary space. Recalling that
$T(\lambda)=T_{1}(\lambda)T_{2}(\lambda)$ we can use equation (39) to find

$\displaystyle\Tr T(\lambda,\\{\nu\\})\ket{C}$
$\displaystyle=\Tr\left(T_{1}(\lambda,\\{\nu\\}_{1})T_{2}(\lambda,\\{\nu\\}_{2})\right)\ket{C}$
$\displaystyle=(-1)^{\frac{L}{2}}\Tr\left(T_{1}(\lambda,\\{\nu\\}_{1})\sigma^{2}T_{1}(-\lambda,\\{-\nu\\}_{2})\sigma^{2}\right)\ket{C}.$
(42)

In order to use the algebraic structure of the monodromy matrix we need to
assume that $T_{1}(\lambda,\\{\nu\\}_{1})$ and
$T_{1}(-\lambda,\\{-\nu\\}_{2})$ are operating on the same spin chain. This
means that the inhomogeneities must satisfy

$\nu_{n+\frac{L}{2}}=-\nu_{n}.$ (43)

Assuming that this is true, we write the product of monodromy matrices using
index notation (with summation over repeated indices) and use equation (10)
twice to get

$\displaystyle\Tr[T_{1}(\lambda),\sigma^{2}T_{1}(-\lambda)\sigma^{2}]$
$\displaystyle=(\sigma^{2})^{j}_{k}(\sigma^{2})^{l}_{i}[(T_{1})^{i}_{j}(\lambda),(T_{1})^{k}_{l}(-\lambda)]$
$\displaystyle=g(\lambda,-\lambda)(\sigma^{2})^{j}_{k}(\sigma^{2})^{l}_{i}$
$\displaystyle\crossproduct\left((T_{1})^{k}_{j}(-\lambda)(T_{1})^{i}_{l}(\lambda)-(T_{1})^{k}_{j}(\lambda)(T_{1})^{i}_{l}(-\lambda)\right)$
$\displaystyle=g(\lambda,-\lambda)(\sigma^{2})^{j}_{k}(\sigma^{2})^{l}_{i}[(T_{1})^{k}_{j}(-\lambda),(T_{1})^{i}_{l}(\lambda)]$
$\displaystyle=g(\lambda,-\lambda)g(-\lambda,\lambda)(\sigma^{2})^{j}_{k}(\sigma^{2})^{l}_{i}$
$\displaystyle\
\crossproduct\left((T_{1})^{i}_{j}(\lambda)(T_{1})^{k}_{l}(-\lambda)-(T_{1})^{i}_{j}(-\lambda)(T_{1})^{k}_{l}(\lambda)\right)$
$\displaystyle=g(\lambda,-\lambda)g(-\lambda,\lambda)(\sigma^{2})^{j}_{k}(\sigma^{2})^{l}_{i}[(T_{1})^{i}_{j}(\lambda),(T_{1})^{k}_{l}(-\lambda)],$
(44)

where we renamed the indices in the third and last steps in order to produce
the commutators. Hence

$\Tr[T_{1}(\lambda),\sigma^{2}T_{1}(-\lambda)\sigma^{2}]=g(\lambda,-\lambda)g(-\lambda,\lambda)\Tr[T_{1}(-\lambda),\sigma^{2}T_{1}(\lambda)\sigma^{2}].$
(45)

Assuming that $g(\lambda,-\lambda)g(-\lambda,\lambda)\neq 1$ this implies that

$\Tr[T_{1}(\lambda),\sigma^{2}T_{1}(-\lambda)\sigma^{2}]=0\iff\Tr\left(T_{1}(\lambda)\sigma^{2}T_{1}(-\lambda)\sigma^{2}\right)=\Tr\left(T_{1}(-\lambda)\sigma^{2}T_{1}(\lambda)\sigma^{2}\right)$
(46)

so finally using equation (39) once more and restoring the inhomogeneities we
obtain

$\displaystyle(-1)^{\frac{L}{2}}\Tr\left(T_{1}(\lambda)\sigma^{2}T_{1}(-\lambda)\sigma^{2}\right)\ket{C}$
$\displaystyle=(-1)^{\frac{L}{2}}\Tr\left(T_{1}(-\lambda)\sigma^{2}T_{1}(\lambda)\sigma^{2}\right)\ket{C}$
$\displaystyle=\Tr\left(T_{1}(-\lambda)\sigma^{2}T_{1}(\lambda)\sigma^{2}\right)\ket{C}$
$\displaystyle=\Tr\left(T_{1}(-\lambda)T_{2}(-\lambda)\right)\ket{C}$
$\displaystyle=\Tr\left(T(-\lambda)\right)\ket{C}.$ (47)

We have now shown that if the inhomogeneities are paired such than
$\nu_{n+\frac{L}{2}}=-\nu_{n}$ then

$(\tau(\lambda)-\tau(-\lambda))\ket{C}=0$ (48)

whenever $g(\lambda,-\lambda)g(-\lambda,\lambda)\neq 1$. Recalling that
$g(\lambda,\mu)=\frac{i}{\lambda-\mu}$ we see that this is equivalent to the
requirement $\lambda\neq\pm\frac{1}{2}$. This is not a commonly occuring value
for rapidities, so it is curious that the proof does not work for in this
case. The proof in [15] does work for $\lambda=\pm\frac{1}{2}$, but not for
$\lambda=\pm i$ (see their equation (4.29) and the ones below), so our proof
complements theirs.

Note that changing the $\sigma^{2}$ to anything other than the identity either
leads to non-integrable states or integrable states that are global rotations
of $\ket{C}$. The states $\ket{C}$ and $\ket{C_{S}}$ therefore represent the
only two classes of integrable crosscap states in the XXX spin-1/2 spin chain.

##  4 Overlap formulae

###  4.1 On-shell overlaps

In [15] it was conjectured on the basis of integrability and the general form
(24) that the crosscap state $\ket{C}$ has the overlap

$\frac{\bra{C}\ket{\\{\lambda\\}}}{\sqrt{\bra{\\{\lambda\\}}\ket{\\{\lambda\\}}}}=\sqrt{\frac{\det
G^{+}}{\det G^{-}}}.$ (49)

This formula has been verified for small spin chains and small numbers of
magnons.

For the crosscap singlet state $\ket{C_{S}}$ no successful guess based on the
form (24) has been made. As was stated in the previous section $\ket{C_{S}}$
is unique among the integrable boundary states by having momentum $\pi$. On
the basis of the discussion below equation (22) this means that $\ket{C_{S}}$
has non-zero overlap only with parity invariant on-shell Bethe states that
also have either rapidities $\pm\frac{i}{2}$ or a rapidity equal to $0$. In
the former case the Bethe state is singular and requires special care [22, 25,
26, 27]. For example, one can show that for singular Bethe states the
rapidities must satisfy the following equation [25]

$\left[-\prod_{k=3}^{N}\frac{\lambda_{k}+\frac{i}{2}}{\lambda_{k}-\frac{i}{2}}\right]^{N}=1$
(50)

in addition to the Bethe equations. This severely restricts the Bethe roots.
Finally exactly half the spins in $\ket{C_{S}}$ are excited, so if it is to
have non-zero overlap with a Bethe state then that Bethe state must have
$\frac{L}{2}$ magnons.

###  4.2 Off-shell formulae for the crosscap overlaps

We now move on to discuss some novel results regarding the crosscap overlaps.

We wish to calculate quantities of the form

$\bra{C}\prod_{k=1}^{N}B(\lambda_{k})\ket{0}.$ (51)

Recall equations (39) and (40).These relations suggest that it is productive
to split the spin chain into two halves. Using equation (2.1) we get

$\displaystyle\bra{C}\prod_{n=1}^{N}B(\lambda_{n})\ket{0}=\bra{C}\sum_{\\{\lambda\\}=\\{\lambda^{I}\\}\cup\\{\lambda^{II}\\}}$
$\displaystyle\prod_{j\in{I}}\prod_{k\in{II}}[a_{2}(\lambda_{j}^{I})d_{1}(\lambda_{k}^{II})f(\lambda_{k}^{II},\lambda_{j}^{I})]$
$\displaystyle\crossproduct\prod_{j\in{I}}B_{1}(\lambda^{I}_{j})\ket{0}_{1}\prod_{k\in{II}}B_{2}(\lambda^{II}_{k})\ket{0}_{2}$
(52)

where the sum is over all partitions of the set of rapidities into two sets.
From equation (39) we see that

$\bra{C}B_{1}(\lambda)=(-1)^{\frac{L}{2}+1}\bra{C}C_{2}(-\lambda).$ (53)

This permits us to turn all $B_{1}$ operators into $C_{2}$ operators under the
assumption that the inhomogeneities satisfy $\nu_{n+\frac{L}{2}}=-\nu_{n}$ as
in the preceding section. Hence

$\displaystyle\bra{C}\prod_{n=1}^{N}B(\lambda_{n})\ket{0}=\bra{C}\sum_{\\{\lambda\\}=\\{\lambda^{I}\\}\cup\\{\lambda^{II}\\}}$
$\displaystyle\prod_{j\in{I}}\prod_{k\in{II}}[a_{2}(\lambda_{j}^{I})d_{1}(\lambda_{k}^{II})f(\lambda_{k}^{II},\lambda_{j}^{I})]$
$\displaystyle\crossproduct(-1)^{n_{I}(\frac{L}{2}+1)}\ket{0}_{1}\prod_{j\in{I}}C_{2}(-\lambda^{I}_{j})\prod_{k\in{II}}B_{2}(\lambda^{II}_{k})\ket{0}_{2},$
(54)

where $n_{I}$ is the cardinality of the set $\\{\lambda^{I}\\}$. The only term
in $\bra{C}$ that contributes is the one where none of the spins on the first
half are excited. From the definition of $\ket{C}$ we see that this term is
$\bra{0}=\bra{0}_{1}\bra{0}_{2}$, so

$\displaystyle\bra{C}\prod_{n=1}^{N}B(\lambda_{n})\ket{0}=\sum_{\\{\lambda\\}=\\{\lambda^{I}\\}\cup\\{\lambda^{II}\\}}$
$\displaystyle\prod_{j\in{I}}\prod_{k\in{II}}[a_{2}(\lambda_{j}^{I})d_{1}(\lambda_{k}^{II})f(\lambda_{k}^{II},\lambda_{j}^{I})]$
$\displaystyle\crossproduct(-1)^{n_{I}(\frac{L}{2}+1)}\bra{0}_{2}\prod_{j\in{I}}C_{2}(-\lambda^{I}_{j})\prod_{k\in{II}}B_{2}(\lambda^{II}_{k})\ket{0}_{2}.$
(55)

The factor involving creation and annihilation operators is the scalar product
between two off-shell Bethe states:

$\bra{0}_{2}\prod_{j\in{I}}C_{2}(-\lambda^{I}_{j})\prod_{k\in{II}}B_{2}(\lambda^{II}_{k})\ket{0}_{2}=S_{N}^{\frac{L}{2}}(\\{-\lambda^{I}\\},\\{\lambda^{II}\\})$
(56)

where the superscript indicates that the scalar product is only on one half of
the spin chain. Hence

$\displaystyle\bra{C}\prod_{n=1}^{N}B(\lambda_{n})\ket{0}=\sum_{\\{\lambda\\}=\\{\lambda^{I}\\}\cup\\{\lambda^{II}\\}}$
$\displaystyle\prod_{j\in{I}}\prod_{k\in{II}}[a_{2}(\lambda_{j}^{I})d_{1}(\lambda_{k}^{II})f(\lambda_{k}^{II},\lambda_{j}^{I})]$
$\displaystyle\crossproduct(-1)^{n_{I}(\frac{L}{2}+1)}S_{N}^{\frac{L}{2}}(\\{-\lambda^{I}\\},\\{\lambda^{II}\\}).$
(57)

Note that if the two partitions $\\{\lambda^{I}\\}$ and $\\{\lambda^{II}\\}$
have different cardinalities the scalar product is $0$, so the sum is over all
partitions that have equal cardinality $n_{I}=n_{II}=\frac{N}{2}$. Due to the
complexity of off-shell scalar products this expressions has not been
simplified further.

A similar calculation can be performed for the crosscap singlet state
$\ket{C_{S}}$. From equation (40) we have

$\bra{C_{S}}B_{1}(\lambda)=(-1)^{\frac{L}{2}}\bra{C_{S}}B_{2}(-\lambda)$ (58)

so

$\displaystyle\bra{C_{S}}\prod_{n=1}^{N}B(\lambda_{n})\ket{0}$
$\displaystyle=\bra{C_{S}}\sum_{\\{\lambda\\}=\\{\lambda^{I}\\}\cup\\{\lambda^{II}\\}}\prod_{j\in{I}}\prod_{k\in{II}}[a_{2}(\lambda_{j}^{I})d_{1}(\lambda_{k}^{II})f(\lambda_{k}^{II},\lambda_{j}^{I})]$
$\displaystyle\crossproduct\prod_{j\in{I}}B_{1}(\lambda^{I}_{j})\ket{0}_{1}\prod_{k\in{II}}B_{2}(\lambda^{II}_{k})\ket{0}_{2}$
$\displaystyle=\bra{C_{S}}\sum_{\\{\lambda\\}=\\{\lambda^{I}\\}\cup\\{\lambda^{II}\\}}\prod_{j\in{I}}\prod_{k\in{II}}[a_{2}(\lambda_{j}^{I})d_{1}(\lambda_{k}^{II})f(\lambda_{k}^{II},\lambda_{j}^{I})]$
$\displaystyle\crossproduct(-1)^{\frac{n_{I}L}{2}}\ket{0}_{1}\prod_{j\in{I}}B_{2}(-\lambda^{I}_{j})\prod_{k\in{II}}B_{2}(\lambda^{II}_{k})\ket{0}_{2}.$
(59)

The only contributing term from $\bra{C_{S}}$ is again the one where none of
the spins on the first half of the spin chain are excited. From the definition
of $\ket{C_{S}}$ we see that this term is $\bra{0}_{1}\bra{0^{\prime}}_{2}$
where $\bra{0^{\prime}}$ is the state where all the spins are excited. Hence

$\displaystyle\bra{C_{S}}\prod_{n=1}^{N}B(\lambda_{n})\ket{0}=\sum_{\\{\lambda\\}=\\{\lambda^{I}\\}\cup\\{\lambda^{II}\\}}$
$\displaystyle\prod_{j\in{I}}\prod_{k\in{II}}[a_{2}(\lambda_{j}^{I})d_{1}(\lambda_{k}^{II})f(\lambda_{k}^{II},\lambda_{j}^{I})]$
$\displaystyle\crossproduct(-1)^{\frac{n_{I}L}{2}}\bra{0^{\prime}}_{2}\prod_{j\in{I}}B_{2}(-\lambda^{I}_{j})\prod_{k\in{II}}B_{2}(\lambda^{II}_{k})\ket{0}_{2}.$
(60)

The factor involving annihilation operators was defined in equation (28):

$\bra{0^{\prime}}_{2}\prod_{j\in{I}}B_{2}(-\lambda^{I}_{j})\prod_{k\in{II}}B_{2}(\lambda^{II}_{k})\ket{0}_{2}=Z_{\frac{L}{2}}(\\{-\lambda^{I}\\}\cup\\{\lambda^{II}\\},\\{\nu\\}_{2}).$
(61)

The overlap formula becomes

$\displaystyle\bra{C_{S}}\prod_{n=1}^{N}B(\lambda_{n})\ket{0}=\sum_{\\{\lambda\\}=\\{\lambda^{I}\\}\cup\\{\lambda^{II}\\}}$
$\displaystyle\prod_{j\in{I}}\prod_{k\in{II}}[a_{2}(\lambda_{j}^{I})d_{1}(\lambda_{k}^{II})f(\lambda_{k}^{II},\lambda_{j}^{I})]$
$\displaystyle\crossproduct(-1)^{\frac{n_{I}L}{2}}Z_{\frac{L}{2}}(\\{-\lambda^{I}\\}\cup\\{\lambda^{II}\\},\\{\nu\\}_{2}).$
(62)

Note that $Z_{\frac{L}{2}}$ is $0$ if the number of magnons differs from
$\frac{L}{2}$. The crosscap singlet overlap is therefore non-zero only if
$N=\frac{L}{2}$.

In contrast to the crosscap state overlap as shown in formula (57) this
formula can be simplified drastically. But first we will discuss some
recursion properties of the crosscap singlet overlap To simplify the notation
we define

$\bra{C_{S}}\prod_{n=1}^{\frac{L}{2}}B(\lambda_{n})\ket{0}=O_{L}(\\{\lambda\\},\\{\nu\\})$
(63)

First, both $a_{2}$ and $d_{1}$ are symmetric in the inhomogeneities. Since
$Z_{L}$ is symmetric in the inhomogeneities this means that $O_{L}$ is also
symmetric in the inhomogeneities. Furthermore, the sum is over all partitions
of the set of rapidities. This coupled with the fact that $Z_{L}$ is symmetric
in the rapidities implies that $O_{L}$ is symmetric in the rapidities as well.

Second, $O_{L}$ is a polynomial of order $L-1$ in all its arguments. To see
why it has order $L-1$ we recall that $Z_{\frac{L}{2}}$ is a polynomial with
order $\frac{L}{2}-1$ in all its arguments. Each vacuum eigenvalue in equation
(4.2) is of order $\frac{L}{2}$ in its argument, so $O_{L}$ is of order
$\frac{L}{2}+\frac{L}{2}-1=L-1$ in each rapidity. Each inhomogeneity appears
once in each vacuum eigenvalue factor in equation (4.2) and there are
$\frac{L}{2}$ such factors, so $O_{L}$ is of order
$\frac{L}{2}+\frac{L}{2}-1=L-1$ in each inhomogeneity. In order to show that
$O_{L}$ is a polynomial we need to show that its residue at each pole is $0$.
We know that $Z_{\frac{L}{2}}$ is a polynomial, so only the factors
$f(\lambda_{k}^{II},\lambda_{j}^{I})$ contribute with poles whenever
$\lambda_{k}\to\lambda_{l}$, $l\neq k$. The singular part involving
$\lambda_{1}$ and $\lambda_{2}$ is

$\displaystyle\frac{i}{\lambda_{2}-\lambda_{1}}$ $\displaystyle
a_{2}(\lambda_{1})d_{1}(\lambda_{2})Z_{\frac{L}{2}}({-\lambda_{1}}\cup\\{\lambda_{2}\\}\\{-\lambda^{I}\\}\cup\\{\lambda^{II}\\},\\{\nu\\}_{2})$
$\displaystyle+\frac{i}{\lambda_{1}-\lambda_{2}}a_{2}(\lambda_{2})d_{1}(\lambda_{1})Z_{\frac{L}{2}}({-\lambda_{2}}\cup\\{\lambda_{1}\\}\\{-\lambda^{I}\\}\cup\\{\lambda^{II}\\},\\{\nu\\}_{2}).$
(64)

In the limit where $\lambda_{1}\to\lambda_{2}$ these two terms cancel, so the
residue at this pole is $0$. Since all other poles have the same form this
means that $O_{L}$ is a polynomial.

Third, we can obtain a recursion relation for $O_{L}$ by setting
$\lambda_{1}=\nu_{1}+\frac{i}{2}$. This sets $d_{1}(\lambda_{1})=0$, as can be
seen in equation (13). The rapidity $\lambda_{1}$ is therefore forced to be in
partition $I$ so

$\displaystyle
O_{L}\bigg{\rvert}_{\lambda_{1}=\nu_{1}+\frac{i}{2}}=a_{2}(\lambda_{1})\sum_{\\{\lambda\\}=\\{\lambda^{I}\\}\cup\\{\lambda^{II}\\}}$
$\displaystyle\prod_{j\in{I}}\prod_{k\in{II}}[a_{2}(\lambda_{j}^{I})d_{1}(\lambda_{k}^{II})f(\lambda_{k}^{II},\lambda_{1})f(\lambda_{k}^{II},\lambda_{j}^{I})]$
$\displaystyle\crossproduct(-1)^{\frac{(n_{I}+1)L}{2}}Z_{\frac{L}{2}}(\\{-\lambda_{1}\\}\cup\\{-\lambda^{I}\\}\cup\\{\lambda^{II}\\},\\{\nu\\}_{2})$
(65)

Identifying the vacuum eigenvalues in equation (29) we can write

$Z_{L}=\frac{\prod_{k=1}^{L}a(\lambda_{k})d(\lambda_{k})}{\prod_{1\leq k<l\leq
L}(\nu_{k}-\nu_{l})(\lambda_{l}-\lambda_{k})}\det\mathcal{M}.$ (66)

Furthermore using the relations
$a_{2}(-\lambda)=(-1)^{\frac{L}{2}}d_{1}(\lambda),\
d_{2}(-\lambda)=(-1)^{\frac{L}{2}}a_{1}(\lambda)$ we then obtain the following
recursion relation for $O_{L}$

$O_{L}\bigg{\rvert}_{\lambda_{1}=\nu_{1}+\frac{i}{2}}=a_{2}(\lambda_{1})d_{1}(\lambda_{1})O_{L-1}(\\{\lambda\\}/\lambda_{1},\\{\nu\\}/\nu_{1})^{\mathrm{mod}_{1}}$
(67)

where the superscript ${}^{\mathrm{mod}_{1}}$ indicates that the vacuum
eigenvalues have been modified in the following way

$\displaystyle a_{1}(\lambda)$ $\displaystyle\to
a_{1}(\lambda)(\lambda-\nu_{1}+\frac{i}{2})$ $\displaystyle d_{1}(\lambda)$
$\displaystyle\to
d_{1}(\lambda)(\lambda+\nu_{1}-\frac{i}{2})f(\lambda,\lambda_{1})$
$\displaystyle a_{2}(\lambda)$ $\displaystyle\to a_{2}(\lambda)$
$\displaystyle d_{2}(\lambda)$ $\displaystyle\to d_{2}(\lambda)$

One can obtain a similar recursion relation by setting
$\lambda_{1}=-\nu_{1}-\frac{i}{2}$:

$O_{L}\bigg{\rvert}_{\lambda_{1}=-\nu_{1}-\frac{i}{2}}=d_{2}(\lambda_{1})a_{1}(\lambda_{1})O_{L-1}(\\{\lambda\\}/\lambda_{1},\\{\nu\\}/\nu_{1})^{\mathrm{mod}_{2}}$
(68)

where ${}^{\mathrm{mod}_{2}}$ indicates the following modification

$\displaystyle a_{1}(\lambda)$ $\displaystyle\to
a_{1}(\lambda)(\lambda-\nu_{1}+\frac{i}{2})f(\lambda_{1},\lambda)$
$\displaystyle d_{1}(\lambda)$ $\displaystyle\to
d_{1}(\lambda)(\lambda+\nu_{1}-\frac{i}{2})$ $\displaystyle a_{2}(\lambda)$
$\displaystyle\to a_{2}(\lambda)$ $\displaystyle d_{2}(\lambda)$
$\displaystyle\to d_{2}(\lambda)$

These recursion relations give us $O_{L}$ at $L$ different values for each of
its arguments, thus determining it completely if $O_{2}$ is known. One can
obtain the initial value $O_{2}$ by direct calculation:

$O_{2}(\lambda,\\{\nu,-\nu\\})=\frac{d_{1}(\lambda)a_{2}(\lambda)d_{2}(\lambda)}{(\lambda+\nu-\frac{i}{2})(\lambda+\nu+\frac{i}{2})}-\frac{a_{2}(\lambda)a_{1}(\lambda)d_{1}(\lambda)}{(-\lambda+\nu-\frac{i}{2})(-\lambda+\nu+\frac{i}{2})}$
(69)

where we used $a_{2}(-\lambda)d_{2}(-\lambda)=a_{1}(\lambda)d_{1}(\lambda)$.
No concise solution to this recuresion has been found.

###  4.3 A determinant formula for the singlet overlap

The singlet overlap $O_{L}$ can be written in a concise form using auxiliary
quantum fields. We begin by carrying out a multi-row Laplace expansion on the
rows corresponding to $\\{-\lambda^{I}\\}$ in the determinant
$\det\mathcal{M}$ in
$Z_{\frac{L}{2}}((\\{-\lambda_{1}\\}\cup\\{-\lambda^{I}\\}\cup\\{\lambda^{II}\\},\\{\nu\\}_{2}))$.
The columns correspond to the inhomogeneities, so we obtain

$\det\mathcal{M}(\\{-\lambda^{I}\\}\cup\\{\lambda^{II}\\},\\{\nu\\})=\sum_{\\{\nu\\}=\\{\nu^{\gamma}\\}\cup\\{\nu^{\delta}\\}}(-1)^{P}\det\mathcal{M}(\\{-\lambda^{I}\\},\\{\nu^{\gamma}\\})\det\mathcal{M}(\\{\lambda^{II}\\},\\{\nu^{\delta}\\}),$
(70)

where the sum is over all partitions of the set of inhomogeneities such that
the cardinality of $\\{\nu^{\gamma}\\}$ is equal to the cardinality of
$\\{\lambda^{I}\\}$. The sign $(-1)^{P}$ is the sign of the permutation
corresponding to the particular partition of $\\{\lambda\\},\\{\nu\\}$.
Inserting this relation into expression (4.2) and factoring out the prefactor
in $Z_{\frac{L}{2}}$ we get

$\displaystyle\bra{C_{S}}\prod_{n=1}^{\frac{L}{2}}B(\lambda_{n})\ket{0}=$
$\displaystyle\frac{\prod_{k=1}^{N}a_{2}(\lambda_{k})d_{2}(\lambda_{k})}{\prod_{1\leq
k<l\leq\frac{L}{2}}(\nu_{l}-\nu_{k})(\lambda_{l}-\lambda_{k})}$
$\displaystyle\crossproduct\sum_{\begin{subarray}{c}\\{\lambda\\}=\\{\lambda^{I}\\}\cup\\{\lambda^{II}\\}\\\
\\{\nu\\}=\\{\nu^{\gamma}\\}\cup\\{\nu^{\delta}\\}\end{subarray}}(-1)^{P}\prod_{j\in{I}}\prod_{k\in{II}}[a_{2}(\lambda_{j}^{I})d_{1}(\lambda_{k}^{II})f(\lambda_{k}^{II},\lambda_{j}^{I})]$
$\displaystyle\crossproduct(-1)^{\frac{n_{I}L}{2}}\prod_{j\in{I}}\prod_{k\in{II}}\left[\frac{\lambda_{j}^{I}-\lambda_{k}^{II}}{\lambda_{j}^{I}+\lambda_{k}^{II}}\right]\prod_{j\in
I}\frac{a_{1}(\lambda^{I}_{j})d_{1}(\lambda^{I}_{j})}{a_{2}(\lambda^{I}_{j})d_{2}(\lambda^{I}_{j})}$
$\displaystyle\crossproduct\det\mathcal{M}(\\{-\lambda^{I}\\},\\{\nu^{\gamma}\\})\det\mathcal{M}(\\{\lambda^{II}\\},\\{\nu^{\delta}\\}).$
(71)

This expression may be simplified by making use of the Laplace formula for the
determinant of a sum of matrices (see appendix IX. 1 in [19]):

$\det(A+B)=\sum_{P_{L},P_{C}}(-1)^{[P_{L}]+[P_{C}]}\det A_{P_{L}P_{C}}\det
B_{P_{L}P_{C}}.$ (72)

Here the sum is over all partitions $P_{L},P_{C}$ of the rows and columns into
partitions of type $A$ and $B$. There are $n_{A}$ elements in partition $A$
and $n_{B}$ elements in partition $B,$ and
$n_{A}+n_{B}=\mathrm{dim}(A)=\mathrm{dim}(B)$. The $A$ partition of the rows
need not contain the same elements as the $A$ partition of the columns. The
determinant $\det A_{P_{L}P_{C}}$ is the determinant of the matrix $A$ where
all the rows and columns corresponding to the partitions of type $B$ have been
removed. Finally, $(-1)^{[P_{L}]}$ is the sign of the permutation that puts
all the rows corresponding to the elements in the partition of type $A$ first
in the full $A$ matrix, while $(-1)^{[P_{C}]}$ is the sign of the permutation
that puts all the elements corresponding to the partition of type $A$ first.

One can show that the sign $(-1)^{P}$ in (4.3) is the same as the sign that is
required to make use of the Laplace formula (72). The extra sign
$(-1)^{\frac{n_{I}L}{2}}$ can be absorbed into the determinants, as can all
vacuum eigenvalue factors, so the only problematic parts are the factors

$f(\lambda^{II}_{k},\lambda^{I}_{j})\frac{\lambda_{j}^{I}-\lambda_{k}^{II}}{\lambda_{j}^{I}+\lambda_{k}^{II}}.$
(73)

They can be dealt with by introducing a pair of auxiliary quantum fields $q,p$
with the only non-trivial commutation relations

$[p(\lambda),q(\mu)]=\ln\left(f(\mu,\lambda)\frac{\lambda-\mu}{\lambda+\mu}\right)$
(74)

and a vacuum $|0)$ satisfying

$p(\lambda)|0)=0,\quad(0|q(\lambda)=0.$ (75)

This kind of manipulation is done in [19] for off-shell overlaps between Bethe
states. Using the auxiliary quantum fields the overlap may be written as

$\bra{C_{S}}\prod_{n=1}^{\frac{L}{2}}B(\lambda_{n})\ket{0}=\frac{\prod_{k=1}^{\frac{L}{2}}a_{2}(\lambda_{k})d_{2}(\lambda_{k})}{\prod_{1\leq
k<l\leq\frac{L}{2}}(\nu_{l}-\nu_{k})(\lambda_{l}-\lambda_{k})}(0|\det S|0)$
(76)

where the matrix $S$ is given by

$\displaystyle
S_{k,l}=-a_{2}(\lambda_{k})\frac{a_{1}(\lambda_{k})d_{1}(\lambda_{k})}{a_{2}(\lambda_{k})d_{2}(\lambda_{k})}$
$\displaystyle\frac{1}{(-\lambda_{k}+\nu_{l}-\frac{i}{2})(-\lambda_{k}+\nu_{l}+\frac{i}{2})}e^{p(\lambda_{k})}$
$\displaystyle+d_{1}(\lambda_{k})\frac{1}{(\lambda_{k}+\nu_{l}-\frac{i}{2})(\lambda_{k}+\nu_{l}+\frac{i}{2})}e^{q(\lambda_{k})}.$
(77)

This expression can be simplified further by factoring out all factors
$e^{p(\lambda_{k})}$. This leads to a new dual vacuum

$(\tilde{0}|=(0|\exp{\sum_{k=1}^{\frac{L}{2}}p(\lambda_{k})}$ (78)

satisfying

$(\tilde{0}|q(\mu)=(\tilde{0}|\sum^{\frac{L}{2}}_{k=1}[p(\lambda_{k}),q(\mu)]=(\tilde{0}|\sum^{\frac{L}{2}}_{k=1}\ln\left(f(\mu,\lambda_{k})\frac{\lambda_{k}-\mu}{\lambda_{k}+\mu}\right).$
(79)

Defining new fields $p^{\prime},q^{\prime},\varphi$ such that

$p^{\prime}(\lambda)=p(\lambda),\quad
q^{\prime}(\mu)=q(\mu)-\sum^{\frac{L}{2}}_{k=1}\ln\left(f(\mu,\lambda_{k})\frac{\lambda_{k}-\mu}{\lambda_{k}+\mu}\right),\quad\varphi(\lambda)=q^{\prime}(\lambda)-p^{\prime}(\lambda)$
(80)

where $\varphi$ has commutation relation

$[\varphi(\lambda),\varphi(\mu)]=-\ln\left(f(\lambda,\mu)f(\mu,\lambda)\frac{1}{(\lambda+\mu)^{2}}\right)$
(81)

the overlap becomes

$\bra{C_{S}}\prod_{n=1}^{\frac{L}{2}}B(\lambda_{n})\ket{0}=\frac{\prod_{k=1}^{\frac{L}{2}}a_{2}(\lambda_{k})d_{2}(\lambda_{k})}{\prod_{1\leq
k<l\leq\frac{L}{2}}(\nu_{l}-\nu_{k})(\lambda_{l}-\lambda_{k})}(\tilde{0}|\det\tilde{S}|0)$
(82)

with the matrix $\tilde{S}$ given by

$\displaystyle\tilde{S}_{k,l}=-a_{2}(\lambda_{k})$
$\displaystyle\frac{a_{1}(\lambda_{k})d_{1}(\lambda_{k})}{a_{2}(\lambda_{k})d_{2}(\lambda_{k})}\frac{1}{(-\lambda_{k}+\nu_{l}-\frac{i}{2})(-\lambda_{k}+\nu_{l}+\frac{i}{2})}$
$\displaystyle+d_{1}(\lambda_{k})\frac{1}{(\lambda_{k}+\nu_{l}-\frac{i}{2})(\lambda_{k}+\nu_{l}+\frac{i}{2})}e^{\varphi(\lambda_{k})}\prod_{n\neq
k}f(\lambda_{k},\lambda_{n})\frac{\lambda_{n}-\lambda_{k}}{\lambda_{n}+\lambda_{k}}.$
(83)

This formula only involves one auxiliary quantum field $\varphi$.

##  5 Conclusions

We have discussed two classes of integrable crosscap states and derived off-
shell formulae for their overlaps with Bethe states. The off-shell formulae we
obtained may potentially be used to derive the on-shell overlaps. For example,
it might be possible to use a procedure like the one that was used in [7] to
perform the on-shell limit of VBS overlaps. The procedure amounts to taking
the limit where the rapidities become parity invariant and deriving a
recursion relation for the on-shell overlap. This was recently done [28] for
the crosscap overlap in the more general $\mathfrak{gl}(n)$ case. The on-shell
limit is expected to be more difficult for the crosscap singlet overlaps,
because if $\frac{L}{2}$ is even they are non-zero only if the Bethe state is
singular, as we discussed in section 4.1. In addition, it is of course not
know if the crosscap singlet overlaps take the expected form (24), and it
would be interesting to see if it does. But even if no such on-shell formula
is derived one can take the parity invariant limit and regularize the
singularities before inserting the Bethe roots into the off-shell formulae.
The on-shell formula would thus still be sums over partitions.

In the on-shell limits these formulae may potentially be of use in the study
of quantum quenches, as was discussed in the introduction. In that case one
would need to take the thermodynamic limit, which can be done for the crosscap
overlap, since it has the standard overlap form (24). The thermodynamic limit
of an overlap with this form was computed in [5].

Finally, integrable overlaps have been useful in the AdS/CFT correspondence,
as we discussed in the introduction. It is not known if crosscap state
overlaps appears in any defect configuration in $\mathcal{N}=4\ \mathrm{SYM},$
but work in this direction is in progress [15].

### Acknowledgement

I am grateful to my former supervisor Kostya Zarembo for the many useful
discussions throughout this project.

### Note added

After this work was completed [28] appeared where equation (57) is derived for
$\mathfrak{gl}(n)$ spin chains in the same way as in the present article.

## References

  * [1] Pasquale Calabrese and John Cardy  “Time Dependence of Correlation Functions Following a Quantum Quench”  In _Physical Review Letters_ 96.13 American Physical Society (APS), 2006  DOI: 10.1103/physrevlett.96.136801
  * [2] Christian Gogolin and Jens Eisert  “Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems”  In _Reports on Progress in Physics_ 79.5 IOP Publishing, 2016, pp. 056001  DOI: 10.1088/0034-4885/79/5/056001
  * [3] J. Eisert, M. Friesdorf and C. Gogolin  “Quantum many-body systems out of equilibrium”  In _Nature Physics_ 11.2 Springer ScienceBusiness Media LLC, 2015, pp. 124–130  DOI: 10.1038/nphys3215
  * [4] Balá zs Pozsgay  “Overlaps between eigenstates of the XXZ spin-1/2 chain and a class of simple product states”  In _Journal of Statistical Mechanics: Theory and Experiment_ 2014.6 IOP Publishing, 2014, pp. P06011  DOI: 10.1088/1742-5468/2014/06/p06011
  * [5] M Brockmann, J De Nardis, B Wouters and J-S Caux  “A Gaudin-like determinant for overlaps of Né el and XXZ Bethe states”  In _Journal of Physics A: Mathematical and Theoretical_ 47.14 IOP Publishing, 2014, pp. 145003  DOI: 10.1088/1751-8113/47/14/145003
  * [6] B. Wouters et al.  “Quenching the Anisotropic Heisenberg Chain: Exact Solution and Generalized Gibbs Ensemble Predictions”  In _Physical Review Letters_ 113.11 American Physical Society (APS), 2014  DOI: 10.1103/physrevlett.113.117202
  * [7] Yunfeng Jiang and Balázs Pozsgay  “On exact overlaps in integrable spin chains”  In _Journal of High Energy Physics_ 2020.6 Springer ScienceBusiness Media LLC, 2020  DOI: 10.1007/jhep06(2020)022
  * [8] Tamás Gombor  “On exact overlaps for $\mathfrak{gl}(N)$ symmetric spin chains”, 2021  arXiv:2110.07960 [hep-th]
  * [9] Tamás Gombor and Balázs Pozsgay  “On factorized overlaps: Algebraic Bethe Ansatz, twists, and separation of variables”  In _Nuclear Physics B_ 967 Elsevier BV, 2021, pp. 115390  DOI: 10.1016/j.nuclphysb.2021.115390
  * [10] O Foda and K Zarembo  “Overlaps of partial Néel states and Bethe states”  In _Journal of Statistical Mechanics: Theory and Experiment_ 2016.2 IOP Publishing, 2016, pp. 023107  DOI: 10.1088/1742-5468/2016/02/023107
  * [11] Lorenzo Piroli, Balázs Pozsgay and Eric Vernier  “What is an integrable quench?”  In _Nuclear Physics B_ 925, 2017, pp. 362–402  DOI: 10.1016/j.nuclphysb.2017.10.012
  * [12] B. Pozsgay  “Overlaps with arbitrary two-site states in the XXZ spin chain”  In _Journal of Statistical Mechanics: Theory and Experiment_ 2018.5 IOP Publishing, 2018, pp. 053103  DOI: 10.1088/1742-5468/aabbe1
  * [13] Marius Leeuw, Charlotte Kristjansen and Konstantin Zarembo  “One-point functions in defect CFT and integrability”  In _Journal of High Energy Physics_ 2015.8 Springer ScienceBusiness Media LLC, 2015  DOI: 10.1007/jhep08(2015)098
  * [14] Isak Buhl-Mortensen, Marius Leeuw, Charlotte Kristjansen and Konstantin Zarembo  “One-point functions in AdS/dCFT from matrix product states”  In _Journal of High Energy Physics_ 2016.2 Springer ScienceBusiness Media LLC, 2016  DOI: 10.1007/jhep02(2016)052
  * [15] João Caetano and Shota Komatsu  “Crosscap States in Integrable Field Theories and Spin Chains”  In _Journal of Statistical Physics_ 187.3 Springer ScienceBusiness Media LLC, 2022  DOI: 10.1007/s10955-022-02914-6
  * [16] C. Kristjansen, D. Müller and K. Zarembo  “Integrable boundary states in D3-D5 dCFT: beyond scalars”  In _Journal of High Energy Physics_ , 2020  DOI: 10.1007/JHEP08(2020)103
  * [17] Joseph A Minahan and Konstantin Zarembo  “The Bethe-ansatz for Script N = 4 super Yang-Mills”  In _Journal of High Energy Physics_ 2003.03 Springer ScienceBusiness Media LLC, 2003, pp. 013  DOI: 10.1088/1126-6708/2003/03/013
  * [18] Marius Leeuw  “One-point functions in AdS/dCFT”  In _Journal of Physics A: Mathematical and Theoretical_ 53.28 IOP Publishing, 2020, pp. 283001  DOI: 10.1088/1751-8121/ab15fb
  * [19] V. Korepin, A. Izergin and N. Bogoliubov  “Quantum inverse scattering method and correlation functions”  Cambridge University Press, 2010
  * [20] N.. Slavnov  “Algebraic Bethe ansatz”  arXiv, 2018  arXiv:1804.07350 [math-ph]
  * [21] L.. Faddeev  “How Algebraic Bethe Ansatz works for integrable model”  In _Les Houches School of Physics: Astrophysical Sources of Gravitational Radiation_ arXiv, 1996  arXiv:hep-th/9605187
  * [22] W. Heisenberg  “Zur Theorie des Ferromagnetismus”  In _Zeitschrift für Physik_ 49.9, 1928, pp. 619–636  DOI: 10.1007/BF01328601
  * [23] Charlotte Kristjansen, Dennis Müller and Konstantin Zarembo  “Duality Relations for Overlaps of Integrable Boundary States in AdS/dCFT”, 2021  arXiv:2106.08116 [hep-th]
  * [24] V.. Korepin  “Calculation of norms of Bethe wave functions”  In _Communications in mathematical physics_ 86.3 Springer-Verlag, 1982, pp. 391–418  DOI: 10.1007/BF01212176
  * [25] Rafael I Nepomechie and Chunguang Wang  “Algebraic Bethe ansatz for singular solutions”  In _Journal of Physics A: Mathematical and Theoretical_ 46.32 IOP Publishing, 2013, pp. 325002  DOI: 10.1088/1751-8113/46/32/325002
  * [26] Rahul Siddharthan  “Singularities in the Bethe solution of the XXX and XXZ Heisenberg spin chains”, 1998  arXiv:9804210 [cond-mat]
  * [27] L.V. Avdeev and A.A. Vladimirov  “Exceptional solutions to the Bethe ansatz equations”  In _Theor Math Phys_ 69, 1986, pp. 1071–1079  DOI: https://doi.org/10.1007/BF01037864
  * [28] Tamas Gombor  “Integrable crosscap states in $\mathfrak{gl}(N)$ spin chains”  arXiv, 2022  arXiv:2207.10598 [het-th]