††thanks: These two authors contributed equally.††thanks: These two authors
contributed equally.

# Extracting the Luttinger parameter from a single wave function

Bi-Yang Tan1 Yueshui Zhang2 Hua-Chen Zhang3, Wei Tang4,
Lei Wang5,6 Hong-Hao Tu2<EMAIL_ADDRESS>Ying-Hai Wu1<EMAIL_ADDRESS>1
School of Physics and Wuhan National High Magnetic Field Center, Huazhong
University of Science and Technology, Wuhan 430074, China
2 Physics Department and Arnold Sommerfeld Center for Theoretical Physics,
Ludwig-Maximilians-Universität München, 80333 Munich, Germany
3 Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C,
Denmark
4 Department of Physics and Astronomy, Ghent University, Krijgslaan 281, S9,
B-9000 Ghent, Belgium
5 Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
6 Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China

###### Abstract

The low-energy physics of Tomonaga-Luttinger liquids (TLLs) is controlled by
the Luttinger parameter. We demonstrate that this parameter can be extracted
from a single wave function for one-component TLLs with periodic boundary
condition. This method relies on the fact that TLLs are described by conformal
field theory in which crosscap states can be constructed. The overlaps between
the crosscap states and the ground state as well as some excited states are
proved to be universal numbers that directly reveal the Luttinger parameter.
In microscopic lattice models, crosscap states are formed by putting each pair
of antipodal sites into a maximally entangled state. Analytical and numerical
calculations are performed in a few representative models to substantiate the
conformal field theory prediction. The extracted Luttinger parameters are
generally quite accurate in finite-size systems with moderate lengths, so
there is no need to perform data fitting and/or finite-size scaling.

Introduction — The theory of Tomonaga-Luttinger liquid (TLL) is a great
triumph of strongly correlated physics [1]. Based on previous work of Tomonaga
[2], Luttinger introduced a model to study interacting fermions in one
dimension (1D) [3]. It clearly exemplifies the peculiarity of reduced
dimensionality because infinitesimal interaction in a 1D system drives it away
from the Fermi liquid. In subsequent works [4, 5, 6, 7], the bosonization
framework was established to provide a unified treatment for many strongly
correlated problems in 1D. Spin chains are usually studied using the Jordan-
Wigner transformation that converts spins to fermions. Gapless phases of
bosons or fermions can be understood using free boson fields that correspond
to density fluctuations. Instabilities of the TLL lead to symmetry-breaking
ordered phases. It is not easy to realize 1D systems, but experimental
investigations have been carried out in solid state and cold atom platforms
[8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. Another context in which TLL thrives
is the edge of certain two-dimensional topological states [18, 19, 20, 21].

The low-energy physics of one-component TLLs is captured by the Hamiltonian

$\displaystyle
H=\frac{v}{8\pi}\int^{L}_{0}\mathrm{d}x\,\left[(\partial_{x}\hat{\varphi})^{2}+(\partial_{x}\hat{\theta})^{2}\right].$
(1)

In the conformal field theory (CFT) language, this is a system of compactified
bosons. Here $L$ is the length of the system, $v$ is the characteristic
velocity, $\hat{\varphi}$ is a compactified boson field with radius $R$ (i.e.,
$\hat{\varphi}\sim\hat{\varphi}+2\pi R$), and $\hat{\theta}$ is the conjugate
boson field of $\hat{\varphi}$ with radius $R^{\prime}=2/R$. The Luttinger
parameter $K$ can be expressed using the radius as $K=R^{2}/4$. Its value is
$1$ for free fermions and deviates from $1$ when interaction is turned on. For
a given TLL, many properties are determined by $K$ so it is routinely computed
in numerical studies. If the Hamiltonian has a microscopic U(1) symmetry
(e.g., particle number conservation), $K$ can be obtained by comparing the
ground-state energy of multiple systems with different U(1) charges [22]. This
is feasible because $K$ can be interpreted as the stiffness of a superfluid.
The powerlaw decay of certain correlation functions can be used to deduce $K$
when their bosonized expressions are known [1]. It has been found that $K$
also appears in the U(1) charge fluctuations of a subsystem [23, 24], the U(1)
symmetry-resolved entanglement spectrum [25, 26], or the entanglement entropy
associated with particle partition [27, 28].

In this Letter, we propose that the Luttinger parameter in microscopic models
can be extracted from the eigenstate crosscap overlap (ESCO). It has three
favorable features: i) a single wave function instead of multiple states is
used; ii) data fitting or finite-size scaling is not needed; iii) microscopic
U(1) symmetry is not required. To the best of our knowledge, no other method
enjoys all three advantages simultaneously. Our method is motivated by
classical results and recent progress in CFT and integrable models. When a CFT
is defined on the Klein bottle, important information can be inferred from the
thermal entropy [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. The
Klein bottle can be viewed as a special cylinder with crosscaps at its ends. A
simple circle would turn into a crosscap if all pairs of antipodal points are
identified. It is possible to define a variety of crosscap states in the
framework of boundary CFT [42]. For a rational CFT, the crosscap states have
universal overlaps (“crosscap coefficients”) with its ground state [43, 44,
45]. In some integrable models, the crosscap states have been identified as
“integrable boundary states” [46, 47, 48, 49, 50]. The overlaps between all
eigenstates of these integrable models and the crosscap states can be
expressed as determinants, but their physical significance is not transparent.
We aim to establish a general framework for studying crosscap states in CFT
(without and with perturbations) such that physical properties can be deduced
from their overlaps with eigenstates.

Figure 1: (a) Schematics of the modified Villain model and its crosscap states
$|\mathcal{C}_{\pm}\rangle$. (b) Graphic representations of an MPS with eight
sites and its crosscap overlap.

General construction of crosscap states — While crosscap is simple to
visualize in the continuum, crosscap states in lattice models have only been
constructed in a somewhat intuitive manner. The main obstacle is how to make a
connection between the degrees of freedom in the continuum and lattice. To
this end, we find it particularly instructive to use the modified Villain
model [51]

$\displaystyle H_{\mathrm{mV}}=\sum_{j=1}^{N}\left[(2\pi
R\,\hat{p}_{j})^{2}+{R^{\prime}}^{2}(\hat{X}_{j+1}-\hat{X}_{j}+2\pi\hat{n}_{j,j+1})^{2}\right]\,,$
(2)

as illustrated in Fig. 1(a). It provides a specific lattice realization of the
compactified boson Hamiltonian in Eq. (1). There are $N$ sites in the chain
(labeled by $j$), the lattice spacing is $a$, and periodic boundary condition
is imposed. $\hat{X}_{j}$ ($\hat{E}_{j,j+1}$) is the site (link) variable at
site $j$ [link $(j,j+1)$] and $\hat{p}_{j}$ ($\hat{n}_{j,j+1}$) is its
canonical momentum. The standard commutation relations
$[\hat{X}_{j},\hat{p}_{l}]=[\hat{E}_{j,j+1},\hat{n}_{l,l+1}]=i\delta_{jl}$ are
satisfied. The local Hilbert space at site $j$ [link $(j,j+1)$], denoted as
$\mathcal{H}_{j}$ ($\mathcal{H}_{j,j+1}$), is spanned by the eigenbasis of
$\hat{X}_{j}$ ($\hat{E}_{j,j+1}$) and expressed as $|X_{j}\rangle$ with
$X_{j}\in\mathbb{R}$ ($|E_{j,j+1}\rangle$ with $E_{j,j+1}\in[0,2\pi)$). The
full Hilbert space is thus the tensor product of local ones on all sites and
links, $\mathcal{H}=\prod^{N}_{j=1}\mathcal{H}_{j}\otimes\mathcal{H}_{j,j+1}$.
However, local operators
$\hat{G}_{j}=e^{i(\hat{E}_{j,j+1}-\hat{E}_{j-1,j}-2\pi\hat{p}_{j})}$ commute
with both terms in the Hamiltonian (2),
$[\hat{G}_{j},\hat{p}_{j^{\prime}}]=[\hat{G}_{j},\hat{X}_{j^{\prime}+1}-\hat{X}_{j^{\prime}}+2\pi\hat{n}_{j^{\prime},j^{\prime}+1}]=0\;\forall
j,j^{\prime}$, and hence also commute with the Hamiltonian,
$[H_{\mathrm{mV}},\hat{G}_{j}]=0\;\forall j$. Thus, we may impose the Gauss
law constraint $\hat{G}_{j}=1$ such that the model becomes a lattice gauge
theory. The physical Hilbert space is the subspace
$\mathcal{H}_{\mathrm{phy}}$ of the full space $\mathcal{H}$ that satisfies
the Gauss law. The projection operator onto the subspace
$\mathcal{H}_{\mathrm{phy}}$ is denoted as $\mathcal{P}_{G}$.

To transform the model from lattice to continuum, we introduce new variables
$\hat{\theta}_{j}=R^{\prime}(\hat{X}_{j}+2\pi\sum_{k=1}^{j}\hat{n}_{k-1,k})$
and $\hat{\varphi}_{j,j+1}=R\hat{E}_{j,j+1}$ [51]. It is obvious that
$[e^{i\hat{\theta}_{j}/R^{\prime}},\hat{G}_{j^{\prime}}]=[e^{i\hat{\varphi}_{j,j+1}/R},\hat{G}_{j^{\prime}}]=0\;\forall
j,j^{\prime}$, so $e^{i\hat{\theta}_{j}/R^{\prime}}$ and
$e^{i\hat{\varphi}_{j,j+1}/R}$ act within $\mathcal{H}_{\mathrm{phy}}$. The
compactification radii of $\hat{\theta}_{j}$ and $\hat{\varphi}_{j,j+1}$ are
$R^{\prime}$ and $R$, respectively. In terms of these new variables, the
Hamiltonian in Eq. (2) becomes

$\displaystyle
H_{\mathrm{mV}}=\sum_{j=1}^{N}\left[(\hat{\varphi}_{j+1,j+2}-\hat{\varphi}_{j,j+1})^{2}+(\hat{\theta}_{j+1}-\hat{\theta}_{j})^{2}\right]\,.$
(3)

The continuum limit is taken by sending $N\rightarrow\infty$ and $a\rightarrow
0$ while $L=Na$ is fixed as a constant. After the mapping
$\hat{\varphi}_{j+1,j+2}-\hat{\varphi}_{j,j+1}\rightarrow
a\partial_{x}\hat{\varphi}(x)$ and
$\hat{\theta}_{j+1}-\hat{\theta}_{j}\rightarrow a\partial_{x}\hat{\theta}(x)$,
Eq. (3) reduces to the TLL Hamiltonian (1).

Motivated by the geometric structure of crosscap, one should try to construct
crosscap states by putting antipodal points into certain maximally entangled
states. In the modified Villain model, this line of thought leads to two
crosscap states

$\displaystyle|\mathcal{C}_{\pm}\rangle=\mathcal{P}_{G}\prod_{j=1}^{N/2}|\mathcal{B}^{\mathrm{V}}_{\pm}\rangle_{j,j+N/2}\otimes|\mathcal{B}^{\mathrm{L}}_{\mp}\rangle_{j,j+N/2}\,,$
(4)

where

$\displaystyle|\mathcal{B}^{\mathrm{L}}_{\mp}\rangle_{j,j+N/2}$
$\displaystyle=\int_{0}^{2\pi}\mathrm{d}E\,|E\rangle_{j,j+1}|\mp
E\rangle_{j+N/2,j+N/2+1}\,,$
$\displaystyle|\mathcal{B}^{\mathrm{V}}_{\pm}\rangle_{j,j+N/2}$
$\displaystyle=\int_{-\infty}^{\infty}\mathrm{d}X\,|X\rangle_{j}|\pm
X\rangle_{j+N/2}$ (5)

are generalized Bell pairs of antipodal links and sites, respectively. It can
be verified that
$e^{i(\hat{\theta}_{j}\mp\hat{\theta}_{j+N/2})/R^{\prime}}|\mathcal{C}_{\pm}\rangle=|\mathcal{C}_{\pm}\rangle$
and
$e^{i(\hat{\varphi}_{j,j+1}\pm\hat{\varphi}_{j+N/2,j+N/2+1})/R}|\mathcal{C}_{\pm}\rangle=|\mathcal{C}_{\pm}\rangle$
for all $j$ (note that $e^{i\hat{\theta}_{j}/R^{\prime}}$ and
$e^{i\hat{\varphi}_{j,j+1}/R}$ commute with $\mathcal{P}_{G}$). In the
continuum limit, they become the “sewing conditions”

$\displaystyle
e^{i[\hat{\theta}(x)\mp\hat{\theta}(x+L/2)]/R^{\prime}}|\mathcal{C}_{\pm}\rangle$
$\displaystyle=|\mathcal{C}_{\pm}\rangle\,,$ $\displaystyle
e^{i[\hat{\varphi}(x)\pm\hat{\varphi}(x+L/2)]/R}|\mathcal{C}_{\pm}\rangle$
$\displaystyle=|\mathcal{C}_{\pm}\rangle\,,$ (6)

that are imposed on the crosscap states by boundary CFT [52, 53, 54]. It is
exhilarating that these conditions can be derived rigorously using our
formalism.

For subsequent calculations, the crosscap states may be rewritten using the
mode expansion [55]:

$\displaystyle\hat{\theta}(x)$ $\displaystyle=\tilde{\varphi}_{0}+\frac{4\pi
x}{L}\pi_{0}+\sum_{k\neq 0}\frac{i}{k}\left[e^{i\frac{2\pi
k}{L}x}a_{k}-e^{-i\frac{2\pi k}{L}x}\bar{a}_{k}\right],$
$\displaystyle\hat{\varphi}(x)$ $\displaystyle=\varphi_{0}+\frac{4\pi
x}{L}\tilde{\pi}_{0}+\sum_{k\neq 0}\frac{i}{k}\left[e^{i\frac{2\pi
k}{L}x}a_{k}+e^{-i\frac{2\pi k}{L}x}\bar{a}_{k}\right]\,,$ (7)

where the zero modes and oscillatory modes satisfy
$[\varphi_{0},\pi_{0}]=[\tilde{\varphi}_{0},\tilde{\pi}_{0}]=i$ and
$[a_{k},a_{l}]=[\bar{a}_{k},\bar{a}_{l}]=k\delta_{k+l,0}$, respectively.
$\varphi_{0}$ ($\tilde{\varphi}_{0}$) is an angular variable with radius $R$
($R^{\prime}$), and $\pi_{0}$ ($\tilde{\pi}_{0}$) is the associated canonical
momentum. The eigenvectors of $\pi_{0}$ and $\tilde{\pi}_{0}$ that can be
annihilated by all bosonic annihilation operators $a_{k>0}$ and
$\bar{a}_{k>0}$ are denoted as $|n,m\rangle$ ($n,m\in\mathbb{Z}$); they are
highest weight states (Virasoro primary states) of the CFT and satisfy
$\pi_{0}|n,m\rangle=\tfrac{n}{R}|n,m\rangle$ and
$\tilde{\pi}_{0}|n,m\rangle=\tfrac{m}{R^{\prime}}|n,m\rangle$. Using the mode
expansion, the compactified boson Hamiltonian (1) becomes

$\displaystyle H=\frac{2\pi
v}{L}\left[\pi_{0}^{2}+\tilde{\pi}_{0}^{2}+\sum_{k=1}^{\infty}(a_{-k}a_{k}+\bar{a}_{-k}\bar{a}_{k})-\frac{1}{12}\right],$
(8)

whose energy eigenstates are just Virasoro primary states $|n,m\rangle$ and
the descendant states obtained by applying the creation operators $a_{k<0}$
and $\bar{a}_{k<0}$ on top of $|n,m\rangle$. The crosscap states
$|\mathcal{C}_{\pm}\rangle$ can be written as

$\displaystyle|\mathcal{C}_{+}\rangle$
$\displaystyle=\kappa_{+}\exp\left[\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}a_{-k}\bar{a}_{-k}\right]\sum_{n\in\mathbb{Z}}|2n,0\rangle\,,$
$\displaystyle|\mathcal{C}_{-}\rangle$
$\displaystyle=\kappa_{-}\exp\left[-\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}a_{-k}\bar{a}_{-k}\right]\sum_{m\in\mathbb{Z}}|0,2m\rangle$
(9)

with undetermined overall factors using the mode expansion [56]. The values of
$\kappa_{\pm}$ cannot be fixed by normalization. Employing the “cut-and-sew”
procedure, we can compare the partition function
$Z^{\mathcal{C}}(L,\beta)=\langle\mathcal{C}|e^{-\beta H}|\mathcal{C}\rangle$
on a cylinder with two crosscap boundaries (“tree channel”) and the partition
function $Z^{\mathcal{K}}(2\beta,L/2)$ on a Klein bottle (“loop channel”).
This analysis yields $\kappa_{+}=\sqrt{R^{\prime}}$ and $\kappa_{-}=\sqrt{R}$
[56].

An appealing property of the crosscap states is that their overlaps with
certain highest weight states $|n,m\rangle$ in the CFT are universal numbers
depending only on the compactification radius (equivalently, Luttinger
parameter):

$\displaystyle|\langle\mathcal{C_{+}}|2n,0\rangle|^{2}=R^{\prime}=\frac{1}{\sqrt{K}},\quad
n\in\mathbb{Z}\,,$
$\displaystyle|\langle\mathcal{C_{-}}|0,2m\rangle|^{2}=R=2\sqrt{K},\quad
m\in\mathbb{Z}\,.$ (10)

It was found in Ref. [33] that the Luttinger parameter appears in the thermal
entropy of compactified boson CFTs on the Klein bottle. In view of the
spacetime symmetry of CFTs, it is natural to expect that the same variable can
also be found using the ground state alone. Our derivation not only puts this
expectation on a firm ground but also provides further insights into crosscap
states. Some excited states are incorporated in the present formalism, and
perturbations to a pure CFT can be investigated [57].

Spin chains with microscopic U(1) symmetry — To apply the general theory in
specific lattice models, the crosscap states should be expressed in terms of
the microscopic degrees of freedom. It is helpful to begin with spin-1/2
chains. The first one is the XXZ model

$\displaystyle
H_{\mathrm{XXZ}}=\sum_{j=1}^{N}\left(S^{x}_{j}S^{x}_{j+1}+S^{y}_{j}S^{y}_{j+1}+\Delta
S^{z}_{j}S^{z}_{j+1}\right)$ (11)

with $\Delta\in(-1,1]$ for which $K=\frac{\pi}{2(\pi-\cos^{-1}\Delta)}$ [1].
The second one is the Haldane-Shastry (HS) model

$\displaystyle H_{\mathrm{HS}}=\left(\frac{\pi}{N}\right)^{2}\sum_{1\leq
j<l\leq
N}\frac{\mathbf{S}_{j}\cdot\mathbf{S}_{l}}{\sin^{2}\left[\frac{\pi}{N}(j-l)\right]}$
(12)

with $K=1/2$ that provides a perfect lattice realization of the SU(2)1 Wess-
Zumino-Witten model [58, 59]. Both models have a U(1) symmetry associated with
the conservation of the $z$-component of the total spin.

The success of Eq. (4) suggests that

$\displaystyle|\mathcal{C}_{\mathrm{latt}}\rangle=\prod_{j=1}^{N/2}\left(|\uparrow\rangle_{j}|\uparrow\rangle_{j+N/2}+|\downarrow\rangle_{j}|\downarrow\rangle_{j+N/2}\right)$
(13)

is likely a crosscap state for spin-1/2 chains with even $N$. However, there
is one subtlety in this definition: it corresponds to
$|\mathcal{C}_{+}\rangle$ when $\mathrm{mod}(N,4)=0$ but does not give the
expected result when $\mathrm{mod}(N,4)=2$. The reason for this can be traced
to the constraint
$(S_{j}^{z}-S_{j+N/2}^{z})|\mathcal{C}_{\mathrm{latt}}\rangle=0$. Using
bosonization, $S^{z}_{j}$ operator reads
$S^{z}_{j}\sim\partial_{x}\hat{\theta}(x)+(-1)^{j}\alpha\cos(\hat{\theta}(x)/R^{\prime})$
for XXZ and HS chains, where $\alpha$ is a model-dependent constant. For
$\mathrm{mod}(N,4)=0$, the lattice constraint
$(S_{j}^{z}-S_{j+N/2}^{z})|\mathcal{C}_{\mathrm{latt}}\rangle=0$ is hence
consistent with the constraints of $|\mathcal{C}_{+}\rangle$ in Eq. (6).

We now proceed to demonstrate the validity of Eq. (13). For the XY model
(i.e., $\Delta=0$ in the XXZ model) and the HS model, explicit wave functions
of many eigenstates are known [60]. This allows us to derive analytical
results to support our theory. In both models, the eigenstates that correspond
to the Virasoro primary states $|n,0\rangle$ have a Jastrow form in the
hardcore boson basis:

$\displaystyle\Psi^{\lambda}(x_{1},\ldots,x_{M})\propto(-1)^{\sum_{l}x_{l}}\prod_{j<k}\left|\sin\frac{\pi(x_{j}-x_{k})}{N}\right|^{\lambda}$
(14)

with $1\leq x_{1}<\cdots<x_{M}\leq N$ denoting the positions of
spin-$\uparrow$ sites (with hardcore bosons occupied), and $\lambda=1$
($\lambda=2$) for the XY (HS) chain. The hardcore boson number $M$ is related
to the U(1) charge $n$ via $M=N/2+n$. As $|\mathcal{C}_{\mathrm{latt}}\rangle$
contains only states with even number of bosons,
$\langle\mathcal{C}_{\mathrm{latt}}|\Psi^{\lambda}(M)\rangle$ vanishes
identically for odd $M$. This implies
$\langle\mathcal{C}_{\mathrm{latt}}|\Psi^{\lambda}(M=N/2+2n)\rangle=0$ when
$\mathrm{mod}(N,4)=2$. As $|\Psi^{\lambda}(M=N/2+2n)\rangle$ corresponds to
the Virasoro primary state $|2n,0\rangle$, this indicates
$|\mathcal{C}_{\mathrm{latt}}\rangle$ does not realize
$|\mathcal{C}_{\pm}\rangle$ in this case [see Eq. (10)]. For
$\mathrm{mod}(N,4)=0$, we have
$\langle\mathcal{C}_{\mathrm{latt}}|\Psi^{\lambda}(M)\rangle=0$ for odd
$M=N/2+2n+1$ and

$\displaystyle|\langle\mathcal{C}_{\mathrm{latt}}|\Psi^{\lambda=1}(M)\rangle|^{2}$
$\displaystyle=1\,,$
$\displaystyle|\langle\mathcal{C}_{\mathrm{latt}}|\Psi^{\lambda=2}(M)\rangle|^{2}$
$\displaystyle=\frac{2^{M}(M!)^{3}}{[(M/2)!]^{2}(2M)!}=\sqrt{2}+\mathcal{O}(1/M)\,$
(15)

for even $M=N/2+2n$ [56]. This is in perfect agreement with the field theory
prediction in Eq. (10), giving a strong indication that
$|\mathcal{C}_{\mathrm{latt}}\rangle$ corresponds to $|\mathcal{C}_{+}\rangle$
for $\mathrm{mod}(N,4)=0$.

Figure 2: Numerically extracted Luttinger parameters in three models. Exact
values are indicated using solid or dashed black lines for comparison. (a) The
XXZ model with $N=100$ and various $\Delta$. (b) The XXZ model with $\Delta=1$
and various $N$. (c) The next-nearest-neighbor model with $N=100$ and various
$J_{2}$. (d) The five-state clock model with various $h$ and $N$.

For the XXZ model with $\Delta{\neq}0$, the ground state can be computed
numerically using the density matrix renormalization group (DMRG) [61]. When
the ground state $|\Psi\rangle$ is expressed as a matrix product state (MPS)
[62, 63], its crosscap overlap $\langle\mathcal{C}_{\rm latt}|\Psi\rangle$ is
given by the tensor contraction shown in Fig. 1(b). Numerical results at many
different $N$’s and $\Delta$’s are displayed in Fig. 2. The MPS bond dimension
is $2000$ for $N=100$ and $6000$ for $N=400$. The accuracy is very good in a
wide range of $\Delta$ for a moderate $N=100$. The relative error of $K$ is of
the order $10^{-4}$ or smaller when $\Delta\in[-0.95,0.8]$. As $\Delta$
approaches $1$, clear deviation from exact values is observed (not visible in
Fig. 2(a) due to its scale) and the relative error reaches $7.20\%$ at
$\Delta=1$.

We attribute the obvious deviation at $\Delta=1$ to the marginal terms in the
low-energy theory [64], which usually cause strong finite-size effects. The
deviation gets smaller as $N$ increases, but the convergence towards the exact
value is very slow [see Fig. 2(b)]. To validate our conjecture, we add a next-
nearest-neighbor coupling to the XXZ model with $\Delta=1$:

$\displaystyle H_{\rm
next}=\sum_{j=1}^{N}(\mathbf{S}_{j}\cdot\mathbf{S}_{j+1}+J_{2}\mathbf{S}_{j}\cdot\mathbf{S}_{j+2})\,.$
(16)

Its low-energy physics contains no marginal term at $J_{2}{\approx}0.24$ [65].
The results for $N=100$ are presented in Fig. 2(c). As $J_{2}$ increases
toward $0.24$, the relative error of $K$ decreases and eventually becomes
$0.11\%$. This confirms that the marginal terms are responsible for the
deviation. Further analysis based on conformal perturbation theory suggests
that the Luttinger parameter should be fitted as $K=a[\ln(N)]^{-2}+b$ [57].
The data points with $N\in[100,400]$ in Fig. 2(b) thus yield $K=0.523$ in the
thermodynamic limit.

Spin chains without microscopic U(1) symmetry — A fundamental improvement of
our method is that microscopic U(1) symmetry is not required. To demonstrate
this advantage, we consider the quantum $q$-state clock model [66, 67, 68, 69,
70, 71]

$\displaystyle H_{q{\rm
SC}}=-\sum_{j=1}^{N}(\sigma^{{\dagger}}_{j}\sigma_{j+1}+\sigma_{j}\sigma^{{\dagger}}_{j+1})-h\sum_{j=1}^{N}(\tau_{j}+\tau_{j}^{{\dagger}})\,,$
(17)

where the $\mathbb{Z}_{q}$ spin operators are defined by
$\sigma|\alpha\rangle=|{\rm mod}(\alpha-1+q,q)\rangle$ and
$\tau|\alpha\rangle=e^{i2\pi\alpha/q}|\alpha\rangle$ with $\alpha=1,\ldots,q$.
This model is self-dual (SD) at $h=1$. For $q{\geq}5$ and intermediate $h$
(including the SD point), this model hosts a Luttinger liquid that is
separated by two Kosterlitz–Thouless (KT) transitions at $h_{\rm KT1}$ and
$h_{\rm KT2}$ from gapped phases at small and large $h$ [72, 73]. The
transition points are related to each other by duality via $h_{\rm KT1}h_{\rm
KT2}=1$. There is no analytical expression for the Luttinger parameter, but
its value at the SD point and the KT transition points was predicted to be
[68, 34]

$\displaystyle K_{\rm KT1}=\frac{q^{2}}{8},\quad K_{\rm SD}=\frac{q}{2},\quad
K_{\rm KT2}=2\,.$ (18)

For the $q$-state clock model, we shall employ the lattice crosscap state
$|\mathcal{C}_{\mathrm{latt}}\rangle=\prod_{j=1}^{N/2}\left(\sum_{\alpha=1}^{q}|\alpha\rangle_{j}|\alpha\rangle_{j+N/2}\right)$.
Numerical results for the $q=5$ model with $N=50,60,\cdots,90$ and various $h$
are presented in Fig. 2(d). The MPS bond dimension is $1500$ for $N=50$ and
$2500$ for $N=90$. At the SD point, finite-size systems are described by CFT
with irrelevant perturbations, and the Luttinger parameter has a relative
error $0.79\%$ at $N=90$. In the Luttinger liquid regime, the crosscap overlap
only exhibits weak dependence on the system size, so the data points for
different $N$ almost coincide. For the small and large $h$ phases, the overlap
strongly depends on $N$. This picture is supported by Fig. 2(d), but the
transition points cannot be located accurately due to the marginal terms at
the KT transition points.

Summary and outlook — To summarize, we have demonstrated that the Luttinger
parameter can be extracted from the overlaps between individual eigenstates
and the crosscap states. The advantages of our method are corroborated by
analytical and numerical results. It is also noteworthy that our method does
not necessarily require a Hamiltonian, so it may be applied to check if a
trial wave function indeed describes the physics of TLL or other CFTs (see,
e.g., Refs. [74, 75]). From the theoretical perspective, our results open up
an exciting new avenue in extracting conformal data and many interesting
directions are yet to be explored. It is clear that the accuracy of our
results is strongly affected by marginal terms, so one would like to improve
the analytical formula to take their impact into account [57]. The crosscap
state used here is constructed from a certain type of Bell states. For
integrable spin chains, other types of crosscap states have been studied [47].
Do they have clear counterparts in CFTs? An extension of our method to include
multi-component Luttinger liquids, such as the fermionic Hubbard model and
SU(3) spin chains, is certainly very desirable. We hope that interesting
results on these topics will be reported in the future.

Acknowledgments — We are grateful to Meng Cheng for stimulating discussions.
B.Y.T. and Y.H.W. are supported by NNSF of China under Grant No. 12174130.
Y.S.Z. is supported by the Sino-German (CSC-DAAD) Postdoc Scholarship Program.
W.T. is supported by the Research Foundation Flanders (FWO) via grant
GOE1520N. L.W. is supported by the NNSF of China under Grants No. T2225018 and
No. 92270107.

## References

  * Giamarchi [2003] T. Giamarchi,  _Quantum Physics in One Dimension_ (Oxford University Press, Oxford, 2003).
  * Tomonaga [1950] S.-i. Tomonaga, Prog. Theor. Phys. 5, 544 (1950).
  * Luttinger [1963] J. M. Luttinger, J. Math. Phys. 4, 1154 (1963).
  * Mattis [1974] D. C. Mattis, J. Math. Phys. 15, 609 (1974).
  * Luther and Peschel [1974] A. Luther and I. Peschel, Phys. Rev. B 9, 2911 (1974).
  * Haldane [1981] F. D. M. Haldane, Phys. Rev. Lett. 47, 1840 (1981).
  * von Delft and Schoeller [1998] J. von Delft and H. Schoeller, Ann. Phys. (Leipzig) 7, 225 (1998).
  * Yacoby  _et al._ [1996] A. Yacoby, H. L. Stormer, N. S. Wingreen, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, Phys. Rev. Lett. 77, 4612 (1996).
  * Schwartz  _et al._ [1998] A. Schwartz, M. Dressel, G. Grüner, V. Vescoli, L. Degiorgi, and T. Giamarchi, Phys. Rev. B 58, 1261 (1998).
  * Bockrath  _et al._ [1999] M. Bockrath, D. H. Cobden, J. Lu, A. G. Rinzler, R. E. Smalley, L. Balents, and P. L. McEuen, Nature 397, 598 (1999).
  * Yao  _et al._ [1999] Z. Yao, H. W. C. Postma, L. Balents, and C. Dekker, Nature 402, 273 (1999).
  * Levy  _et al._ [2006] E. Levy, A. Tsukernik, M. Karpovski, A. Palevski, B. Dwir, E. Pelucchi, A. Rudra, E. Kapon, and Y. Oreg, Phys. Rev. Lett. 97, 196802 (2006).
  * Kono  _et al._ [2015] Y. Kono, T. Sakakibara, C. P. Aoyama, C. Hotta, M. M. Turnbull, C. P. Landee, and Y. Takano, Phys. Rev. Lett. 114, 037202 (2015).
  * Hofferberth  _et al._ [2008] S. Hofferberth, I. Lesanovsky, T. Schumm, A. Imambekov, V. Gritsev, E. Demler, and J. Schmiedmayer, Nat. Phys. 4, 489 (2008).
  * Haller  _et al._ [2010] E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsöllner, M. Gustavsson, M. Dalmonte, G. Pupillo, and H.-C. Nägerl, Nature 466, 597 (2010).
  * Yang  _et al._ [2017] B. Yang, Y.-Y. Chen, Y.-G. Zheng, H. Sun, H.-N. Dai, X.-W. Guan, Z.-S. Yuan, and J.-W. Pan, Phys. Rev. Lett. 119, 165701 (2017).
  * Vijayan  _et al._ [2020] J. Vijayan, P. Sompet, G. Salomon, J. Koepsell, S. Hirthe, A. Bohrdt, F. Grusdt, I. Bloch, and C. Gross, Science 367, 186 (2020).
  * Wen [1995] X.-G. Wen, Adv. Phys. 44, 405 (1995).
  * Chang [2003] A. M. Chang, Rev. Mod. Phys. 75, 1449 (2003).
  * Wu  _et al._ [2006] C. Wu, B. A. Bernevig, and S.-C. Zhang, Phys. Rev. Lett. 96, 106401 (2006).
  * Li  _et al._ [2015a] T. Li, P. Wang, H. Fu, L. Du, K. A. Schreiber, X. Mu, X. Liu, G. Sullivan, G. A. Csáthy, X. Lin, and R.-R. Du, Phys. Rev. Lett. 115, 136804 (2015a).
  * Alcaraz  _et al._ [1987] F. C. Alcaraz, M. N. Barber, and M. T. Batchelor, Phys. Rev. Lett. 58, 771 (1987).
  * Song  _et al._ [2010] H. F. Song, S. Rachel, and K. Le Hur, Phys. Rev. B 82, 012405 (2010).
  * Nishimoto [2011] S. Nishimoto, Phys. Rev. B 84, 195108 (2011).
  * [25] A. M. Läuchli,  arXiv:1303.0741 .
  * Vanderstraeten  _et al._ [2019] L. Vanderstraeten, B. Vanhecke, A. M. Läuchli, and F. Verstraete, Phys. Rev. E 100, 062136 (2019).
  * Herdman and Del Maestro [2015] C. M. Herdman and A. Del Maestro, Phys. Rev. B 91, 184507 (2015).
  * Barghathi  _et al._ [2017] H. Barghathi, E. Casiano-Diaz, and A. Del Maestro, J. Stat. Mech. 2017, 083108 (2017).
  * Tu [2017] H.-H. Tu, Phys. Rev. Lett. 119, 261603 (2017).
  * Tang  _et al._ [2017] W. Tang, L. Chen, W. Li, X. C. Xie, H.-H. Tu, and L. Wang, Phys. Rev. B 96, 115136 (2017).
  * Chen  _et al._ [2017] L. Chen, H.-X. Wang, L. Wang, and W. Li, Phys. Rev. B 96, 174429 (2017).
  * Wang  _et al._ [2018] H.-X. Wang, L. Chen, H. Lin, and W. Li, Phys. Rev. B 97, 220407 (2018).
  * Tang  _et al._ [2019] W. Tang, X. C. Xie, L. Wang, and H.-H. Tu, Phys. Rev. B 99, 115105 (2019).
  * Li  _et al._ [2020] Z.-Q. Li, L.-P. Yang, Z. Y. Xie, H.-H. Tu, H.-J. Liao, and T. Xiang, Phys. Rev. E 101, 060105 (2020).
  * Tang  _et al._ [2020] W. Tang, H.-H. Tu, and L. Wang, Phys. Rev. Lett. 125, 170604 (2020).
  * García-Compeán and Quiroz [2021] H. García-Compeán and N. Quiroz, Euro. Phys. J. Plus 136, 881 (2021).
  * Chen  _et al._ [2020] Y. Chen, K. Ji, Z. Y. Xie, and J. F. Yu, Phys. Rev. B 101, 165123 (2020).
  * Li and Yang [2021] H. Li and L.-P. Yang, Phys. Rev. E 104, 024118 (2021).
  * Vanhove  _et al._ [2022] R. Vanhove, L. Lootens, H.-H. Tu, and F. Verstraete, J. Phys. A: Math. Theor. 55, 235002 (2022).
  * Zhang  _et al._ [2023] Y. Zhang, A. Hulsch, H.-C. Zhang, W. Tang, L. Wang, and H.-H. Tu, Phys. Rev. Lett. 130, 151602 (2023).
  * [41] H. Shimizu and A. Ueda,  arXiv:2402.15507 .
  * Ishibashi [1989] N. Ishibashi, Mod. Phys. Lett. A 04, 251 (1989).
  * Fioravanti  _et al._ [1994] D. Fioravanti, G. Pradisi, and A. Sagnotti, Phys. Lett. B 321, 349 (1994).
  * Pradisi  _et al._ [1996] G. Pradisi, A. Sagnotti, and Y. S. Stanev, Phys. Lett. B 381, 97 (1996).
  * Fuchs  _et al._ [2000] J. Fuchs, L. Huiszoon, A. Schellekens, C. Schweigert, and J. Walcher, Phys. Lett. B 495, 427 (2000).
  * Caetano and Komatsu [2022] J. Caetano and S. Komatsu, J. Stat. Phys. 187, 30 (2022).
  * [47] C. Ekman,  arXiv:2207.12354 .
  * Gombor [2022] T. Gombor, J. High Energy Phys. 10, 096 (2022).
  * Gombor [2023] T. Gombor, J. High Energy Phys. 03, 146 (2023).
  * He and Jiang [2023] M. He and Y. Jiang, J. High Energy Phys. 08, 079 (2023).
  * Cheng and Seiberg [2023] M. Cheng and N. Seiberg, SciPost Phys. 15, 051 (2023).
  * Blumenhagen and Plauschinn [2009] R. Blumenhagen and E. Plauschinn,  _Introduction to Conformal Field Theory_ (Springer Berlin, Heidelberg, 2009).
  * Brunner and Hori [2004] I. Brunner and K. Hori, J. High Energy Phys. 2004, 023 (2004).
  * Cho  _et al._ [2015] G. Y. Cho, C.-T. Hsieh, T. Morimoto, and S. Ryu, Phys. Rev. B 91, 195142 (2015).
  * Francesco  _et al._ [1997] P. D. Francesco, P. Mathieu, and D. Sénéchal,  _Conformal Field Theory_ (Springer-Verlag, New York, 1997).
  * [56] See the Appendices for details about the derivation of crosscap states and the analytical calculation of the crosscap state overlap for certain Jastrow-type eigenstates of the spin-1/2 XY and Haldane-Shastry chains .
  * [57] Y. Zhang et al., work in progress .
  * Haldane [1988] F. D. M. Haldane, Phys. Rev. Lett. 60, 635 (1988).
  * Shastry [1988] B. S. Shastry, Phys. Rev. Lett. 60, 639 (1988).
  * Kuramoto and Kato [2009] Y. Kuramoto and Y. Kato,  _Dynamics of One-Dimensional Quantum Systems: Inverse-Square Interaction Models_ (Cambridge University Press, 2009).
  * White [1992] S. R. White, Phys. Rev. Lett. 69, 2863 (1992).
  * Östlund and Rommer [1995] S. Östlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (1995).
  * Schollwöck [2011] U. Schollwöck, Ann. Phys. 326, 96 (2011).
  * Affleck  _et al._ [1989] I. Affleck, D. Gepner, H. J. Schulz, and T. Ziman, J. Phys. A: Math. Gen. 22, 511 (1989).
  * Affleck [1986] I. Affleck, Nucl. Phys. B 265, 409 (1986).
  * Elitzur  _et al._ [1979] S. Elitzur, R. B. Pearson, and J. Shigemitsu, Phys. Rev. D 19, 3698 (1979).
  * Fröhlich and Spencer [1981] J. Fröhlich and T. Spencer, Commun. Math. Phys. 81, 527 (1981).
  * Ortiz  _et al._ [2012] G. Ortiz, E. Cobanera, and Z. Nussinov, Nucl. Phys. B 854, 780 (2012).
  * Li  _et al._ [2015b] W. Li, S. Yang, H.-H. Tu, and M. Cheng, Phys. Rev. B 91, 115133 (2015b).
  * Sun  _et al._ [2019] G. Sun, T. Vekua, E. Cobanera, and G. Ortiz, Phys. Rev. B 100, 094428 (2019).
  * [71] D. Berenstein and P. N. T. Lloyd,  arXiv:2311.00057 .
  * Wiegmann [1978] P. B. Wiegmann, J. Phys. C 11, 1583 (1978).
  * Matsuo and Nomura [2006] H. Matsuo and K. Nomura, J. Phys. A 39, 2953 (2006).
  * Cirac and Sierra [2010] J. I. Cirac and G. Sierra, Phys. Rev. B 81, 104431 (2010).
  * Patil  _et al._ [2017] P. Patil, Y. Tang, E. Katz, and A. W. Sandvik, Phys. Rev. B 96, 045140 (2017).
  * Zhang  _et al._ [2022] H.-C. Zhang, Y.-H. Wu, T. Xiang, and H.-H. Tu, Nucl. Phys. B 976, 115712 (2022).

##  Appendix A Appendix A: Tomonaga-Luttinger liquid and compactified boson
conformal field theory

The bosonized version of the (single-component) Tomonaga-Luttinger liquid is
formulated as a free massless compactified boson field theory in
(1+1)-dimensional spacetime. This section is devoted to a brief review of the
operator formalism of the compactified boson field theory, and we mostly
follow the notation used in Refs. [55, 76]. We assume that the spatial
direction is compact with length $L$, and the system admits periodic boundary
condition along this direction. Therefore, the spacetime manifold (or “world
sheet”), parametrized by coordinates $(x~{}\mathrm{mod}~{}L,t)$, is
effectively a cylinder of circumference $L$. The Hamiltonian reads

$\displaystyle
H=\frac{v}{8\pi}\int^{L}_{0}\mathrm{d}x\,\left[(\partial_{x}\hat{\varphi})^{2}+(\partial_{x}\hat{\theta})^{2}\right],$
(A1)

where $\hat{\varphi}$ and $\hat{\theta}$ are compactified boson fields, and
$v$ is the “speed of light” in this theory. Apart from $v$, the theory is
characterized by the compactification radius $R$ of $\hat{\varphi}$, which is
related to the Luttinger parameter via $K=R^{2}/4$. The field $\hat{\theta}$
is dual to $\hat{\varphi}$ in the sense that
$\partial_{x}\hat{\theta}(x,t)=v^{-1}\partial_{t}\hat{\varphi}(x,t)$.

On this manifold, the mode expansion of the $\hat{\varphi}$-field is given by

$\displaystyle\hat{\varphi}(x,t)=\varphi_{0}+\frac{4\pi}{L}\left(\pi_{0}vt+\tilde{\pi}_{0}x\right)+i\sum_{k\neq
0}\frac{1}{k}\left(a_{k}e^{2\pi ik\frac{x-vt}{L}}+\bar{a}_{k}e^{-2\pi
ik\frac{x+vt}{L}}\right),$ (A2)

where the zero-mode operators satisfy canonical commutation relations (we have
formally introduced $\tilde{\varphi}_{0}$ as the operator canonically
conjugate to $\tilde{\pi}_{0}$),

$\displaystyle[\varphi_{0},\pi_{0}]=[\tilde{\varphi}_{0},\tilde{\pi}_{0}]=i,$
(A3)

whilst the other modes generate the Heisenberg algebra

$\displaystyle[a_{k},a_{k^{\prime}}]=[\bar{a}_{k},\bar{a}_{k^{\prime}}]=k\delta_{k+k^{\prime}},\quad[a_{k},\bar{a}_{k^{\prime}}]=0.$
(A4)

The fact the $\hat{\varphi}$-field is compactified with radius $R$ means that
the field configurations $\hat{\varphi}$ and $\hat{\varphi}+2\pi R$ are
identified with each other. This implies that the field can have non-trivial
winding around the cylinder, i.e.,
$\hat{\varphi}(x+L,t)\equiv\hat{\varphi}(x,t)-2\pi mR$ with $m\in\mathbb{Z}$;
Eq. (A2) then shows that the operator $\tilde{\pi}_{0}$ has eigenvalues
$-mR/2$.

Before proceeding, let us remark on the physical significance of the zero-mode
operators $\varphi_{0}$ and $\pi_{0}$. By definition, the “canonical momentum”
(density) corresponding to the $\hat{\varphi}$-field is

$\displaystyle\Pi(x,t)\equiv\frac{1}{4\pi
v}\partial_{t}\hat{\varphi}(x,t)=\frac{\pi_{0}}{L}+\frac{1}{2L}\sum_{k\neq
0}\left(a_{k}e^{2\pi ik\frac{x-vt}{L}}+\bar{a}_{k}e^{-2\pi
ik\frac{x+vt}{L}}\right).$ (A5)

Indeed, the canonical equal-time commutation relation,
$[\hat{\varphi}(x,0),\Pi(x^{\prime},0)]=i\delta(x-x^{\prime})$, can readily be
verified. The resulting $\mathrm{U}(1)$ conserved charge,
$\int_{0}^{L}\mathrm{d}x~{}\Pi(x,t)$, is precisely $\pi_{0}$; this indicates
that the latter admits an interpretation as the “centre-of-mass momentum” of
the field. In particular, the single-valuedness of the operator $e^{2\pi
iR\pi_{0}}$ translating around the compactified dimension (in the space of
field configurations; not to be confused with the worldsheet) enforces that
the eigenvalues of $\pi_{0}$ are quantized to be $n/R$ with $n\in\mathbb{Z}$.
As $\varphi_{0}$ is conjugate to $\pi_{0}$, the former is naturally
interpreted as the “centre-of-mass coordinate” operator of the field.

Next, we recombine these zero-mode operators as follows:

$\displaystyle Q$
$\displaystyle=\frac{1}{2}(\varphi_{0}-\tilde{\varphi}_{0}),\quad
P=\pi_{0}-\tilde{\pi}_{0};$ $\displaystyle\bar{Q}$
$\displaystyle=\frac{1}{2}(\varphi_{0}+\tilde{\varphi}_{0}),\quad\bar{P}=\pi_{0}+\tilde{\pi}_{0}.$
(A6)

It is easily verified that these new operators still obey the canonical
commutation relations,

$\displaystyle[Q,P]=[\bar{Q},\bar{P}]=i.$ (A7)

In terms of this new set of zero-mode operators as well as the “light-cone”
coordinates $z\equiv-(x-vt)/L$ and $\bar{z}\equiv-(x+vt)/L$, one arrives at
the chiral decomposition of the $\hat{\varphi}$-field,
$\hat{\varphi}(x,t)=\phi(z)+\bar{\phi}(\bar{z})$, where

$\displaystyle\phi(z)$ $\displaystyle=Q+2\pi Pz+i\sum_{k\neq
0}\frac{1}{k}a_{k}e^{-2\pi ikz},$ $\displaystyle\bar{\phi}(\bar{z})$
$\displaystyle=\bar{Q}-2\pi\bar{P}\bar{z}+i\sum_{k\neq
0}\frac{1}{k}\bar{a}_{k}e^{2\pi ik\bar{z}}.$ (A8)

Using the relations $\partial_{x}=-L^{-1}(\partial+\bar{\partial})$ and
$\partial_{t}=vL^{-1}(\partial-\bar{\partial})$ with
$\partial\equiv\partial_{z}$ and $\bar{\partial}\equiv\partial_{\bar{z}}$, one
finds that $\hat{\theta}(x,t)=\bar{\phi}(\bar{z})-\phi(z)$, we obtain

$\displaystyle\hat{\theta}(x,t)=\tilde{\varphi}_{0}+\frac{4\pi}{L}\left(\pi_{0}x+\tilde{\pi}_{0}vt\right)-i\sum_{k\neq
0}\frac{1}{k}\left(a_{k}e^{2\pi ik\frac{x-vt}{L}}-\bar{a}_{k}e^{-2\pi
ik\frac{x+vt}{L}}\right).$ (A9)

As $\hat{\theta}(x+L,t)=\hat{\theta}(x,t)+4\pi\pi_{0}$, the quantization of
the eigenvalues of $\pi_{0}$ implies that the $\hat{\theta}$-field is also
compactified, for which the radius is $2/R$.

Finally, substituting the mode expansions (A2) and (A9) into the expression
(A1) of the Hamiltonian, and making use of (A4) as well as the zeta-function
regularization (“$1+2+3+\cdots=-1/12$”), one finds after a straightforward
computation that

$\displaystyle H=\frac{2\pi v}{L}\left(L_{0}+\bar{L}_{0}-\frac{1}{12}\right),$
(A10)

where

$\displaystyle L_{0}$
$\displaystyle=\frac{1}{2}a_{0}^{2}+\sum_{k=1}^{\infty}a_{-k}a_{k},$
$\displaystyle\bar{L}_{0}$
$\displaystyle=\frac{1}{2}\bar{a}_{0}^{2}+\sum_{k=1}^{\infty}\bar{a}_{-k}\bar{a}_{k},$
(A11)

with $a_{0}\equiv P,~{}\bar{a}_{0}\equiv\bar{P}$, are the zeroth Virasoro
generators. The Virasoro primary states (which are normalized by assumption),
$|n,m\rangle$, are labelled by the eigenvalues of $a_{0}$ and $\bar{a}_{0}$ as

$\displaystyle a_{0}|n,m\rangle$
$\displaystyle=\left(\frac{n}{R}+\frac{1}{2}mR\right)|n,m\rangle,$
$\displaystyle\bar{a}_{0}|n,m\rangle$
$\displaystyle=\left(\frac{n}{R}-\frac{1}{2}mR\right)|n,m\rangle.$ (A12)

By definition, they are annihilated by the modes $a_{k}$ and $\bar{a}_{k}$
with $k\in\mathbb{Z}^{+}$; other states in the corresponding conformal tower,
which are orthogonal to each other and properly normalized, are constructed by
acting modes with negative indices on the primary states:

$\displaystyle|n,m;\\{n_{k}\\},\\{\bar{n}_{k}\\}\rangle=\prod_{k=1}^{\infty}\left[\frac{1}{\sqrt{n_{k}!\bar{n}_{k}!}}\left(\frac{a_{-k}}{\sqrt{k}}\right)^{n_{k}}\left(\frac{\bar{a}_{-k}}{\sqrt{k}}\right)^{\bar{n}_{k}}\right]|n,m\rangle,\quad
n_{k},\bar{n}_{k}\in\mathbb{Z}^{+}_{0}\,.$ (A13)

One obviously has $|n,m;\\{0\\},\\{0\\}\rangle\equiv|n,m\rangle$. This basis
trivializes the computation of partition functions, as $L_{0}$ and
$\bar{L}_{0}$ act diagonally:

$\displaystyle L_{0}|n,m;\\{n_{k}\\},\\{\bar{n}_{k}\\}\rangle$
$\displaystyle=\left[\frac{1}{2}\left(\frac{n}{R}+\frac{1}{2}mR\right)^{2}+\sum_{k=1}^{\infty}kn_{k}\right]|n,m;\\{n_{k}\\},\\{\bar{n}_{k}\\}\rangle\,,$
$\displaystyle\bar{L}_{0}|n,m;\\{n_{k}\\},\\{\bar{n}_{k}\\}\rangle$
$\displaystyle=\left[\frac{1}{2}\left(\frac{n}{R}-\frac{1}{2}mR\right)^{2}+\sum_{k=1}^{\infty}k\bar{n}_{k}\right]|n,m;\\{n_{k}\\},\\{\bar{n}_{k}\\}\rangle\,.$
(A14)

##  Appendix B Appendix B: Determining crosscap states from constraints

The mode expansion form of the crosscap states $|\mathcal{C}_{\pm}\rangle$ is
given in Eq. (9) of the main text. Below we derive it in two steps:

i) Substitute the mode expansions of the compactified boson fields (A8) into
the constraints

$\displaystyle
e^{i[\hat{\theta}(x)\mp\hat{\theta}(x+L/2)]/R^{\prime}}|\mathcal{C}_{\pm}\rangle$
$\displaystyle=|\mathcal{C}_{\pm}\rangle\,,$ $\displaystyle
e^{i[\hat{\varphi}(x)\pm\hat{\varphi}(x+L/2)]/R}|\mathcal{C}_{\pm}\rangle$
$\displaystyle=|\mathcal{C}_{\pm}\rangle$ (A15)

to determine the crosscap states $|\mathcal{C}_{\pm}\rangle$ up to overall
factors $\kappa_{\pm}$.

ii) Compare the cylinder partition function with two crosscap boundaries
(“tree-channel”),
$Z^{\mathcal{C}_{\pm}}(L,\beta)=\langle\mathcal{C}_{\pm}|e^{-\beta
H}|\mathcal{C}_{\pm}\rangle$, with the Klein bottle partition function (“loop
channel”),
$Z^{\mathcal{K}_{\pm}}(2\beta,L/2)=\mathrm{Tr}(\Omega_{\pm}e^{-\beta H})$ to
fix the factors $\kappa_{\pm}$.

Step i):

Consider the mode expansions of the compactified boson fields in Eqs. (A2) and
(A9) at $t=0$:

$\displaystyle\hat{\theta}(x)=\tilde{\varphi}_{0}+\frac{4\pi
x}{L}\pi_{0}-i\sum_{k\neq 0}\frac{1}{k}\left(a_{k}e^{2\pi
ik\frac{x}{L}}-\bar{a}_{k}e^{-2\pi ik\frac{x}{L}}\right),$
$\displaystyle\hat{\varphi}(x)=\varphi_{0}+\frac{4\pi
x}{L}\tilde{\pi}_{0}+i\sum_{k\neq 0}\frac{1}{k}\left(a_{k}e^{2\pi
ik\frac{x}{L}}+\bar{a}_{k}e^{-2\pi ik\frac{x}{L}}\right).$ (A16)

To analyze the constraints in Eq. (A15), we consider

$\displaystyle\hat{\theta}(x)-\hat{\theta}(x+L/2)$
$\displaystyle=-2\pi\pi_{0}-2i\sum_{k\neq
0,\,k\;\mathrm{odd}}\frac{1}{k}\left(a_{k}e^{2\pi
ik\frac{x}{L}}-\bar{a}_{k}e^{-2\pi ik\frac{x}{L}}\right),$
$\displaystyle\hat{\varphi}(x)+\hat{\varphi}(x+L/2)$
$\displaystyle=2\varphi_{0}+4\pi\tilde{\pi}_{0}\left(\frac{2x}{L}+\frac{1}{2}\right)+2i\sum_{k\neq
0,\,k\;\mathrm{even}}\frac{1}{k}\left(a_{k}e^{2\pi
ik\frac{x}{L}}+\bar{a}_{k}e^{-2\pi ik\frac{x}{L}}\right),$ (A17)

and

$\displaystyle\hat{\theta}(x)+\hat{\theta}(x+L/2)$
$\displaystyle=2\tilde{\varphi}_{0}+4\pi\pi_{0}\left(\frac{2x}{L}+\frac{1}{2}\right)-2i\sum_{k\neq
0,\,k\;\mathrm{even}}\frac{1}{k}\left(a_{k}e^{2\pi
ik\frac{x}{L}}-\bar{a}_{k}e^{-2\pi ik\frac{x}{L}}\right),$
$\displaystyle\hat{\varphi}(x)-\hat{\varphi}(x+L/2)$
$\displaystyle=-2\pi\tilde{\pi}_{0}+2i\sum_{k\neq
0,\,k\;\mathrm{odd}}\frac{1}{k}\left(a_{k}e^{2\pi
ik\frac{x}{L}}+\bar{a}_{k}e^{-2\pi ik\frac{x}{L}}\right).$ (A18)

By inserting these into Eq. (A15) and noting that the constraints (A15) are
valid for all $x\in[0,L]$, it is then obvious that the oscillatory part should
satisfy

$\displaystyle\left[a_{k}\mp(-1)^{k}\bar{a}_{-k}\right]|\mathcal{C}_{\pm}\rangle$
$\displaystyle=0,\quad k\neq 0\,,$ (A19)

which fixes the form of $|\mathcal{C}_{\pm}\rangle$ to be

$\displaystyle|\mathcal{C}_{\pm}\rangle$
$\displaystyle\propto\exp\left[\pm\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}a_{-k}\bar{a}_{-k}\right]|\mathcal{C}^{0}_{\pm}\rangle\,.$
(A20)

Here
$|\mathcal{C}^{0}_{\pm}\rangle=\sum_{n,m\in\mathbb{Z}}c^{\pm}_{n,m}|n,m\rangle$
encode the zero mode information of the crosscap states, where $c^{\pm}_{n,m}$
are the superposition coefficients that should be determined by using the zero
mode constraints from Eq. (A15):

$\displaystyle e^{-2\pi
i\pi_{0}/R^{\prime}}|\mathcal{C}^{0}_{+}\rangle=|\mathcal{C}^{0}_{+}\rangle\,,$
$\displaystyle e^{2i\varphi_{0}/R}e^{4\pi
i\left(\frac{2x}{L}+\frac{1}{2}\right)\tilde{\pi}_{0}/R}|\mathcal{C}^{0}_{+}\rangle=|\mathcal{C}^{0}_{+}\rangle\,,$
(A21)

and

$\displaystyle e^{2i\tilde{\varphi}_{0}/R^{\prime}}e^{4\pi
i\left(\frac{2x}{L}+\frac{1}{2}\right)\pi_{0}/R^{\prime}}|\mathcal{C}^{0}_{-}\rangle=|\mathcal{C}^{0}_{-}\rangle\,,$
$\displaystyle e^{-2\pi
i\tilde{\pi}_{0}/R}|\mathcal{C}^{0}_{-}\rangle=|\mathcal{C}^{0}_{-}\rangle\,.$
(A22)

For
$|\mathcal{C}^{0}_{+}\rangle=\sum_{n,m\in\mathbb{Z}}c^{+}_{n,m}|n,m\rangle$,
the first equation in Eq. (A21) requires $c^{+}_{2n+1,m}=0$, since $e^{-2\pi
i\pi_{0}/R^{\prime}}|n,m\rangle=(-1)^{n}|n,m\rangle$ (note that
$\pi_{0}|n,m\rangle=\tfrac{n}{R}|n,m\rangle$ and $R^{\prime}=2/R$); the second
equation in Eq. (A21) requires $c^{+}_{n,m\neq 0}=0$ and
$c^{+}_{2n,0}=c^{+}_{0,0}\;\forall n\in\mathbb{Z}$, since $e^{4\pi
i(\frac{2x}{L}+\frac{1}{2})\tilde{\pi}_{0}/R}|n,m\rangle=e^{2\pi
i(\frac{2x}{L}+\frac{1}{2})m}|n,m\rangle$ and
$e^{2i\varphi_{0}/R}|n,m\rangle=|n+2,m\rangle$ (note that
$\tilde{\pi}_{0}|n,m\rangle=-\tfrac{1}{2}mR|n,m\rangle$ and
$[\varphi_{0},\pi_{0}]=i$). Thus, we obtain

$\displaystyle|\mathcal{C}^{0}_{+}\rangle\propto\sum_{n\in\mathbb{Z}}|2n,0\rangle\,.$
(A23)

The determination of $|\mathcal{C}^{0}_{-}\rangle$ is similar and will not be
repeated. It is given by

$\displaystyle|\mathcal{C}^{0}_{-}\rangle\propto\sum_{m\in\mathbb{Z}}|0,2m\rangle\,.$
(A24)

Altogether, the constraints in Eq. (A15) determine the mode expansion form of
the crosscap states

$\displaystyle|\mathcal{C}_{+}\rangle$
$\displaystyle=\kappa_{+}\exp\left[\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}a_{-k}\bar{a}_{-k}\right]\sum_{n\in\mathbb{Z}}|2n,0\rangle\,,$
$\displaystyle|\mathcal{C}_{-}\rangle$
$\displaystyle=\kappa_{-}\exp\left[-\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}a_{-k}\bar{a}_{-k}\right]\sum_{m\in\mathbb{Z}}|0,2m\rangle$
(A25)

up to overall factors $\kappa_{\pm}$. Without loss of generality, we choose
$\kappa_{\pm}\in\mathbb{R}$ and $\kappa_{\pm}>0$ below.

Step ii):

To fix the factors $\kappa_{\pm}$ in Eq. (A25), we first calculate the (“tree
channel”) cylinder partition function with crosscap boundary states

$\displaystyle
Z^{\mathcal{C}_{\pm}}(L,\beta)=\langle\mathcal{C}_{\pm}|e^{-\beta
H}|\mathcal{C}_{\pm}\rangle\,,$ (A26)

where the Hamiltonian $H$ is given in Eq. (A10). The crosscap states in Eq.
(A25) can be expanded in the orthonormal basis states in Eq. (A13) as

$\displaystyle|\mathcal{C}_{+}\rangle$
$\displaystyle=\kappa_{+}\prod_{k=1}^{\infty}\sum_{n_{k}=0}^{\infty}\frac{1}{n_{k}!}\left[\frac{(-1)^{k}a_{-k}\bar{a}_{-k}}{k}\right]^{n_{k}}\sum_{n\in\mathbb{Z}}|2n,0\rangle$
$\displaystyle=\kappa_{+}\sum_{n\in\mathbb{Z}}\sum_{n_{1},n_{2},\ldots=0}^{\infty}(-1)^{\sum\limits_{k=1}^{\infty}kn_{k}}|2n,0;\\{n_{k}\\},\\{n_{k}\\}\rangle\,,$
(A27)

and

$\displaystyle|\mathcal{C}_{-}\rangle$
$\displaystyle=\kappa_{-}\prod_{k=1}^{\infty}\sum_{n_{k}=0}^{\infty}\frac{1}{n_{k}!}\left[\frac{(-1)^{k+1}a_{-k}\bar{a}_{-k}}{k}\right]^{n_{k}}\sum_{m\in\mathbb{Z}}|0,2m\rangle$
$\displaystyle=\kappa_{-}\sum_{m\in\mathbb{Z}}\sum_{n_{1},n_{2},\ldots=0}^{\infty}(-1)^{\sum\limits_{k=1}^{\infty}(k+1)n_{k}}|0,2m;\\{n_{k}\\};\\{n_{k}\\}\rangle\,.$
(A28)

As $|n,m;\\{n_{k}\\},\\{\bar{n}_{k}\\}\rangle$ is the Hamiltonian eigenbasis
[Eq. (A14)], $Z^{\mathcal{C}_{\pm}}(L,\beta)$ are easy to calculate

$\displaystyle Z^{\mathcal{C}_{+}}(L,\beta)$
$\displaystyle=\kappa_{+}^{2}\sum_{n\in\mathbb{Z}}\sum_{n_{1},n_{2},\ldots=0}^{\infty}\langle
2n,0;\\{n_{k}\\},\\{n_{k}\\}|q^{L_{0}+\bar{L}_{0}-1/12}|2n,0;\\{n_{k}\\},\\{n_{k}\\}\rangle$
$\displaystyle=\kappa_{+}^{2}\;q^{-1/12}\sum_{n\in\mathbb{Z}}q^{{R^{\prime}}^{2}n^{2}}\sum_{n_{1},n_{2},\ldots=0}^{\infty}q^{2\sum_{k=1}^{\infty}kn_{k}}$
$\displaystyle=\kappa_{+}^{2}\sum_{n\in\mathbb{Z}}q^{{R^{\prime}}^{2}n^{2}}\cdot
q^{-1/12}\prod_{k=1}^{\infty}\frac{1}{1-q^{2k}}$
$\displaystyle=\kappa_{+}^{2}\,\frac{\vartheta_{3}(2{R^{\prime}}^{2}\tau)}{\eta(2\tau)}\,,$
(A29)

and similarly

$\displaystyle
Z^{\mathcal{C}_{-}}(L,\beta)=\kappa_{-}^{2}\,\frac{\vartheta_{3}(2{R}^{2}\tau)}{\eta(2\tau)}\,,$
(A30)

where $\tau=iv\beta/L$ is the modular parameter and $q=e^{2\pi i\tau}$. The
results are expressed in terms of the Dedekind eta function
$\eta(\tau)=q^{1/24}\prod_{k=1}^{\infty}(1-q^{k})$ and one of the standard
Jacobi’s theta functions
$\vartheta_{3}(\tau)=\sum_{k\in\mathbb{Z}}q^{k^{2}/2}$. Under the modular $S$
transformation ($\tau\rightarrow-1/\tau$), Dedekind eta function and Jacobi’s
theta function transform as
$\sqrt{-i\tau}\vartheta_{3}(\tau)=\vartheta_{3}(-1/\tau)$ and
$\sqrt{-i\tau}\eta(\tau)=\eta(-1/\tau)$, respectively. This allows us to
rewrite $Z^{\mathcal{C}_{\pm}}$ as

$\displaystyle Z^{\mathcal{C}_{+}}(L,\beta)$
$\displaystyle=\kappa_{+}^{2}\,\frac{1}{R^{\prime}}\,\frac{\vartheta_{3}(-1/2{R^{\prime}}^{2}\tau)}{\eta(-1/2\tau)}\,,$
$\displaystyle Z^{\mathcal{C}_{-}}(L,\beta)$
$\displaystyle=\kappa_{-}^{2}\,\frac{1}{R}\,\frac{\vartheta_{3}(-1/2{R^{\prime}}^{2}\tau)}{\eta(-1/2\tau)}\,.$
(A31)

Now we turn to the (“loop channel”) Klein bottle partition function [33]

$\displaystyle
Z^{\mathcal{K}_{\pm}}(2\beta,L/2)=\mathrm{Tr}\left(\Omega_{\pm}e^{-\tilde{H}L/2}\right),$
(A32)

where $\tilde{H}=\frac{2\pi}{\tilde{L}}(L_{0}+\bar{L}_{0}-\frac{1}{12})$ is
the compactified boson Hamiltonian [c.f. Eqs. (A1) and (A10)] defined on a
circle (in the imaginary time direction) with circumference
$\tilde{L}=2v\beta$, which generates evolution along the spatial direction (it
can be viewed as the generator of the quantum transfer matrix). The action of
the “reflection operator” $\Omega_{\pm}$ on oscillatory modes is given by
$\Omega^{-1}_{\pm}a_{k}\Omega_{\pm}=\mp\bar{a}_{k}$ and
$\Omega^{-1}_{\pm}\bar{a}_{k}\Omega_{\pm}=\mp a_{k}$ for $k\neq 0$. Applying
them to Virasoro primary states gives

$\displaystyle\Omega_{+}|n,m\rangle$ $\displaystyle=|-n,m\rangle\,,$
$\displaystyle\Omega_{-}|n,m\rangle$ $\displaystyle=|n,-m\rangle\,.$ (A33)

Evaluating the Klein bottle partition function $Z^{\mathcal{K}_{\pm}}$ in the
Hamiltonian eigenbasis [Eq. (A14)] and noticing that only “left-right-
symmetric” states $|0,m;\\{n_{k}\\},\\{n_{k}\\}\rangle$
($|n,0;\\{n_{k}\\},\\{n_{k}\\}\rangle$) contribute to $Z^{\mathcal{K}_{+}}$
($Z^{\mathcal{K}_{-}}$), we obtain

$\displaystyle Z^{\mathcal{K}_{+}}(2\beta,L/2)$
$\displaystyle=\sum_{m\in\mathbb{Z}}\sum_{n_{1},n_{2},\ldots=0}^{\infty}\langle
0,m;\\{n_{k}\\},\\{n_{k}\\}|\tilde{q}^{2L_{0}-1/12}|0,m;\\{n_{k}\\},\\{n_{k}\\}\rangle$
$\displaystyle=\tilde{q}^{-1/12}\sum_{m\in\mathbb{Z}}\tilde{q}^{m^{2}/{R^{\prime}}^{2}}\sum_{n_{1},n_{2},\ldots=0}^{\infty}\tilde{q}^{2\sum_{k=1}^{\infty}kn_{k}}$
$\displaystyle=\frac{\vartheta_{3}(-1/2{R^{\prime}}^{2}\tau)}{\eta(-1/2\tau)}\,,$
(A34)

and similarly

$\displaystyle Z^{\mathcal{K}_{-}}(2\beta,L/2)$
$\displaystyle=\sum_{n\in\mathbb{Z}}\sum_{n_{1},n_{2},\ldots=0}^{\infty}\langle
n,0;\\{n_{k}\\},\\{n_{k}\\}|\tilde{q}^{2L_{0}-1/12}|n,0;\\{n_{k}\\},\\{n_{k}\\}\rangle$
$\displaystyle=\frac{\vartheta_{3}(-1/2{R}^{2}\tau)}{\eta(-1/2\tau)}$ (A35)

with $\tilde{q}=e^{-\pi L/2v\beta}=e^{-\pi i/2\tau}$.

Considering the spacetime symmetry, the CFT partition function evaluated in
tree and loop channels should be equal (“loop channel-tree channel
equivalence”), so we expect

$\displaystyle
Z^{\mathcal{C}_{\pm}}(L,\beta)=Z^{\mathcal{K}_{\pm}}(2\beta,L/2)\,.$ (A36)

Comparing Eqs. (A31), (A34) and (A35), the factors in the crosscap states
(A25) are fixed to be

$\displaystyle\kappa_{+}$ $\displaystyle=\sqrt{R^{\prime}}\,,$
$\displaystyle\kappa_{-}$ $\displaystyle=\sqrt{R}\,.$ (A37)

##  Appendix C Appendix C: Crosscap state overlap of Jastrow wave functions

In this section, we calculate the overlap between the lattice crosscap state

$\displaystyle|\mathcal{C}_{\mathrm{latt}}\rangle=\prod_{j=1}^{N/2}\left(|\uparrow\rangle_{j}|\uparrow\rangle_{j+N/2}+|\downarrow\rangle_{j}|\downarrow\rangle_{j+N/2}\right)$
(A38)

and the Jastrow wave function

$\displaystyle|\Psi^{\lambda}(M)\rangle=\sum_{1\leq x_{1}<\cdots<x_{M}\leq
N}\Psi^{\lambda}(x_{1},\ldots,x_{M})\sigma_{x_{1}}^{+}\cdots\sigma_{x_{M}}^{+}|\downarrow\cdots\downarrow\rangle$
(A39)

with

$\displaystyle\Psi^{\lambda}(x_{1},\ldots,x_{M})=\mathcal{N}_{N,M,\lambda}\,(-1)^{\sum_{l=1}^{M}x_{l}}\prod_{1\leq
j<k\leq M}\left|\sin\frac{\pi(x_{j}-x_{k})}{N}\right|^{\lambda}$ (A40)

for $\mathrm{mod}(N,4)=0$. The number of hardcore bosons, $M$, is taken to be
even. For $\lambda=1$ ($\lambda=2$), $|\Psi^{\lambda}(M)\rangle$ is an exact
eigenstate of the spin-1/2 XY (Haldane-Shastry) chain.
$\mathcal{N}_{N,M,\lambda}$ ensures normalization of
$|\Psi^{\lambda}(M)\rangle$ and is chosen to be positive.

To calculate the overlap
$\langle\mathcal{C}_{\mathrm{latt}}|\Psi^{\lambda}(M)\rangle$, we first
introduce a unitary
$U=\exp\left[i\tfrac{\pi}{2}\sum_{j:\mathrm{odd}}(\sigma^{z}_{j}+1)\right]$ to
remove the Marshall sign $(-1)^{\sum_{l=1}^{M}x_{l}}$ in
$|\Psi^{\lambda}(M)\rangle$:

$\displaystyle|\tilde{\Psi}^{\lambda}(M)\rangle=U|\Psi^{\lambda}(M)\rangle$
(A41)

with

$\displaystyle\tilde{\Psi}^{\lambda}(x_{1},\ldots,x_{M})$
$\displaystyle=\mathcal{N}_{N,M,\lambda}\,\prod_{1\leq j<k\leq
M}\left|\sin\frac{\pi(x_{j}-x_{k})}{N}\right|^{\lambda}\,$
$\displaystyle=\mathcal{N}_{N,M,\lambda}\cdot 2^{-\lambda
M(M-1)/2}\prod_{1\leq j<k\leq M}\left|z_{x_{j}}-z_{x_{k}}\right|^{\lambda}\,$
$\displaystyle=\tilde{\mathcal{N}}_{N,M,\lambda}\,\prod_{1\leq j<k\leq
M}\left|z_{x_{j}}-z_{x_{k}}\right|^{\lambda}\,,$ (A42)

where $z_{x_{j}}=\exp(i\frac{2\pi}{N}x_{j})$ and
$\tilde{\mathcal{N}}_{N,M,\lambda}=2^{-\lambda
M(M-1)/2}\mathcal{N}_{N,M,\lambda}$.

For $\mathrm{mod}(N,4)=0$, it is obvious that the lattice crosscap state
$|\mathcal{C}_{\mathrm{latt}}\rangle$ in Eq. (A38) is invariant under the
action of $U$,
$U|\mathcal{C}_{\mathrm{latt}}\rangle=|\mathcal{C}_{\mathrm{latt}}\rangle$.
The crosscap state overlap is then rewritten as

$\displaystyle\langle\mathcal{C}_{\mathrm{latt}}|\Psi^{\lambda}(M)\rangle=\langle\mathcal{C}_{\mathrm{latt}}|U^{{\dagger}}U|\Psi^{\lambda}(M)\rangle=\langle\mathcal{C}_{\mathrm{latt}}|\tilde{\Psi}^{\lambda}(M)\rangle\,.$
(A43)

For even $M$, the overlap
$\langle\mathcal{C}_{\mathrm{latt}}|\tilde{\Psi}^{\lambda}(M)\rangle$ can be
simplified as follows:

$\displaystyle\langle\mathcal{C}_{\mathrm{latt}}|\tilde{\Psi}^{\lambda}(M)\rangle$
$\displaystyle=\tilde{\mathcal{N}}_{N,M,\lambda}\sum_{1\leq
x_{1}<\cdots<x_{M/2}\leq
N/2}\Psi^{\lambda}(x_{1},\ldots,x_{M/2},x_{1}+N/2,\ldots,x_{M/2}+N/2)$
$\displaystyle=\tilde{\mathcal{N}}_{N,M,\lambda}\sum_{1\leq
x_{1}<\cdots<x_{M/2}\leq N/2}\,\prod_{1\leq j<k\leq
M/2}\left|(z_{x_{j}}-z_{x_{k}})(z_{x_{j+N/2}}-z_{x_{k+N/2}})\right|^{\lambda}\cdot\prod_{1\leq
j,k\leq M/2}\left|z_{x_{j}}-z_{x_{k+N/2}}\right|^{\lambda}$
$\displaystyle=\tilde{\mathcal{N}}_{N,M,\lambda}\sum_{1\leq
x_{1}<\cdots<x_{M/2}\leq N/2}\,\prod_{1\leq j<k\leq
M/2}\left|z_{x_{j}}-z_{x_{k}}\right|^{2\lambda}\cdot\prod_{1\leq j<k\leq
M/2}\left|z_{x_{j}}+z_{x_{k}}\right|^{2\lambda}\cdot 2^{\lambda M/2}$
$\displaystyle=2^{\lambda M/2}\tilde{\mathcal{N}}_{N,M,\lambda}\sum_{1\leq
x_{1}<\cdots<x_{M/2}\leq N/2}\,\prod_{1\leq j<k\leq
M/2}\left|z^{2}_{x_{j}}-z^{2}_{x_{k}}\right|^{2\lambda}$
$\displaystyle=2^{\lambda
M/2}\tilde{\mathcal{N}}_{N,M,\lambda}\cdot\frac{1}{|\tilde{\mathcal{N}}_{N/2,M/2,\lambda}|^{2}}\,.$
(A44)

For the last step, we have used $\sum_{1\leq x_{1}<\ldots<x_{M/2}\leq
N/2}\,\prod_{1\leq j<k\leq
M/2}|z_{x_{j}}^{2}-z_{x_{k}}^{2}|^{2}=|\tilde{\mathcal{N}}_{N/2,M/2,\lambda}|^{-2}$,
which is the normalization condition of $|\tilde{\Psi}^{\lambda}(M/2)\rangle$
defined on a chain with length $N/2$. Remarkably, the crosscap overlap for the
Jastrow wave function $|\Psi^{\lambda}(M)\rangle$ defined in Eqs. (A39) and
(A40) is almost solely determined by the normalization factor. Although this
interesting result applies to generic $\lambda$, analytical solutions for the
normalization factor $\tilde{\mathcal{N}}_{N,M,\lambda}$ are available only
for $\lambda=1$ and $2$ (see, e.g., Ref. [60]). For completeness, we provide a
self-contained derivation of $\tilde{\mathcal{N}}_{N,M,\lambda}$ for
$\lambda=1$ and $2$ below.

###  C.1 C1: XY chain

For the XY chain with $\lambda=1$, we use the normalization condition
$\langle\tilde{\Psi}^{\lambda=1}(M)|\tilde{\Psi}^{\lambda=1}(M)\rangle=1$ to
obtain

$\displaystyle|\tilde{\mathcal{N}}_{N,M,\lambda=1}|^{-2}$
$\displaystyle=\sum_{1\leq x_{1}<\cdots<x_{M}\leq
N}|\Psi^{\lambda=1}(x_{1},\ldots,x_{M})|^{2}$ $\displaystyle=\sum_{1\leq
x_{1}<\cdots<x_{M}\leq N}\prod_{1\leq j<k\leq M}|z_{x_{j}}-z_{x_{k}}|^{2}$
$\displaystyle=\frac{1}{M!}\sum_{1\leq x_{1},\ldots,x_{M}\leq N}\prod_{1\leq
j<k\leq M}|z_{x_{j}}-z_{x_{k}}|^{2}\,.$ (A45)

The Jastrow factor $\prod_{1\leq i<j\leq M}(z_{x_{j}}-z_{x_{i}})$ can be
written as a Vandermonde determinant

$\displaystyle\prod_{1\leq j<k\leq
M}(z_{x_{k}}-z_{x_{j}})=\mathrm{det}\begin{pmatrix}1&z_{x_{1}}&\cdots&z_{x_{1}}^{M-1}\\\
1&z_{x_{2}}&\cdots&z_{x_{2}}^{M-1}\\\ \vdots&\vdots&\ddots&\vdots\\\
1&z_{x_{M}}&\cdots&z_{x_{M}}^{M-1}\end{pmatrix}\equiv\mathrm{det}(\mathbb{V})\,,$
(A46)

so we obtain

$\displaystyle|\tilde{\mathcal{N}}_{N,M,\lambda=1}|^{-2}$
$\displaystyle=\frac{1}{M!}\sum_{1\leq x_{1},\ldots,x_{M}\leq
N}\mathrm{det}(\mathbb{V})\;\mathrm{det}(\mathbb{V^{*}})$
$\displaystyle=\frac{1}{M!}\sum_{1\leq x_{1},\ldots,x_{M}\leq
N}\sum_{\sigma\in S_{M}}\mathrm{sgn}(\sigma)z_{x_{1}}^{\sigma(1)-1}\cdots
z_{x_{M}}^{\sigma(M)-1}\sum_{\tau\in
S_{M}}\mathrm{sgn}(\tau){(z_{x_{1}}^{*})}^{\tau(1)-1}\cdots{(z_{x_{M}}^{*})}^{\tau(M)-1}$
$\displaystyle=\frac{1}{M!}\sum_{\sigma\in S_{M}}\sum_{\tau\in
S_{M}}\mathrm{sgn}(\sigma)\,\mathrm{sgn}(\tau)\left(\sum_{x_{1}=1}^{N}z_{x_{1}}^{\sigma(1)-\tau(1)}\right)\cdots\left(\sum_{x_{M}=1}^{N}z_{x_{M}}^{\sigma(M)-\tau(M)}\right)$
$\displaystyle=\frac{1}{M!}\sum_{\sigma\in S_{M}}\sum_{\tau\in
S_{M}}\mathrm{sgn}(\sigma)\,\mathrm{sgn}(\tau)\,\left(N\delta_{\sigma(1),\tau(1)}\right)\cdots\left(N\delta_{\sigma(M),\tau(M)}\right)$
$\displaystyle=\frac{1}{M!}\sum_{\sigma\in S_{M}}N^{M}$
$\displaystyle=N^{M}\,,$ (A47)

where we have used the definition of the determinant [$S_{M}$ is the
permutation of $\\{1,2,\ldots,M\\}$ and $\mathrm{sgn}(\sigma)$ denotes the
signature of the permutation $\sigma$] and the identity
$\sum_{x=1}^{N}z_{x}^{k}=N\delta_{k,0}$ for $k\in\mathbb{Z}$. Thus, we obtain
the normalization factor $\tilde{\mathcal{N}}_{N,M,\lambda=1}=N^{-M/2}$. By
using this result and Eq. (A44), we arrive at

$\displaystyle\langle\mathcal{C}_{\mathrm{latt}}|\Psi^{\lambda=1}(M)\rangle=2^{M/2}\cdot
N^{-M/2}\cdot\frac{1}{[(N/2)^{-M/4}]^{2}}=1\,.$ (A48)

###  C.2 C2: Haldane-Shastry chain

We now turn to the Haldane-Shastry chain ($\lambda=2$) and determine the
normalization factor $\tilde{\mathcal{N}}_{N,M,\lambda=2}$. Using the
normalization condition
$\langle\tilde{\Psi}^{\lambda=2}(M)|\tilde{\Psi}^{\lambda=2}(M)\rangle=1$, we
obtain

$\displaystyle|\tilde{\mathcal{N}}_{N,M,\lambda=2}|^{-2}$
$\displaystyle=\sum_{1\leq x_{1}<\cdots<x_{M}\leq
N}|\Psi^{\lambda=2}(x_{1},\ldots,x_{M})|^{2}$
$\displaystyle=\frac{1}{M!}\sum_{1\leq x_{1},\ldots,x_{M}\leq N}\prod_{1\leq
j<k\leq M}|z_{x_{j}}-z_{x_{k}}|^{4}$ $\displaystyle=\frac{1}{M!}\sum_{1\leq
x_{1},\ldots,x_{M}\leq N}\prod_{1\leq j<k\leq
M}(z_{x_{j}}-z_{x_{k}})^{4}\frac{1}{z_{x_{j}}^{2}z_{x_{k}}^{2}}$
$\displaystyle=\frac{1}{M!}\sum_{1\leq x_{1},\ldots,x_{M}\leq
N}\prod_{i=1}^{M}z_{x_{i}}^{-2(M-1)}\prod_{1\leq j<k\leq
M}(z_{x_{j}}-z_{x_{k}})^{4}\,.$ (A49)

The product $\prod_{1\leq j<k\leq M}(z_{x_{j}}-z_{x_{k}})^{4}$ also has a
determinant representation called “confluent alternant” [60]:

$\displaystyle\prod_{1\leq j<k\leq
M}(z_{x_{j}}-z_{x_{k}})^{4}=\mathrm{det}\begin{pmatrix}1&z_{x_{1}}&z_{x_{1}}^{2}&z_{x_{1}}^{3}&\cdots&z_{x_{1}}^{2M-1}\\\
0&1&2z_{x_{1}}&3z_{x_{1}}^{2}&\cdots&(2M-1)z_{x_{1}}^{2M-2}\\\
1&z_{x_{2}}&z_{x_{1}}^{2}&z_{x_{1}}^{3}&\cdots&z_{x_{2}}^{2M-1}\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
1&z_{x_{M}}&z_{x_{M}}^{2}&z_{x_{M}}^{3}&\cdots&z_{x_{M}}^{2M-1}\\\
0&1&2z_{x_{M}}&3z_{x_{M}}^{2}&\cdots&(2M-1)z_{x_{M}}^{2M-2}\end{pmatrix}\equiv\mathrm{det}(\mathbb{M})\,.$
(A50)

After expanding the “confluent alternant” $\mathrm{det}(\mathbb{M})$ as a
summation of permutation terms (definition of the determinant), the last term
in Eq. (A49) can be calculated as follows:

$\displaystyle\phantom{=}\;\sum_{1\leq x_{1},\ldots,x_{M}\leq
N}\prod_{i=1}^{M}z_{x_{i}}^{-2(M-1)}\prod_{1\leq j<k\leq
M}(z_{x_{j}}-z_{x_{k}})^{4}$ $\displaystyle=\sum_{1\leq x_{1},\ldots,x_{M}\leq
N}\prod_{i=1}^{M}z_{x_{i}}^{-2(M-1)}\;\mathrm{det}(\mathbb{M})$
$\displaystyle=\sum_{1\leq x_{1},\ldots,x_{M}\leq
N}\prod_{i=1}^{M}z_{x_{i}}^{-2(M-1)}\sum_{\sigma\in
S_{2M}}\mathrm{sgn}(\sigma)\,z_{x_{1}}^{\sigma(1)-1}(\sigma(2)-1)z_{x_{1}}^{\sigma(2)-2}\cdots
z_{x_{M}}^{\sigma(2M-1)-1}(\sigma(2M)-1)z_{x_{M}}^{\sigma(2M)-2}$
$\displaystyle=\sum_{\sigma\in
S_{2M}}\mathrm{sgn}(\sigma)\prod_{j=1}^{M}(\sigma(2j)-1)\prod_{j=1}^{M}\left(\sum_{x_{j}=1}^{N}z_{x_{j}}^{\sigma(2j-1)+\sigma(2j)-(2M+1)}\right)$
$\displaystyle=N^{M}\sum_{\sigma\in
S_{2M}}\mathrm{sgn}(\sigma)\prod_{j=1}^{M}(\sigma(2j)-1)\prod_{j=1}^{M}\delta_{\sigma(2j-1)+\sigma(2j),2M+1}\,.$
(A51)

To evaluate the sum, one observes that nonvanishing contributions come from
the permutation elements satisfying $\sigma(2j-1)+\sigma(2j)=2M+1$. All such
permutation elements are generated from
$\sigma=\\{1,2M,2,2M-1,\ldots,M,M+1\\}$ [which has $\mathrm{sgn}(\sigma)=+1$]
by using two types of permutations: i) $\sigma(2j-1)\leftrightarrow\sigma(2j)$
[which has $\mathrm{sgn}(\sigma)=-1$] and ii)
$\\{\sigma(2j-1),\sigma(2j)\\}\leftrightarrow\\{\sigma(2l-1),\sigma(2l)\\}$
[which has $\mathrm{sgn}(\sigma)=+1$]. Making use of the first type of
permutation, the sum can be restricted to $\sigma(2j-1)<\sigma(2j)\;\forall
j=1,\ldots,M$ and is then reduced to a simple sum over
$\\{\sigma(1),\sigma(3),\ldots,\sigma(2M-1)\\}\in S_{M}$:

$\displaystyle\phantom{=}\;\sum_{\sigma\in
S_{2M}}\mathrm{sgn}(\sigma)\prod_{j=1}^{M}(\sigma(2j)-1)\prod_{j=1}^{M}\delta_{\sigma(2j-1)+\sigma(2j),2M+1}$
$\displaystyle=\sum_{\sigma\in
S_{2M},\,\sigma(2j-1)<\sigma(2j)}\mathrm{sgn}(\sigma)\prod_{j=1}^{M}[(\sigma(2j)-1)-(\sigma(2j-1)-1)]\prod_{j=1}^{M}\delta_{\sigma(2j-1)+\sigma(2j),2M+1}$
$\displaystyle=\sum_{\sigma\in
S_{2M},\,\sigma(2j-1)<\sigma(2j)}\mathrm{sgn}(\sigma)\prod_{j=1}^{M}[\sigma(2j)-\sigma(2j-1)]\prod_{j=1}^{M}\delta_{\sigma(2j-1)+\sigma(2j),2M+1}$
$\displaystyle=\sum_{\\{\sigma(1),\sigma(3),\ldots,\sigma(2M-1)\\}\in
S_{M}}(+1)\prod_{j=1}^{M}[2M+1-2\,\sigma(2j-1)]$
$\displaystyle=M!\;(2M-1)(2M-3)\cdots 3\cdot 1$
$\displaystyle=M!\;\frac{(2M)!}{(2M)(2M-2)\cdots 2}$
$\displaystyle=\frac{(2M)!}{2^{M}}\,.$ (A52)

Combining the above results, we obtain

$\displaystyle|\tilde{\mathcal{N}}_{N,M,\lambda=2}|^{-2}=\frac{1}{M!}\cdot
N^{M}\cdot\frac{(2M)!}{2^{M}}=\left(\frac{N}{2}\right)^{M}\frac{(2M)!}{M!}\,,$
(A53)

and
$\tilde{\mathcal{N}}_{N,M,\lambda=2}=\left(\tfrac{2}{N}\right)^{M/2}\sqrt{\tfrac{M!}{(2M)!}}$.
By using this result and Eq. (A44), the crosscap state overlap for the
Haldane-Shastry chain is written as

$\displaystyle\langle\mathcal{C}_{\mathrm{latt}}|\Psi^{\lambda=2}(M)\rangle$
$\displaystyle=2^{M}\cdot\left(\frac{2}{N}\right)^{M/2}\sqrt{\frac{M!}{(2M)!}}\cdot\left(\frac{N}{4}\right)^{M/2}\frac{M!}{(M/2)!}$
$\displaystyle=\frac{2^{M/2}(M!)^{3/2}}{(M/2)!\sqrt{(2M)!}}\,.$ (A54)

Using the Stirling approximation $M!=\sqrt{2\pi
M}\left(\tfrac{M}{e}\right)^{M}\left(1+\tfrac{1}{12M}+\mathcal{O}(M^{-2})\right)$
(valid for $M$ large), we obtain

$\displaystyle\langle\mathcal{C}_{\mathrm{latt}}|\Psi^{\lambda=2}(M)\rangle=2^{\frac{1}{4}}\left(1-\frac{1}{16M}+\mathcal{O}\left(\frac{1}{M^{2}}\right)\right)=2^{\frac{1}{4}}+\mathcal{O}(M^{-1}).$
(A55)


#  0.7 MW Yb:YAG pumped degenerate optical parametric oscillator at 2.06
$\mu$m

Anni Li  Max Planck Institute for the Science of Light, Staudstrasse 2,
Erlangen, 91058, Germany.  Friedrich-Alexander-Universität Erlangen-Nürnberg,
Staudstrasse 7, 91058 Erlangen, Germany.  Mehran Bahri  Max Planck Institute
for the Science of Light, Staudstrasse 2, Erlangen, 91058, Germany.
Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudstrasse 7, 91058
Erlangen, Germany.  Robert M. Gray  Department of Electrical Engineering,
California Institute of Technology, Pasadena, CA, 91125, USA  Seowon Choi  Max
Planck Institute for the Science of Light, Staudstrasse 2, Erlangen, 91058,
Germany.  Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudstrasse 7,
91058 Erlangen, Germany.  Sajjad Hoseinkhani  Max Planck Institute for the
Science of Light, Staudstrasse 2, Erlangen, 91058, Germany.  Anchit Srivastava
Max Planck Institute for the Science of Light, Staudstrasse 2, Erlangen,
91058, Germany.  Friedrich-Alexander-Universität Erlangen-Nürnberg,
Staudstrasse 7, 91058 Erlangen, Germany.  Alireza Marandi  Department of
Electrical Engineering, California Institute of Technology, Pasadena, CA,
91125, USA  Hanieh Fattahi<EMAIL_ADDRESS>Max Planck Institute
for the Science of Light, Staudstrasse 2, Erlangen, 91058, Germany.
Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudstrasse 7, 91058
Erlangen, Germany.

###### Abstract

Frequency comb and field-resolved broadband absorption spectroscopy are
promising techniques for rapid, precise, and sensitive detection of short-
lived atmospheric pollutants on-site. Enhancing detection sensitivity in
absorption spectroscopy hinges on bright sources that cover molecular
resonances and fast signal modulation techniques to implement lock-in
detection schemes efficiently. Yb:YAG thin-disk lasers, combined with optical
parametric oscillators (OPO), present a compelling solution to fulfill these
requirements. In this work, we report on a bright OPO pumped by a Yb:YAG thin-
disk Kerr-lens mode-locked oscillator delivering 2.8 W, 114 fs pulses at 2.06
$\mu$m with an averaged energy of 90 nJ. The OPO cavity operates at 30.9 MHz
pulse repetition rates, the second harmonic of the pump cavity, allowing for
broadband, efficient, and dispersion-free modulation of the OPO output pulses
at 15.45 MHz rate. With 13% optical-to-optical conversion efficiency and a
high-frequency intra-cavity modulation, this scalable scheme holds promise to
advance the detection sensitivity and frontiers of field-resolved
spectroscopic techniques.

##  I Introduction

Highly sensitive and precise monitoring of short-lived climate pollutants in
the atmosphere is critical for advanced studies on the carbon cycle,
greenhouse gas balances, and disruptions. For example, methane is a key player
among greenhouse gases for its substantial role in amplifying global warming
and altering climatic patterns Nisbet, Dlugokencky, and Bousquet (2014);
Caulton  _et al._ (2014). Although its presence in the atmosphere is
considerably less than that of carbon dioxide —about 1.8 parts per million
compared to carbon dioxide’s roughly 400 ppm—methane’s effect on global
warming is markedly more intense, with a warming potential of 25 times greater
than carbon dioxide, accounting for about 15% of future global warming
scenarios. Additionally, methane is a critical component in the intricate
feedback loops of atmospheric chemistry, acting as a signal of the Earth’s
atmospheric dynamics on a large scale Rodhe (1990); Isaksen  _et al._ (2011).

Time-domain broadband absorption spectroscopy offers great potential for on-
site, highly precise, and sensitive short-lived pollutant detection. It
benefits from fast, self-calibrating assessments that do not require sample
preparation, facilitating real-time atmospheric monitoring Mappé  _et al._
(2013); Berman  _et al._ (2012); Grossel  _et al._ (2007); Liu  _et al._
(2015). Atmospheric pollutants cover a broad fundamental vibration–rotation
transition in the mid-infrared and fingerprint region. Moreover, their
overtone and combination bands at short-wavelength infrared (SWIR) offer a
sensitive detection window due to the lower absorption cross-section of water
in this spectral range. For instance, methane exhibit distinct absorption at
1.6 $\mu$m, and 2.2 $\mu$m, where the water response is negligible Douglass
_et al._ (2011).

Based on the Beer-Lambert law, the detection signal-to-noise ratio in
absorption spectroscopy is proportional to the brightness of the illumination
source. Therefore, the detection sensitivity can be significantly enhanced by
employing high-power sources operating at megahertz repetition rates.
Moreover, when coupled with broadband, high-frequency modulation lock-in
techniques, such frontends facilitate a further boost in detection signal-to-
noise and sensitivity. Recent progress in frequency comb spectroscopy Herman
_et al._ (2021); Malarich  _et al._ (2023); Fürst  _et al._ (2024); Zhong  _et
al._ (2024) and field-resolved spectroscopy Herbst  _et al._ (2022);
Srivastava  _et al._ (2023); Scheffter  _et al._ (2024) from the SWIR range up
to ultraviolet spectral range have demonstrated an unparalleled ability to
identify and classify molecular responses with exceptional sensitivity and
accuracy. The detection sensitivity in both techniques benefits directly from
the availability of bright sources with intrinsic broadband modulation.

Femtosecond optical parametric oscillators (OPO) synchronously pumped by Ti:Sa
oscillators or fiber lasers have been a crucial tool for expanding wavelength
coverage of frequency combs, as the generated sub-harmonics are locked in
frequency and phase to the input pump pulses Cizmeciyan  _et al._ (2009); Sun,
Gale, and Reid (2007); Wong, Vodopyanov, and Byer (2010); Adler  _et al._
(2009); Sorokin  _et al._ (2007); Vasilyev  _et al._ (2019). Mainly when
operated at degeneracy, the signal and idler pulses with the same polarization
become indistinguishable, exhibiting self-phase-locked behavior through mutual
injection, providing a single coherent broadband output centered at degeneracy
Leindecker  _et al._ (2012); Marandi  _et al._ (2012). On the other hand,
diode-pumped Yb:YAG thin-disk oscillators are capable of delivering sub-100 fs
pulses with tens of microjoule energy and hundreds of watts average power der
Au  _et al._ (2000); Saraceno  _et al._ (2012); Fattahi  _et al._ (2014);
Goncharov, Fritsch, and Pronin (2023); Barbiero  _et al._ (2021); Fischer  _et
al._ (2021). When used to pump OPOs, these advanced sources could potentially
enable SWIR femtosecond sources to achieve peak powers in the MW range and
average powers in the tens of watts, operating at megahertz repetition rates.
The spectral coverage of such powerful OPOs can be extended into fingerprint
regions or even terahertz frequencies through down-conversion processes.
Furthermore, this scheme allows for high-bandwidth, dispersion-free modulation
of output pulses at megahertz rates by precisely controlling the cavity length
mismatch between the OPO and its pump laser Esteban-Martin  _et al._ (2009);
Jiang and Hasama (2002). This capability holds significant promise for
enhancing detection sensitivity in spectroscopic applications, particularly in
real-time atmospheric monitoring.

In this work, we present a degenerate OPO pumped by a Kerr-lens mode-locked
Yb:YAG thin-disk oscillator delivering 114 fs, 2.8 W pulses at 30.9 MHz
repetition rates with 90 nJ averaged pulse energy. By operating the OPO cavity
at second-harmonics of the repetition rate of the pump cavity, broadband
modulation of the output pulses is achieved.

##  II Experimental results

Figure 1:  (a) The temporal profile of the Yb:YAG thin-disk oscillator
retrieved from a second-harmonic, frequency-resolved optical gating and its
spectrum. (b) The degenerate OPO operates at the second harmonic of the pump
laser repetition rate at 30.9 MHz, while the output pulses are modulated at
the pump repetition rate. (c) Schematic of the OPO cavity. M1, M4: in and
output coupler of the pump laser; M2 and M3: curved mirror; M5, M6, M8, M9,
and M10: folding silver-coated mirrors; M8: folding mirror clamped on a piezo
stage; M7: cavity output coupler; LPF: low pass filter.

The degenerate femtosecond OPO is pumped collinearly by a Kerr-lens mode-
locked Yb:YAG oscillator, delivering 25 W, 15.45 MHz, 384 fs pulses at full-
width at half maximum (FWHM), at 1030 nm (see Fig. 1-a)Fattahi  _et al._
(2016). The pump mode is matched to the OPO cavity by a Kepler telescope with
a ratio of 170%, resulting in a focus size of 74 $\mu$m at FWHM. This
configuration corresponds to 61 GW/$cm^{2}$ pump peak intensity at the focus.

The OPO is designed for a ring cavity at the second harmonic of the repetition
rate of the pump with a total length of 10 m (Fig. 1-b). The ring cavity
allows for easy in and out-coupling of the pump beam. Two concave sliver
mirrors (M2 and M3 in Fig.1-c) with a radius curvature of 700 mm, are used to
focus the cavity mode to the beam waist of 83 $\mu$m (FWHM). By having the
cavity mode 1.1 times larger than the pump beam, the efficient energy transfer
from the pump to the signal is ensured. Two dielectric plane mirrors with low
group delay dispersion are used for in-coupling and out-coupling of the pump
beam (M1 and M4 in Fig.1-c). A dielectric mirror with 30% transmission is used
to couple out the generated sub-harmonic pulses. Several highly reflecting
silver mirrors with a protective coating fold the cavity. A 1-mm thick 5%
magnesium-doped periodically poled $LiNbO_{3}$ (PPLN) crystal at type-0 phase
matching is used for degenerate down-conversion of the pump pulses to 2.06
$\mu$m. The PPLN has a period of 30.8 $\mu$m and is placed in an oven at the
temperature of 116 ${}^{o}C$. The end-faces of the crystal are coated by
broadband anti-reflection coating with less than 0.5% reflectivity. The PPLN
is placed behind the focus of the pump beam to avoid optical damage in the
crystal.

Due to the large cavity length and environmental effects, active stabilization
of the cavity is required, which is achieved through Pound–Drever–Hall
modulationLuda  _et al._ (2019) using a red-pitaya-based proportional-
integral-derivative (PID) lock-in system. The feedback signal is measured at
the OPO output using a photodiode. The lock-in system comprises the
microcontroller board (STEMlab 125-14 board), voltage controller, main
piezoelectric transducer (PZT), main piezo controller, and a supplementary PZT
chip. Both PZT are affixed and bonded to one of the folding mirrors in the OPO
cavity (mirror M8 in Fig.1-c). The primary purpose of the first PZT is to
coarsely scan the cavity length on a micrometer range at an approximate speed
of $\sim$ 0.2 $\mu$m/s via the Red Pitaya interface.

Figure 2:  (a) The measured excited cavity modes of OPO versus the cavity
length scan. The time axis corresponds to the ramping signal of the PZT. The
green-shaded area shows the degenerate modes, while the violet-shaded area
shows the non-degenerate modes. (b) The corresponding spectra of the OPO
output of various excited modes in the cavity via cavity length tunning. The
green-shaded area shows the degenerate modes, while the violet-shaded area
shows the non-degenerate modes. (c) The simulated output tunability of the
cavity by temperature tuning. At a crystal temperature of 425 K, the
degeneracy at 2.06 $\mu$m occurs with a period pitch of 30.8 $\mu$m. At a
crystal temperature of 325 K, the degeneracy at 2.06 $\mu$m occurs with a
period pitch of 31.4 $\mu$m. The red curves show the idlers’ wavelengths,
while the blue curves indicate the signals’ wavelengths.

Fig. 2-a shows the excited cavity modes versus the scanning range of the
second PZT. The first three modes indicated by the green-shaded area are
degenerate, while the subsequent non-degenerate modes are shown in the violet-
shaded area. The proportional and integral functions of the PID controller are
used to generate the feedback signal, with the derivative function deactivated
to prevent the possible amplification of undesirable noise. Fig. 2-b presents
the spectra of various excited modes within the cavity as its length is tuned.
Adjusting the cavity length allows the output pulse spectrum to be tuned from
1900 nm to 2200 nm, constrained by the reflectivity of the cavity’s internal
optics. The wavelength tunability of the cavity can also be achieved via
different phase-matchings in the nonlinear crystal. Fig. 2-c shows the output
tunability of the cavity by phase matching of the PPLN at two periods of 30.8
$\mu$m and 31.4 $\mu$m, and various temperatures.

Figure 3: (a) The measured and reconstructed spectrograms of the OPO pulses
characterized by the SHG-FROG. (b) The retrieved temporal profile and the
spectrum of the OPO output at degeneracy. (c) The output power scaling of the
OPO versus input power corresponds to 13% optical-to-optical efficiency.

At the degeneracy, the cavity operates at negative dispersion with an
estimated total group velocity dispersion of -88 fs2. For intra-cavity
compression of the degenerate OPO pulses, a 0.9 mm-thick Borosilicate plate is
inserted into the cavity at a Brewster angle. The output pulses of the OPO are
characterized by using second-harmonic generation frequency-resolved optical
gating (SHG-FROG) comprising a 20 $\mu$m-thick BBO crystal. Fig. 3-a and Fig.
3-b shows the measured and retrieved spectrograms along the temporal profile
of the OPO pulses corresponding to 114 fs at FWHM when the cavity runs at
degeneracy. The spectral bandwidth of the OPO pulse supports 110 fs Fourier
transform-limited pulses. Fig. 3-c shows the output power of the OPO versus
the input pump power. The output power of the OPO is measured after two long-
pass filters to exclude the contribution of the other nonlinear parasitic
effects in the crystal, which are co-propagating with the OPO pulses. With a
30% output coupler, the OPO has 13% optical-to-optical efficiency and a lasing
threshold at 4 W pump power. The purple area in the curve indicates that the
cavity is approaching saturation.

The cavity length of the OPO is half that of the pump laser, resulting in the
generation of an OPO pulse upon the first incident pump pulse on the nonlinear
crystal. This initial OPO pulse travels within the cavity until the arrival of
the second pump pulse. During each round trip within the cavity, the OPO
output pulses are coupled out through the output coupler. As the cavity
operates at the second harmonics of the pump pulses, the OPO pulse meets the
pump pulse and obtains parametric gain after traveling two round trips in the
OPO cavity. Therefore, the output pulse train of the OPO has a repetition rate
of 30.9 MHz, which is determined by the cavity length of the OPO. The time
interval between every two consecutive pulses in the train is 324 $\mu$s. At
this regime, the cavity output is modulated at 15.45 MHz with a broadband
modulation depth of 41% (Fig. 1-b). The modulation depth between the two
adjunct pulses can be tuned by varying the cavity loss in the OPO cavity. For
small cavity losses, the energy of the succeeding pulses will be only a few
percent less than that of the previous one. The inherent capability for
adjustable broadband modulation makes such cavities an ideal frontend for
highly sensitive spectroscopic applications.

##  III Conclusion

Figure 4: The state-of-the-art megahertz sources at overtone and combination
band spectral range Marandi, Ingold, and Byer (2015); Wang  _et al._ (2022);
Petersen, Zuegel, and Bromage (2017); Heidrich  _et al._ (2023); Hoghooghi
_et al._ (2022); Yang  _et al._ (2023); Yao  _et al._ (2023, 2022); Shang  _et
al._ (2023).

The availability of broadband, bright sources from ultraviolet to terahertz is
crucial for sensitive, precise, and live monitoring of short-lived climate
pollutants in the atmosphere Kwaśny and Bombalska (2023). The fundamental
resonances of molecules have a unique fingerprint in the mid-infrared spectral
range, allowing for precise spectroscopic selectivity. Overtone and
combination bands of molecules at SWIR provide similar information with the
advantage of lower water absorption cross-section. Among various down-
conversion schemes aimed at generating broadband pulses for absorption
spectroscopy across overtone, fundamental, and rotational resonances, OPO
stands out for its higher conversion efficiency Reiger  _et al._ (2024);
Fattahi  _et al._ (2016); Fattahi (2016); Fattahi  _et al._ (2013); Alismail
_et al._ (2017). Yb:YAG thin-disk oscillators offer great potential to pump
the OPOs and to elevate the brightness, repetition rates, and peak power of
the femtosecond pulses in these spectral ranges. Moreover, enhancing detection
sensitivity in absorption spectroscopy requires implementing lock-in systems,
necessitating high-rate signal modulation. Existing techniques like acousto-
optic Tournois (1997) or mechanical modulation suffer from limitations such as
insufficient bandwidth, low-frequency modulation rate, high spectral
dispersion, or low throughput.

This work presented the first Yb:YAG thin-disk pumped OPO containing a
broadband, dispersion-free modulator. The OPO comprised a ring cavity and a 1
mm-thick PPLN crystal operating at the second harmonics of the Yb:YAG cavity
at 30.9 MHz repetition rates. The frontend delivers 114 fs, 2.8 W pulses at
2.06 $\mu$m with an average energy of 90 nJ. The wavelength tunability of the
cavity from 1900 nm to 2200 nm is demonstrated by adjusting the cavity length,
overlapping with several crucial atmospheric gasses resonances in SWIR,
particularly with the water-free window of Methane. Fig. 4 compares the
performance of the frontend with the state-of-the-art sources at MHz
repetition rates at overtone and combination band spectral regions. The
intrinsic broadband, dispersion-free signal modulation at 15.45 MHz offers a
great opportunity for enhancing detection sensitivity in field-resolved
spectroscopy. The system holds potential for further wavelength tunability by
optimizing the intracavity dispersion or employing other nonlinear crystals.
At the same time, the average power and peak power of the OPO pulses can be
scaled by implementing higher energy Yb:YAG oscillators at MHz repetition
rates, allowing for the down-conversion of the OPO pulses to terahertz and
mid-infrared spectral range Wang  _et al._ (2023); Barbiero  _et al._ (2020).
With an intrinsic modulator, this scalable scheme holds promise to pave the
path for atmospheric monitoring of short-lived pollutants and novel
spectroscopic techniques Herbst  _et al._ (2022).

###### Acknowledgements.

We thank Kilian Scheffter for the insightful discussion. Anni Li and Mehran
Bahri acknowledge scholarship from the Max Planck School of Photonics, which
is supported by the German Federal Ministry of Education and Research (BMBF),
the Max Planck Society, and the Fraunhofer Society.

## DECLARATIONS

  * •

This work was supported by research funding from the Max Planck Society.

  * •

Conflict of interest/Competing interests: The authors do not declare any
competing interests.

## DATA AVAILABILITY

The data supporting this study’s findings are available from the corresponding
author upon reasonable request.

## References

  * Nisbet, Dlugokencky, and Bousquet (2014) E. G. Nisbet, E. J. Dlugokencky, and P. Bousquet, “Methane on the rise—again,” Science 343, 493–495 (2014).
  * Caulton  _et al._ (2014) D. R. Caulton, P. B. Shepson, R. L. Santoro, J. P. Sparks, R. W. Howarth, A. R. Ingraffea, M. O. Cambaliza, C. Sweeney, A. Karion, K. J. Davis, _et al._ , “Toward a better understanding and quantification of methane emissions from shale gas development,” Proceedings of the National Academy of Sciences 111, 6237–6242 (2014).
  * Rodhe (1990) H. Rodhe, “A comparison of the contribution of various gases to the greenhouse effect,” Science 248, 1217–1219 (1990).
  * Isaksen  _et al._ (2011) I. S. Isaksen, M. Gauss, G. Myhre, K. M. Walter Anthony, and C. Ruppel, “Strong atmospheric chemistry feedback to climate warming from arctic methane emissions,” Global Biogeochemical Cycles 25 (2011).
  * Mappé  _et al._ (2013) I. Mappé, L. Joly, G. Durry, X. Thomas, T. Decarpenterie, J. Cousin, N. Dumelie, E. Roth, A. Chakir, and P. Grillon, “A quantum cascade laser absorption spectrometer devoted to the in situ measurement of atmospheric N2O and CH4 emission fluxes,” Review of scientific instruments 84 (2013).
  * Berman  _et al._ (2012) E. S. Berman, M. Fladeland, J. Liem, R. Kolyer, and M. Gupta, “Greenhouse gas analyzer for measurements of carbon dioxide, methane, and water vapor aboard an unmanned aerial vehicle,” Sensors and Actuators B: Chemical 169, 128–135 (2012).
  * Grossel  _et al._ (2007) A. Grossel, V. Zeninari, B. Parvitte, L. Joly, and D. Courtois, “Optimization of a compact photoacoustic quantum cascade laser spectrometer for atmospheric flux measurements: application to the detection of methane and nitrous oxide,” Applied Physics B 88, 483–492 (2007).
  * Liu  _et al._ (2015) K. Liu, L. Wang, T. Tan, G. Wang, W. Zhang, W. Chen, and X. Gao, “Highly sensitive detection of methane by near-infrared laser absorption spectroscopy using a compact dense-pattern multipass cell,” Sensors and Actuators B: Chemical 220, 1000–1005 (2015).
  * Douglass  _et al._ (2011) K. O. Douglass, S. E. Maxwell, D. F. Plusquellic, J. T. Hodges, R. D. van Zee, D. V. Samarov, and J. R. Whetstone, “Construction of a high power opo laser system for differential absorption lidar,” in  _Lidar Remote Sensing for Environmental Monitoring XII_ , Vol. 8159 (SPIE, 2011) pp. 115–123.
  * Herman  _et al._ (2021) D. I. Herman, C. Weerasekara, L. C. Hutcherson, F. R. Giorgetta, K. C. Cossel, E. M. Waxman, G. M. Colacion, N. R. Newbury, S. M. Welch, B. D. DePaola, I. Coddington, E. A. Santos, and B. R. Washburn, “Precise multispecies agricultural gas flux determined using broadband open-path dual-comb spectroscopy,” Science Advances 7, eabe9765 (2021), https://www.science.org/doi/pdf/10.1126/sciadv.abe9765 .
  * Malarich  _et al._ (2023) N. A. Malarich, B. R. Washburn, K. C. Cossel, G. J. Mead, F. R. Giorgetta, D. I. Herman, N. R. Newbury, and I. Coddington, “Validation of open-path dual-comb spectroscopy against an O2 background,” Opt. Express 31, 5042–5055 (2023).
  * Fürst  _et al._ (2024) L. Fürst, A. Kirchner, A. Eber, F. Siegrist, R. di Vora, and B. Bernhardt, “Broadband near-ultraviolet dual comb spectroscopy,” Optica 11, 471–477 (2024).
  * Zhong  _et al._ (2024) W. Zhong, Y. Liu, Q. Yin, R. Zhao, Y. Ding, C. Wang, T. Chen, X. Dou, and X. Xue, “A photon-level broadband dual-comb interferometer for turbulent open-air trace gases detection application,” arXiv preprint arXiv:2401.11657 (2024).
  * Herbst  _et al._ (2022) A. Herbst, K. Scheffter, M. Bidhendi, M. Kieker, A. Srivastava, and H. Fattahi, “Recent advances in petahertz electric field sampling,” Journal of Physics B: Atomic, Molecular and Optical Physics 55, 172001 (2022).
  * Srivastava  _et al._ (2023) A. Srivastava, A. Herbst, M. M. Bidhendi, M. Kieker, F. Tani, and H. Fattahi, “Near-petahertz fieldoscopy of liquid,” arXiv preprint arXiv:2310.20512 (2023).
  * Scheffter  _et al._ (2024) K. Scheffter, J. Will, C. Riek, H. Jousselin, S. Coudreau, N. Forget, and H. Fattahi, “Compressed sensing of field-resolved molecular fingerprints beyond the nyquist frequency,” Ultrafast Science 4, 0062 (2024).
  * Cizmeciyan  _et al._ (2009) M. N. Cizmeciyan, H. Cankaya, A. Kurt, and A. Sennaroglu, “Kerr-lens mode-locked femtosecond Cr2+: ZnSe laser at 2420 nm,” Optics letters 34, 3056–3058 (2009).
  * Sun, Gale, and Reid (2007) J. Sun, B. Gale, and D. Reid, “Composite frequency comb spanning 0.4-2.4 $\mu$m from a phase-controlled femtosecond Ti: sapphire laser and synchronously pumped optical parametric oscillator,” Optics letters 32, 1414–1416 (2007).
  * Wong, Vodopyanov, and Byer (2010) S. T. Wong, K. L. Vodopyanov, and R. L. Byer, “Self-phase-locked divide-by-2 optical parametric oscillator as a broadband frequency comb source,” JOSA B 27, 876–882 (2010).
  * Adler  _et al._ (2009) F. Adler, K. C. Cossel, M. J. Thorpe, I. Hartl, M. E. Fermann, and J. Ye, “Phase-stabilized, 1.5 w frequency comb at 2.8–4.8 $\mu$m,” Optics letters 34, 1330–1332 (2009).
  * Sorokin  _et al._ (2007) E. Sorokin, I. T. Sorokina, J. Mandon, G. Guelachvili, and N. Picqué, “Sensitive multiplex spectroscopy in the molecular fingerprint 2.4 $\mu$m region with Cr2+: ZnSe femtosecond laser,” Optics express 15, 16540–16545 (2007).
  * Vasilyev  _et al._ (2019) S. Vasilyev, V. Smolski, J. Peppers, I. Moskalev, M. Mirov, Y. Barnakov, S. Mirov, and V. Gapontsev, “Middle-IR frequency comb based on Cr: ZnS laser,” Optics express 27, 35079–35087 (2019).
  * Leindecker  _et al._ (2012) N. Leindecker, A. Marandi, R. L. Byer, K. L. Vodopyanov, J. Jiang, I. Hartl, M. Fermann, and P. G. Schunemann, “Octave-spanning ultrafast OPO with 2.6 - 6.1 $\mu$m instantaneous bandwidth pumped by femtosecond Tm-fiber laser,” Optics express 20, 7046–7053 (2012).
  * Marandi  _et al._ (2012) A. Marandi, N. C. Leindecker, V. Pervak, R. L. Byer, and K. L. Vodopyanov, “Coherence properties of a broadband femtosecond mid-IR optical parametric oscillator operating at degeneracy,” Optics express 20, 7255–7262 (2012).
  * der Au  _et al._ (2000) J. A. der Au, G. J. Spühler, T. Südmeyer, R. Paschotta, R. Hövel, M. Moser, S. Erhard, M. Karszewski, A. Giesen, and U. Keller, “16.2-W average power from a diode-pumped femtosecond Yb:YAG thin disk laser,” Opt. Lett. 25, 859–861 (2000).
  * Saraceno  _et al._ (2012) C. J. Saraceno, F. Emaury, O. H. Heckl, C. R. E. Baer, M. Hoffmann, C. Schriber, M. Golling, T. Südmeyer, and U. Keller, “275 W average output power from a femtosecond thin disk oscillator operated in a vacuum environment,” Opt. Express 20, 23535–23541 (2012).
  * Fattahi  _et al._ (2014) H. Fattahi, H. G. Barros, M. Gorjan, T. Nubbemeyer, B. Alsaif, C. Y. Teisset, M. Schultze, S. Prinz, M. Haefner, M. Ueffing, A. Alismail, L. Vámos, A. Schwarz, O. Pronin, J. Brons, X. T. Geng, G. Arisholm, M. Ciappina, V. S. Yakovlev, D.-E. Kim, A. M. Azzeer, N. Karpowicz, D. Sutter, Z. Major, T. Metzger, and F. Krausz, “Third-generation femtosecond technology,” Optica 1, 45–63 (2014).
  * Goncharov, Fritsch, and Pronin (2023) S. Goncharov, K. Fritsch, and O. Pronin, “110 MW thin-disk oscillator,” Opt. Express 31, 25970–25977 (2023).
  * Barbiero  _et al._ (2021) G. Barbiero, H. Wang, M. Graßl, S. Gröbmeyer, D. Kimbaras, M. Neuhaus, V. Pervak, T. Nubbemeyer, H. Fattahi, and M. F. Kling, “Efficient nonlinear compression of a thin-disk oscillator to 8.5 fs at 55 w average power,” Optics letters 46, 5304–5307 (2021).
  * Fischer  _et al._ (2021) J. Fischer, J. Drs, N. Modsching, F. Labaye, V. J. Wittwer, and T. Südmeyer, “Efficient 100-MW, 100-W, 50-fs-class Yb:YAG thin-disk laser oscillator,” Opt. Express 29, 42075–42081 (2021).
  * Esteban-Martin  _et al._ (2009) A. Esteban-Martin, O. Kokabee, K. Moutzouris, and M. Ebrahim-Zadeh, “High-harmonic-repetition-rate, 1 GHz femtosecond optical parametric oscillator pumped by a 76 MHz Ti:sapphire laser,” Opt. Lett. 34, 428–430 (2009).
  * Jiang and Hasama (2002) J. Jiang and T. Hasama, “Harmonic repetition-rate femtosecond optical parametric oscillator,” Applied Physics B 74, 313–317 (2002).
  * Fattahi  _et al._ (2016) H. Fattahi, H. Wang, A. Alismail, G. Arisholm, V. Pervak, A. M. Azzeer, and F. Krausz, “Near-PHz-bandwidth, phase-stable continua generated from a Yb:YAG thin-disk amplifier,” Opt. Express 24, 24337–24346 (2016).
  * Luda  _et al._ (2019) M. A. Luda, M. Drechsler, C. T. Schmiegelow, and J. Codnia, “Compact embedded device for lock-in measurements and experiment active control,” Review of Scientific Instruments 90 (2019).
  * Marandi, Ingold, and Byer (2015) A. Marandi, K. A. Ingold, and R. L. Byer, “Quarter-harmonic generation of femtosecond pulses at 4.18 µm from a mode-locked Yb:fiber laser,” in  _CLEO: 2015_ (Optica Publishing Group, 2015) p. SW4O.8.
  * Wang  _et al._ (2022) Y. Wang, F. Fleming, R. A. McCracken, C. Liebig, S. McDaniel, G. Cook, P. Laporta, A. K. Kar, and G. Galzerano, “Hot-isostatic-pressed Cr:ZnSe ultrafast laser at 2.4 µm,” Optics and Laser Technology 154, 108300 (2022).
  * Petersen, Zuegel, and Bromage (2017) T. Petersen, J. D. Zuegel, and J. Bromage, “High-average-power, 2-$\mu$m femtosecond optical parametric oscillator synchronously pumped by a thin-disk, mode-locked laser,” Opt. Express 25, 8840–8844 (2017).
  * Heidrich  _et al._ (2023) J. Heidrich, A. Barh, S. L. Camenzind, B. Willenberg, M. Gaulke, M. Golling, C. R. Phillips, and U. Keller, “Low-Noise Femtosecond SESAM Modelocked Diode-Pumped Cr:ZnS oscillator,” IEEE Journal of Quantum Electronics 59, 1–7 (2023).
  * Hoghooghi  _et al._ (2022) N. Hoghooghi, S. Xing, P. Chang, D. Lesko, A. Lind, G. Rieker, and S. Diddams, “1-GHz mid-infrared frequency comb spanning 3 to 13 $\mu$m,” arXiv preprint arXiv:2201.07134 (2022).
  * Yang  _et al._ (2023) J. Yang, C. Wang, W. Ling, J. Xue, X. Du, W. Wang, Y. Zhao, F. Lu, X. Li, J. Song, Z. Wang, and Z. Wei, “High power, widely tunable femtosecond MgO:PPLN optical parametric oscillator,” Chinese Physics B 32, 074204 (2023).
  * Yao  _et al._ (2023) W. Yao, M. Khalili, Y. Wang, M. Hoffmann, M. van Delden, T. Musch, and C. J. Saraceno, “GHz repetition rate, sub-100-fs Ho: CALGO laser at 2.1 um with watt-level average power,” arXiv preprint arXiv:2310.13928 (2023).
  * Yao  _et al._ (2022) W. Yao, Y. Wang, S. Tomilov, M. Hoffmann, S. Ahmed, C. Liebald, D. Rytz, M. Peltz, V. Wesemann, and C. J. Saraceno, “8.7-W average power, in-band pumped femtosecond Ho: CALGO laser at 2.1 $\mu$m,” Optics Express 30, 41075–41083 (2022).
  * Shang  _et al._ (2023) J. Shang, C. Mei, S. Zhao, Y. Liu, K. Yang, C. Wang, T. Li, and T. Feng, “Short mid-infrared watt-level all-fiber nonlinear pulse compressor above 100-MHz pulse repetition rate,” High Power Laser Science and Engineering 11, e14 (2023).
  * Kwaśny and Bombalska (2023) M. Kwaśny and A. Bombalska, “Optical methods of methane detection,” Sensors 23, 2834 (2023).
  * Reiger  _et al._ (2024) S. Reiger, M. Mamaikin, D. Kormin, K. Golyari, H. Kassab, M. Seeger, V. Pervak, N. Karpowicz, and T. Nubbemeyer, “Ultra-phase-stable infrared light source at the watt level,” Opt. Lett. 49, 1049–1052 (2024).
  * Fattahi (2016) H. Fattahi, “Sub-cycle light transients for attosecond, x-ray, four-dimensional imaging,” Contemporary Physics 57, 580–595 (2016).
  * Fattahi  _et al._ (2013) H. Fattahi, A. Schwarz, S. Keiber, and N. Karpowicz, “Efficient, octave-spanning difference-frequency generation using few-cycle pulses in simple collinear geometry,” Optics letters 38, 4216–4219 (2013).
  * Alismail  _et al._ (2017) A. Alismail, H. Wang, N. Altwaijry, and H. Fattahi, “Carrier-envelope phase stable, 5.4 $\mu$j, broadband, mid-infrared pulse generation from a 1-ps, yb: Yag thin-disk laser,” Applied Optics 56, 4990–4994 (2017).
  * Tournois (1997) P. Tournois, “Acousto-optic programmable dispersive filter for adaptive compensation of group delay time dispersion in laser systems,” Optics communications 140, 245–249 (1997).
  * Wang  _et al._ (2023) Y. Wang, T. Vogel, M. Khalili, S. Mansourzadeh, K. Hasse, S. Suntsov, D. Kip, and C. J. Saraceno, “High-power intracavity single-cycle thz pulse generation using thin lithium niobate,” Optica 10, 1719–1722 (2023).
  * Barbiero  _et al._ (2020) G. Barbiero, H. Wang, J. Brons, B.-H. Chen, V. Pervak, and H. Fattahi, “Broadband terahertz solid-state emitter driven by yb: Yag thin-disk oscillator,” Journal of Physics B: Atomic, Molecular and Optical Physics 53, 125601 (2020).